Econometrics

ECONOMETRICS Bruce E. Hansen c 2000, 20131 University of Wisconsin www.ssc.wisc.edu/~bhansen This Revision: January 18, 2013 Comments Welcome 1 This manuscript may be printed and reproduced for individual or instructional use, but may not be printed for commercial purposes. Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii 1 Introduction 1.1 What is Econometrics? . . . . . . . . . . . . 1.2 The Probability Approach to Econometrics 1.3 Econometric Terms and Notation . . . . . . 1.4 Observational Data . . . . . . . . . . . . . . 1.5 Standard Data Structures . . . . . . . . . . 1.6 Sources for Economic Data . . . . . . . . . 1.7 Econometric Software . . . . . . . . . . . . 1.8 Reading the Manuscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 2 3 4 5 6 7 2 Conditional Expectation and Projection 2.1 Introduction . . . . . . . . . . . . . . . . 2.2 The Distribution of Wages . . . . . . . . 2.3 Conditional Expectation . . . . . . . . . 2.4 Log Di¤erences* . . . . . . . . . . . . . 2.5 Conditional Expectation Function . . . 2.6 Continuous Variables . . . . . . . . . . . 2.7 Law of Iterated Expectations . . . . . . 2.8 CEF Error . . . . . . . . . . . . . . . . . 2.9 Regression Variance . . . . . . . . . . . 2.10 Best Predictor . . . . . . . . . . . . . . 2.11 Conditional Variance . . . . . . . . . . . 2.12 Homoskedasticity and Heteroskedasticity 2.13 Regression Derivative . . . . . . . . . . 2.14 Linear CEF . . . . . . . . . . . . . . . . 2.15 Linear CEF with Nonlinear E¤ects . . . 2.16 Linear CEF with Dummy Variables . . . 2.17 Best Linear Predictor . . . . . . . . . . 2.18 Linear Predictor Error Variance . . . . . 2.19 Regression Coe¢ cients . . . . . . . . . . 2.20 Regression Sub-Vectors . . . . . . . . . 2.21 Coe¢ cient Decomposition . . . . . . . . 2.22 Omitted Variable Bias . . . . . . . . . . 2.23 Best Linear Approximation . . . . . . . 2.24 Normal Regression . . . . . . . . . . . . 2.25 Regression to the Mean . . . . . . . . . 2.26 Reverse Regression . . . . . . . . . . . . 2.27 Limitations of the Best Linear Predictor 2.28 Random Coe¢ cient Model . . . . . . . . 2.29 Causal E¤ects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 8 8 10 12 13 14 15 17 18 19 20 21 22 23 24 24 27 32 33 33 34 35 36 36 37 38 39 40 41 i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CONTENTS ii 2.30 Expectation: Mathematical Details* . . . . . . . . . . . . . 2.31 Existence and Uniqueness of the Conditional Expectation* 2.32 Identi…cation* . . . . . . . . . . . . . . . . . . . . . . . . . . 2.33 Technical Proofs* . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Algebra of Least Squares 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 3.2 Random Samples . . . . . . . . . . . . . . . . . . . 3.3 Least Squares Estimator . . . . . . . . . . . . . . . 3.4 Solving for Least Squares with One Regressor . . . 3.5 Solving for Least Squares with Multiple Regressors 3.6 Illustration . . . . . . . . . . . . . . . . . . . . . . 3.7 Least Squares Residuals . . . . . . . . . . . . . . . 3.8 Model in Matrix Notation . . . . . . . . . . . . . . 3.9 Projection Matrix . . . . . . . . . . . . . . . . . . 3.10 Orthogonal Projection . . . . . . . . . . . . . . . . 3.11 Estimation of Error Variance . . . . . . . . . . . . 3.12 Analysis of Variance . . . . . . . . . . . . . . . . . 3.13 Regression Components . . . . . . . . . . . . . . . 3.14 Residual Regression . . . . . . . . . . . . . . . . . 3.15 Prediction Errors . . . . . . . . . . . . . . . . . . . 3.16 In‡uential Observations . . . . . . . . . . . . . . . 3.17 Normal Regression Model . . . . . . . . . . . . . . 3.18 Technical Proofs* . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Least Squares Regression 4.1 Introduction . . . . . . . . . . . . . . 4.2 Sample Mean . . . . . . . . . . . . . 4.3 Linear Regression Model . . . . . . . 4.4 Mean of Least-Squares Estimator . . 4.5 Variance of Least Squares Estimator 4.6 Gauss-Markov Theorem . . . . . . . 4.7 Residuals . . . . . . . . . . . . . . . 4.8 Estimation of Error Variance . . . . 4.9 Mean-Square Forecast Error . . . . . 4.10 Covariance Matrix Estimation Under 4.11 Covariance Matrix Estimation Under 4.12 Standard Errors . . . . . . . . . . . . 4.13 Measures of Fit . . . . . . . . . . . . 4.14 Empirical Example . . . . . . . . . . 4.15 Multicollinearity . . . . . . . . . . . 4.16 Normal Regression Model . . . . . . Exercises . . . . . . . . . . . . . . . . . . 5 An 5.1 5.2 5.3 5.4 5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Homoskedasticity Heteroskedasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction to Large Sample Asymptotics Introduction . . . . . . . . . . . . . . . . . . . . . Asymptotic Limits . . . . . . . . . . . . . . . . . Convergence in Probability . . . . . . . . . . . . Weak Law of Large Numbers . . . . . . . . . . . Almost Sure Convergence and the Strong Law* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 46 46 47 51 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 53 53 54 55 55 57 58 58 60 61 62 63 64 65 67 68 69 70 72 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 75 75 76 77 78 80 82 83 84 85 86 89 90 91 92 95 97 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 98 99 100 101 102 CONTENTS 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 Vector-Valued Moments . . . Convergence in Distribution . Higher Moments . . . . . . . Functions of Moments . . . . Delta Method . . . . . . . . . Stochastic Order Symbols . . Uniform Stochastic Bounds* . Semiparametric E¢ ciency . . Technical Proofs* . . . . . . . iii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Asymptotic Theory for Least Squares 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Consistency of Least-Squares Estimation . . . . . . . . . . . . . . . . 6.3 Asymptotic Normality . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Joint Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Consistency of Error Variance Estimators . . . . . . . . . . . . . . . 6.6 Homoskedastic Covariance Matrix Estimation . . . . . . . . . . . . . 6.7 Heteroskedastic Covariance Matrix Estimation . . . . . . . . . . . . 6.8 Alternative Covariance Matrix Estimators* . . . . . . . . . . . . . . 6.9 Functions of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 6.10 Asymptotic Standard Errors . . . . . . . . . . . . . . . . . . . . . . . 6.11 t statistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.12 Con…dence Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.13 Regression Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.14 Forecast Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.15 Wald Statistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.16 Con…dence Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.17 Semiparametric E¢ ciency in the Projection Model . . . . . . . . . . 6.18 Semiparametric E¢ ciency in the Homoskedastic Regression Model* . 6.19 Uniformly Consistent Residuals* . . . . . . . . . . . . . . . . . . . . 6.20 Asymptotic Leverage* . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Restricted Estimation 7.1 Introduction . . . . . . . . . . . . . . . . 7.2 Constrained Least Squares . . . . . . . . 7.3 Exclusion Restriction . . . . . . . . . . . 7.4 Minimum Distance . . . . . . . . . . . . 7.5 Asymptotic Distribution . . . . . . . . . 7.6 E¢ cient Minimum Distance Estimator . 7.7 Exclusion Restriction Revisited . . . . . 7.8 Variance and Standard Error Estimation 7.9 Misspeci…cation . . . . . . . . . . . . . . 7.10 Nonlinear Constraints . . . . . . . . . . 7.11 Inequality Constraints . . . . . . . . . . 7.12 Constrained MLE . . . . . . . . . . . . . 7.13 Technical Proofs* . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 104 105 106 109 110 112 113 116 . . . . . . . . . . . . . . . . . . . . . 120 . 120 . 121 . 122 . 125 . 128 . 129 . 130 . 131 . 132 . 134 . 134 . 135 . 136 . 139 . 139 . 140 . 142 . 143 . 144 . 145 . 147 . . . . . . . . . . . . . . 149 . 149 . 150 . 151 . 151 . 152 . 154 . 155 . 156 . 156 . 158 . 159 . 160 . 160 . 162 CONTENTS iv 8 Hypothesis Testing 8.1 Hypotheses and Tests . . . . . . . . . . . . . . 8.2 t tests . . . . . . . . . . . . . . . . . . . . . . . 8.3 t-ratios and the Abuse of Testing . . . . . . . . 8.4 Wald Tests . . . . . . . . . . . . . . . . . . . . 8.5 Minimum Distance Tests . . . . . . . . . . . . . 8.6 F Tests . . . . . . . . . . . . . . . . . . . . . . 8.7 Likelihood Ratio Test . . . . . . . . . . . . . . 8.8 Problems with Tests of NonLinear Hypotheses 8.9 Monte Carlo Simulation . . . . . . . . . . . . . 8.10 Con…dence Intervals by Test Inversion . . . . . 8.11 Asymptotic Power . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 163 166 168 169 170 171 172 173 176 179 180 182 9 Regression Extensions 9.1 Generalized Least Squares . . . . . 9.2 Testing for Heteroskedasticity . . . 9.3 NonLinear Least Squares . . . . . 9.4 Testing for Omitted NonLinearity . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 184 187 187 189 191 10 The Bootstrap 10.1 De…nition of the Bootstrap . . . . . . . . . 10.2 The Empirical Distribution Function . . . . 10.3 Nonparametric Bootstrap . . . . . . . . . . 10.4 Bootstrap Estimation of Bias and Variance 10.5 Percentile Intervals . . . . . . . . . . . . . . 10.6 Percentile-t Equal-Tailed Interval . . . . . . 10.7 Symmetric Percentile-t Intervals . . . . . . 10.8 Asymptotic Expansions . . . . . . . . . . . 10.9 One-Sided Tests . . . . . . . . . . . . . . . 10.10Symmetric Two-Sided Tests . . . . . . . . . 10.11Percentile Con…dence Intervals . . . . . . . 10.12Bootstrap Methods for Regression Models . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 192 192 194 194 195 197 197 198 200 201 202 203 205 . . . . . . . . . . . . Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 206 206 208 209 210 212 215 218 218 219 . . . . . . . . . . . . . . . 11 NonParametric Regression 11.1 Introduction . . . . . . . . . . . . . . . . 11.2 Binned Estimator . . . . . . . . . . . . . 11.3 Kernel Regression . . . . . . . . . . . . . 11.4 Local Linear Estimator . . . . . . . . . . 11.5 Nonparametric Residuals and Regression 11.6 Cross-Validation Bandwidth Selection . 11.7 Asymptotic Distribution . . . . . . . . . 11.8 Conditional Variance Estimation . . . . 11.9 Standard Errors . . . . . . . . . . . . . . 11.10Multiple Regressors . . . . . . . . . . . . . . . . . CONTENTS v 12 Series Estimation 12.1 Approximation by Series . . . . . . . . . . . . 12.2 Splines . . . . . . . . . . . . . . . . . . . . . . 12.3 Partially Linear Model . . . . . . . . . . . . . 12.4 Additively Separable Models . . . . . . . . . 12.5 Uniform Approximations . . . . . . . . . . . . 12.6 Runge’s Phenomenon . . . . . . . . . . . . . . 12.7 Approximating Regression . . . . . . . . . . . 12.8 Residuals and Regression Fit . . . . . . . . . 12.9 Cross-Validation Model Selection . . . . . . . 12.10Convergence in Mean-Square . . . . . . . . . 12.11Uniform Convergence . . . . . . . . . . . . . . 12.12Asymptotic Normality . . . . . . . . . . . . . 12.13Asymptotic Normality with Undersmoothing 12.14Regression Estimation . . . . . . . . . . . . . 12.15Kernel Versus Series Regression . . . . . . . . 12.16Technical Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 222 222 224 224 224 226 226 229 229 230 231 232 233 234 235 235 13 Quantile Regression 241 13.1 Least Absolute Deviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 13.2 Quantile Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 14 Generalized Method of Moments 14.1 Overidenti…ed Linear Model . . . . . . . . . 14.2 GMM Estimator . . . . . . . . . . . . . . . 14.3 Distribution of GMM Estimator . . . . . . 14.4 Estimation of the E¢ cient Weight Matrix . 14.5 GMM: The General Case . . . . . . . . . . 14.6 Over-Identi…cation Test . . . . . . . . . . . 14.7 Hypothesis Testing: The Distance Statistic 14.8 Conditional Moment Restrictions . . . . . . 14.9 Bootstrap GMM Inference . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 247 248 249 250 251 251 252 253 254 256 15 Empirical Likelihood 15.1 Non-Parametric Likelihood . . . . . . . . 15.2 Asymptotic Distribution of EL Estimator 15.3 Overidentifying Restrictions . . . . . . . . 15.4 Testing . . . . . . . . . . . . . . . . . . . . 15.5 Numerical Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 258 260 261 262 263 16 Endogeneity 16.1 Instrumental Variables . . . 16.2 Reduced Form . . . . . . . 16.3 Identi…cation . . . . . . . . 16.4 Estimation . . . . . . . . . 16.5 Special Cases: IV and 2SLS 16.6 Bekker Asymptotics . . . . 16.7 Identi…cation Failure . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 266 267 268 268 268 270 271 273 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CONTENTS vi 17 Univariate Time Series 17.1 Stationarity and Ergodicity . . . . . . 17.2 Autoregressions . . . . . . . . . . . . . 17.3 Stationarity of AR(1) Process . . . . . 17.4 Lag Operator . . . . . . . . . . . . . . 17.5 Stationarity of AR(k) . . . . . . . . . 17.6 Estimation . . . . . . . . . . . . . . . 17.7 Asymptotic Distribution . . . . . . . . 17.8 Bootstrap for Autoregressions . . . . . 17.9 Trend Stationarity . . . . . . . . . . . 17.10Testing for Omitted Serial Correlation 17.11Model Selection . . . . . . . . . . . . . 17.12Autoregressive Unit Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 275 277 278 278 279 279 280 281 281 282 283 283 18 Multivariate Time Series 18.1 Vector Autoregressions (VARs) . . . . 18.2 Estimation . . . . . . . . . . . . . . . 18.3 Restricted VARs . . . . . . . . . . . . 18.4 Single Equation from a VAR . . . . . 18.5 Testing for Omitted Serial Correlation 18.6 Selection of Lag Length in an VAR . . 18.7 Granger Causality . . . . . . . . . . . 18.8 Cointegration . . . . . . . . . . . . . . 18.9 Cointegrated VARs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 285 286 286 286 287 287 288 288 289 . . . . 291 . 291 . 292 . 293 . 294 19 Limited Dependent Variables 19.1 Binary Choice . . . . . . . . 19.2 Count Data . . . . . . . . . 19.3 Censored Data . . . . . . . 19.4 Sample Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Panel Data 296 20.1 Individual-E¤ects Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 20.2 Fixed E¤ects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 20.3 Dynamic Panel Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 21 Nonparametric Density Estimation 299 21.1 Kernel Density Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 21.2 Asymptotic MSE for Kernel Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 301 A Matrix Algebra A.1 Notation . . . . . . . . . . . A.2 Matrix Addition . . . . . . A.3 Matrix Multiplication . . . A.4 Trace . . . . . . . . . . . . . A.5 Rank and Inverse . . . . . . A.6 Determinant . . . . . . . . . A.7 Eigenvalues . . . . . . . . . A.8 Positive De…niteness . . . . A.9 Matrix Calculus . . . . . . . A.10 Kronecker Products and the A.11 Vector and Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vec Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 . 304 . 305 . 305 . 306 . 307 . 308 . 309 . 310 . 311 . 311 . 312 CONTENTS vii A.12 Matrix Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 B Probability B.1 Foundations . . . . . . . . . . . . . . . . . . B.2 Random Variables . . . . . . . . . . . . . . B.3 Expectation . . . . . . . . . . . . . . . . . . B.4 Gamma Function . . . . . . . . . . . . . . . B.5 Common Distributions . . . . . . . . . . . . B.6 Multivariate Random Variables . . . . . . . B.7 Conditional Distributions and Expectation . B.8 Transformations . . . . . . . . . . . . . . . B.9 Normal and Related Distributions . . . . . B.10 Inequalities . . . . . . . . . . . . . . . . . . B.11 Maximum Likelihood . . . . . . . . . . . . . . . . . . . . . . . . 316 . 316 . 318 . 318 . 319 . 320 . 322 . 324 . 326 . 327 . 329 . 332 C Numerical Optimization C.1 Grid Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2 Gradient Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.3 Derivative-Free Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 . 337 . 337 . 339 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preface This book is intended to serve as the textbook for a …rst-year graduate course in econometrics. It can be used as a stand-alone text, or be used as a supplement to another text. Students are assumed to have an understanding of multivariate calculus, probability theory, linear algebra, and mathematical statistics. A prior course in undergraduate econometrics would be helpful, but not required. Two excellent undergraduate textbooks are Wooldridge (2009) and Stock and Watson (2010). For reference, some of the basic tools of matrix algebra, probability, and statistics are reviewed in the Appendix. For students wishing to deepen their knowledge of matrix algebra in relation to their study of econometrics, I recommend Matrix Algebra by Abadir and Magnus (2005). An excellent introduction to probability and statistics is Statistical Inference by Casella and Berger (2002). For those wanting a deeper foundation in probability, I recommend Ash (1972) or Billingsley (1995). For more advanced statistical theory, I recommend Lehmann and Casella (1998), van der Vaart (1998), Shao (2003), and Lehmann and Romano (2005). For further study in econometrics beyond this text, I recommend Davidson (1994) for asymptotic theory, Hamilton (1994) for time-series methods, Wooldridge (2002) for panel data and discrete response models, and Li and Racine (2007) for nonparametrics and semiparametric econometrics. Beyond these texts, the Handbook of Econometrics series provides advanced summaries of contemporary econometric methods and theory. The end-of-chapter exercises are important parts of the text and are meant to help teach students of econometrics. Answers are not provided, and this is intentional. I would like to thank Ying-Ying Lee for providing research assistance in preparing some of the empirical examples presented in the text. As this is a manuscript in progress, some parts are quite incomplete, and there are many topics which I plan to add. In general chapters 1-8 are the most complete, the remaining need signi…cant work and revision. viii Chapter 1 Introduction 1.1 What is Econometrics? The term “econometrics” is believed to have been crafted by Ragnar Frisch (1895-1973) of Norway, one of the three principle founders of the Econometric Society, …rst editor of the journal Econometrica, and co-winner of the …rst Nobel Memorial Prize in Economic Sciences in 1969. It is therefore …tting that we turn to Frisch’s own words in the introduction to the …rst issue of Econometrica for an explanation of the discipline. A word of explanation regarding the term econometrics may be in order. Its de…nition is implied in the statement of the scope of the [Econometric] Society, in Section I of the Constitution, which reads: “The Econometric Society is an international society for the advancement of economic theory in its relation to statistics and mathematics.... Its main object shall be to promote studies that aim at a uni…cation of the theoreticalquantitative and the empirical-quantitative approach to economic problems....” But there are several aspects of the quantitative approach to economics, and no single one of these aspects, taken by itself, should be confounded with econometrics. Thus, econometrics is by no means the same as economic statistics. Nor is it identical with what we call general economic theory, although a considerable portion of this theory has a de…ninitely quantitative character. Nor should econometrics be taken as synonomous with the application of mathematics to economics. Experience has shown that each of these three view-points, that of statistics, economic theory, and mathematics, is a necessary, but not by itself a su¢ cient, condition for a real understanding of the quantitative relations in modern economic life. It is the uni…cation of all three that is powerful. And it is this uni…cation that constitutes econometrics. Ragnar Frisch, Econometrica, (1933), 1, pp. 1-2. This de…nition remains valid today, although some terms have evolved somewhat in their usage. Today, we would say that econometrics is the uni…ed study of economic models, mathematical statistics, and economic data. Within the …eld of econometrics there are sub-divisions and specializations. Econometric theory concerns the development of tools and methods, and the study of the properties of econometric methods. Applied econometrics is a term describing the development of quantitative economic models and the application of econometric methods to these models using economic data. 1.2 The Probability Approach to Econometrics The unifying methodology of modern econometrics was articulated by Trygve Haavelmo (19111999) of Norway, winner of the 1989 Nobel Memorial Prize in Economic Sciences, in his seminal 1 CHAPTER 1. INTRODUCTION 2 paper “The probability approach in econometrics”, Econometrica (1944). Haavelmo argued that quantitative economic models must necessarily be probability models (by which today we would mean stochastic). Deterministic models are blatently inconsistent with observed economic quantities, and it is incoherent to apply deterministic models to non-deterministic data. Economic models should be explicitly designed to incorporate randomness; stochastic errors should not be simply added to deterministic models to make them random. Once we acknowledge that an economic model is a probability model, it follows naturally that an appropriate tool way to quantify, estimate, and conduct inferences about the economy is through the powerful theory of mathematical statistics. The appropriate method for a quantitative economic analysis follows from the probabilistic construction of the economic model. Haavelmo’s probability approach was quickly embraced by the economics profession. Today no quantitative work in economics shuns its fundamental vision. While all economists embrace the probability approach, there has been some evolution in its implementation. The structural approach is the closest to Haavelmo’s original idea. A probabilistic economic model is speci…ed, and the quantitative analysis performed under the assumption that the economic model is correctly speci…ed. Researchers often describe this as “taking their model seriously.” The structural approach typically leads to likelihood-based analysis, including maximum likelihood and Bayesian estimation. A criticism of the structural approach is that it is misleading to treat an economic model as correctly speci…ed. Rather, it is more accurate to view a model as a useful abstraction or approximation. In this case, how should we interpret structural econometric analysis? The quasistructural approach to inference views a structural economic model as an approximation rather than the truth. This theory has led to the concepts of the pseudo-true value (the parameter value de…ned by the estimation problem), the quasi-likelihood function, quasi-MLE, and quasi-likelihood inference. Closely related is the semiparametric approach. A probabilistic economic model is partially speci…ed but some features are left unspeci…ed. This approach typically leads to estimation methods such as least-squares and the Generalized Method of Moments. The semiparametric approach dominates contemporary econometrics, and is the main focus of this textbook. Another branch of quantitative structural economics is the calibration approach. Similar to the quasi-structural approach, the calibration approach interprets structural models as approximations and hence inherently false. The di¤erence is that the calibrationist literature rejects mathematical statistics as inappropriate for approximate models, and instead selects parameters by matching model and data moments using non-statistical ad hoc 1 methods. 1.3 Econometric Terms and Notation In a typical application, an econometrician has a set of repeated measurements on a set of variables. For example, in a labor application the variables could include weekly earnings, educational attainment, age, and other descriptive characteristics. We call this information the data, dataset, or sample. We use the term observations to refer to the distinct repeated measurements on the variables. An individual observation often corresponds to a speci…c economic unit, such as a person, household, corporation, …rm, organization, country, state, city or other geographical region. An individual observation could also be a measurement at a point in time, such as quarterly GDP or a daily interest rate. Economists typically denote variables by the italicized roman characters y, x; and/or z: The convention in econometrics is to use the character y to denote the variable to be explained, while 1 Ad hoc means “for this purpose” – a method designed for a speci…c problem – and not based on a generalizable principle. CHAPTER 1. INTRODUCTION 3 the characters x and z are used to denote the conditioning (explaining) variables. Following mathematical convention, real numbers (elements of the real line R) are written using lower case italics such as y, and vectors (elements of Rk ) by lower case bold italics such as x; e.g. 0 1 x1 B x2 C B C x = B . C: @ .. A xk Upper case bold italics such as X are used for matrices. We typically denote the number of observations by the natural number n; and subscript the variables by the index i to denote the individual observation, e.g. yi ; xi and z i . In some contexts we use indices other than i, such as in time-series applications where the index t is common, and in panel studies we typically use the double index it to refer to individual i at a time period t. The i’th observation is the set (yi ; xi ; z i ): It is proper mathematical practice to use upper case X for random variables and lower case x for realizations or speci…c values. This practice is not commonly followed in econometrics because instead we use upper case to denote matrices. Thus the notation yi will in some places refer to a random variable, and in other places a speci…c realization. Hopefully there will be no confusion as the use should be evident from the context. We typically use Greek letters such as ; and 2 to denote unknown parameters of an econometric model, and will use boldface, e.g. or , when these are vector-valued. Estimates are typically denoted by putting a hat “^”, tilde “~” or bar “-” over the corresponding letter, e.g. ^ and ~ are estimates of : The covariance matrix of an econometric estimator will typically be written using the capital p b boldface V ; often with a subscript to denote the estimator, e.g. V b = var n as the p b covariance matrix for n : Hopefully without causing confusion, we will use the notation p V = avar( b ) to denote the asymptotic covariance matrix of n b (the variance of the asymptotic distribution). Estimates will be denoted by appending hats or tildes, e.g. Vb is an estimate of V . 1.4 Observational Data A common econometric question is to quantify the impact of one set of variables on another variable. For example, a concern in labor economics is the returns to schooling – the change in earnings induced by increasing a worker’s education, holding other variables constant. Another issue of interest is the earnings gap between men and women. Ideally, we would use experimental data to answer these questions. To measure the returns to schooling, an experiment might randomly divide children into groups, mandate di¤erent levels of education to the di¤erent groups, and then follow the children’s wage path after they mature and enter the labor force. The di¤erences between the groups would be direct measurements of the effects of di¤erent levels of education. However, experiments such as this would be widely condemned as immoral! Consequently, we see few non-laboratory experimental data sets in economics. Instead, most economic data is observational. To continue the above example, through data collection we can record the level of a person’s education and their wage. With such data we can measure the joint distribution of these variables, and assess the joint dependence. But from CHAPTER 1. INTRODUCTION 4 observational data it is di¢ cult to infer causality, as we are not able to manipulate one variable to see the direct e¤ect on the other. For example, a person’s level of education is (at least partially) determined by that person’s choices. These factors are likely to be a¤ected by their personal abilities and attitudes towards work. The fact that a person is highly educated suggests a high level of ability, which suggests a high relative wage. This is an alternative explanation for an observed positive correlation between educational levels and wages. High ability individuals do better in school, and therefore choose to attain higher levels of education, and their high ability is the fundamental reason for their high wages. The point is that multiple explanations are consistent with a positive correlation between schooling levels and education. Knowledge of the joint distibution alone may not be able to distinguish between these explanations. Most economic data sets are observational, not experimental. This means that all variables must be treated as random and possibly jointly determined. This discussion means that it is di¢ cult to infer causality from observational data alone. Causal inference requires identi…cation, and this is based on strong assumptions. We will return to a discussion of some of these issues in Chapter 16. 1.5 Standard Data Structures There are three major types of economic data sets: cross-sectional, time-series, and panel. They are distinguished by the dependence structure across observations. Cross-sectional data sets have one observation per individual. Surveys are a typical source for cross-sectional data. In typical applications, the individuals surveyed are persons, households, …rms or other economic agents. In many contemporary econometric cross-section studies the sample size n is quite large. It is conventional to assume that cross-sectional observations are mutually independent. Most of this text is devoted to the study of cross-section data. Time-series data are indexed by time. Typical examples include macroeconomic aggregates, prices and interest rates. This type of data is characterized by serial dependence so the random sampling assumption is inappropriate. Most aggregate economic data is only available at a low frequency (annual, quarterly or perhaps monthly) so the sample size is typically much smaller than in cross-section studies. The exception is …nancial data where data are available at a high frequency (weekly, daily, hourly, or by transaction) so sample sizes can be quite large. Panel data combines elements of cross-section and time-series. These data sets consist of a set of individuals (typically persons, households, or corporations) surveyed repeatedly over time. The common modeling assumption is that the individuals are mutually independent of one another, but a given individual’s observations are mutually dependent. This is a modi…ed random sampling environment. Data Structures Cross-section Time-series Panel CHAPTER 1. INTRODUCTION 5 Some contemporary econometric applications combine elements of cross-section, time-series, and panel data modeling. These include models of spatial correlation and clustering. As we mentioned above, most of this text will be devoted to cross-sectional data under the assumption of mutually independent observations. By mutual independence we mean that the i’th observation (yi ; xi ; z i ) is independent of the j’th observation (yj ; xj ; z j ) for i 6= j. (Sometimes the label “independent”is misconstrued. It is a statement about the relationship between observations i and j, not a statement about the relationship between yi and xi and/or z i :) Furthermore, if the data is randomly gathered, it is reasonable to model each observation as a random draw from the same probability distribution. In this case we say that the data are independent and identically distributed or iid. We call this a random sample. For most of this text we will assume that our observations come from a random sample. De…nition 1.5.1 The observations (yi ; xi ; z i ) are a random sample if they are mutually independent and identically distributed (iid) across i = 1; :::; n: In the random sampling framework, we think of an individual observation (yi ; xi ; z i ) as a realization from a joint probability distribution F (y; x; z) which we can call the population. This “population” is in…nitely large. This abstraction can be a source of confusion as it does not correspond to a physical population in the real world. It’s an abstraction since the distribution F is unknown, and the goal of statistical inference is to learn about features of F from the sample. The assumption of random sampling provides the mathematical foundation for treating economic statistics with the tools of mathematical statistics. The random sampling framework was a major intellectural breakthrough of the late 19th century, allowing the application of mathematical statistics to the social sciences. Before this conceptual development, methods from mathematical statistics had not been applied to economic data as they were viewed as inappropriate. The random sampling framework enabled economic samples to be viewed as homogenous and random, a necessary precondition for the application of statistical methods. 1.6 Sources for Economic Data Fortunately for economists, the internet provides a convenient forum for dissemination of economic data. Many large-scale economic datasets are available without charge from governmental agencies. An excellent starting point is the Resources for Economists Data Links, available at rfe.org. From this site you can …nd almost every publically available economic data set. Some speci…c data sources of interest include Bureau of Labor Statistics US Census Current Population Survey Survey of Income and Program Participation Panel Study of Income Dynamics Federal Reserve System (Board of Governors and regional banks) National Bureau of Economic Research CHAPTER 1. INTRODUCTION 6 U.S. Bureau of Economic Analysis CompuStat International Financial Statistics Another good source of data is from authors of published empirical studies. Most journals in economics require authors of published papers to make their datasets generally available. For example, in its instructions for submission, Econometrica states: Econometrica has the policy that all empirical, experimental and simulation results must be replicable. Therefore, authors of accepted papers must submit data sets, programs, and information on empirical analysis, experiments and simulations that are needed for replication and some limited sensitivity analysis. The American Economic Review states: All data used in analysis must be made available to any researcher for purposes of replication. The Journal of Political Economy states: It is the policy of the Journal of Political Economy to publish papers only if the data used in the analysis are clearly and precisely documented and are readily available to any researcher for purposes of replication. If you are interested in using the data from a published paper, …rst check the journal’s website, as many journals archive data and replication programs online. Second, check the website(s) of the paper’s author(s). Most academic economists maintain webpages, and some make available replication …les complete with data and programs. If these investigations fail, email the author(s), politely requesting the data. You may need to be persistent. As a matter of professional etiquette, all authors absolutely have the obligation to make their data and programs available. Unfortunately, many fail to do so, and typically for poor reasons. The irony of the situation is that it is typically in the best interests of a scholar to make as much of their work (including all data and programs) freely available, as this only increases the likelihood of their work being cited and having an impact. Keep this in mind as you start your own empirical project. Remember that as part of your end product, you will need (and want) to provide all data and programs to the community of scholars. The greatest form of ‡attery is to learn that another scholar has read your paper, wants to extend your work, or wants to use your empirical methods. In addition, public openness provides a healthy incentive for transparency and integrity in empirical analysis. 1.7 Econometric Software Economists use a variety of econometric, statistical, and programming software. STATA (www.stata.com) is a powerful statistical program with a broad set of pre-programmed econometric and statistical tools. It is quite popular among economists, and is continuously being updated with new methods. It is an excellent package for most econometric analysis, but is limited when you want to use new or less-common econometric methods which have not yet been programed. R (www.r-project.org), GAUSS (www.aptech.com), MATLAB (www.mathworks.com), and Ox (www.oxmetrics.net) are high-level matrix programming languages with a wide variety of built-in statistical functions. Many econometric methods have been programed in these languages and are available on the web. The advantage of these packages is that you are in complete control of your CHAPTER 1. INTRODUCTION 7 analysis, and it is easier to program new methods than in STATA. Some disadvantages are that you have to do much of the programming yourself, programming complicated procedures takes signi…cant time, and programming errors are hard to prevent and di¢ cult to detect and eliminate. Of these languages, Gauss used to be quite popular among econometricians, but now Matlab is more popular. A smaller but growing group of econometricians are enthusiastic fans of R, which of these languages is uniquely open-source, user-contributed, and best of all, completely free! For highly-intensive computational tasks, some economists write their programs in a standard programming language such as Fortran or C. This can lead to major gains in computational speed, at the cost of increased time in programming and debugging. As these di¤erent packages have distinct advantages, many empirical economists end up using more than one package. As a student of econometrics, you will learn at least one of these packages, and probably more than one. 1.8 Reading the Manuscript Chapter 2 is a review of moment estimation and asymptotic distribution theory. This material should be familiar from an earlier course in statistics, but I have included this at the beginning because of its central importance in econometric distribution theory. Chapters 3 through 9 deal with the core linear regression and projection models. Chapter 10 introduces the bootstrap. Chapters 11 through 13 deal with the Generalized Method of Moments, empirical likelihood and endogeneity. Chapters 14 and 15 cover time series, and Chapters 16, 17 and 18 cover limited dependent variables, panel data, and nonparametrics. Reviews of matrix algebra, probability theory, maximum likelihood, and numerical optimization can be found in the appendix. Technical sections which may not be of interest to all readers are marked with an asterisk (*). Chapter 2 Conditional Expectation and Projection 2.1 Introduction The most commonly applied econometric tool is least-squares estimation, also known as regression. As we will see, least-squares is a tool to estimate an approximate conditional mean of one variable (the dependent variable) given another set of variables (the regressors, conditioning variables, or covariates). In this chapter we abstract from estimation, and focus on the probabilistic foundation of the conditional expectation model and its projection approximation. 2.2 The Distribution of Wages Suppose that we are interested in wage rates in the United States. Since wage rates vary across workers, we cannot describe wage rates by a single number. Instead, we can describe wages using a probability distribution. Formally, we view the wage of an individual worker as a random variable wage with the probability distribution F (u) = Pr(wage u): When we say that a person’s wage is random we mean that we do not know their wage before it is measured, and we treat observed wage rates as realizations from the distribution F: Treating unobserved wages as random variables and observed wages as realizations is a powerful mathematical abstraction which allows us to use the tools of mathematical probability. A useful thought experiment is to imagine dialing a telephone number selected at random, and then asking the person who responds to tell us their wage rate. (Assume for simplicity that all workers have equal access to telephones, and that the person who answers your call will respond honestly.) In this thought experiment, the wage of the person you have called is a single draw from the distribution F of wages in the population. By making many such phone calls we can learn the distribution F of the entire population. When a distribution function F is di¤erentiable we de…ne the probability density function f (u) = d F (u): du The density contains the same information as the distribution function, but the density is typically easier to visually interpret. 8 9 Wage Density 0.6 0.5 0.4 0.0 0.1 0.2 0.3 Wage Distribution 0.7 0.8 0.9 1.0 CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 0 10 20 30 40 50 60 70 0 Dollars per Hour 10 20 30 40 50 60 70 80 90 100 Dollars per Hour Figure 2.1: Wage Distribution and Density. All full-time U.S. workers In Figure 2.1 we display estimates1 of the probability distribution function (on the left) and density function (on the right) of U.S. wage rates in 2009. We see that the density is peaked around $15, and most of the probability mass appears to lie between $10 and $40. These are ranges for typical wage rates in the U.S. population. Important measures of central tendency are the median and the mean. The median m of a continuous2 distribution F is the unique solution to 1 F (m) = : 2 The median U.S. wage ($19.23) is indicated in the left panel of Figure 2.1 by the arrow. The median is a robust3 measure of central tendency, but it is tricky to use for many calculations as it is not a linear operator. The expectation or mean of a random variable y with density f is Z 1 uf (u)du: = E (y) = 1 A general de…nition of the mean is presented in Section 2.30. The mean U.S. wage ($23.90) is indicated in the right panel of Figure 2.1 by the arrow. Here we have used the common and convenient convention of using the single character y to denote a random variable, rather than the more cumbersome label wage. The mean is a convenient measure of central tendency because it is a linear operator and arises naturally in many economic models. A disadvantage of the mean is that it is not robust4 especially in the presence of substantial skewness or thick tails, which are both features of the wage distribution as can be seen easily in the right panel of Figure 2.1. Another way of viewing this is that 64% of workers earn less that the mean wage of $23.90, suggesting that it is incorrect to describe the mean as a “typical” wage rate. 1 The distribution and density are estimated nonparametrically from the sample of 50,742 full-time non-military wage-earners reported in the March 2009 Current Population Survey. The wage rate is constructed as individual wage and salary earnings divided by hours worked. 1 2 If F is not continuous the de…nition is m = inffu : F (u) g 2 3 The median is not sensitive to pertubations in the tails of the distribution. 4 The mean is sensitive to pertubations in the tails of the distribution. 10 Log Wage Density CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 1 2 3 4 5 6 Log Dollars per Hour Figure 2.2: Log Wage Density In this context it is useful to transform the data by taking the natural logarithm5 . Figure 2.2 shows the density of log hourly wages log(wage) for the same population, with its mean 2.95 drawn in with the arrow. The density of log wages is much less skewed and fat-tailed than the density of the level of wages, so its mean E (log(wage)) = 2:95 is a much better (more robust) measure6 of central tendency of the distribution. For this reason, wage regressions typically use log wages as a dependent variable rather than the level of wages. Another useful way to summarize the probability distribution F (u) is in terms of its quantiles. For any 2 (0; 1); the ’th quantile of the continuous7 distribution F is the real number q which satis…es F (q ) = : The quantile function q ; viewed as a function of ; is the inverse of the distribution function F: The most commonly used quantile is the median, that is, q0:5 = m: We sometimes refer to quantiles by the percentile representation of ; and in this case they are often called percentiles, e.g. the median is the 50th percentile. 2.3 Conditional Expectation We saw in Figure 2.2 the density of log wages. Is this distribution the same for all workers, or does the wage distribution vary across subpopulations? To answer this question, we can compare wage distributions for di¤erent groups – for example, men and women. The plot on the left in Figure 2.3 displays the densities of log wages for U.S. men and women with their means (3.05 and 2.81) indicated by the arrows. We can see that the two wage densities take similar shapes but the density for men is somewhat shifted to the right with a higher mean. The values 3.05 and 2.81 are the mean log wages in the subpopulations of men and women workers. They are called the conditional means (or conditional expectations) of log wages 5 Throughout the text, we will use log(y) to denote the natural logarithm of y: More precisely, the geometric mean exp (E (log w)) = $19:11 is a robust measure of central tendency. 7 If F is not continuous the de…nition is q = inffu : F (u) g 6 CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 11 Women 0 1 Log Wage Density Log Wage Density white men white women black men black women Men 2 3 4 5 6 Log Dollars per Hour 1 2 3 4 5 Log Dollars per Hour Figure 2.3: Left: Log Wage Density for Women and Men. Right: Log Wage Density by Gender and Race given gender. We can write their speci…c values as E (log(wage) j gender = man) = 3:05 (2.1) E (log(wage) j gender = woman) = 2:81: (2.2) We call these means conditional as they are conditioning on a …xed value of the variable gender. While you might not think of a person’s gender as a random variable, it is random from the viewpoint of econometric analysis. If you randomly select an individual, the gender of the individual is unknown and thus random. (In the population of U.S. workers, the probability that a worker is a woman happens to be 43%.) In observational data, it is most appropriate to view all measurements as random variables, and the means of subpopulations are then conditional means. As the two densities in Figure 2.3 appear similar, a hasty inference might be that there is not a meaningful di¤erence between the wage distributions of men and women. Before jumping to this conclusion let us examine the di¤erences in the distributions of Figure 2.3 more carefully. As we mentioned above, the primary di¤erence between the two densities appears to be their means. This di¤erence equals E (log(wage) j gender = man) E (log(wage) j gender = woman) = 3:05 = 0:24 2:81 (2.3) A di¤erence in expected log wages of 0.24 implies an average 24% di¤erence between the wages of men and women, which is quite substantial. (For an explanation of logarithmic and percentage di¤erences see Section 2.4.) Consider further splitting the men and women subpopulations by race, dividing the population into whites, blacks, and other races. We display the log wage density functions of four of these groups on the right in Figure 2.3. Again we see that the primary di¤erence between the four density functions is their central tendency. Focusing on the means of these distributions, Table 2.1 reports the mean log wage for each of the six sub-populations. CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION men 3.07 2.86 3.03 white black other 12 women 2.82 2.73 2.86 Table 2.1: Mean Log Wages by Sex and Race The entries in Table 2.1 are the conditional means of log(wage) given gender and race. For example E (log(wage) j gender = man; race = white) = 3:07 and E (log(wage) j gender = woman; race = black) = 2:73 One bene…t of focusing on conditional means is that they reduce complicated distributions to a single summary measure, and thereby facilitate comparisons across groups. Because of this simplifying property, conditional means are the primary interest of regression analysis and are a major focus in econometrics. Table 2.1 allows us to easily calculate average wage di¤erences between groups. For example, we can see that the wage gap between men and women continues after disaggregation by race, as the average gap between white men and white women is 25%, and that between black men and black women is 13%. We also can see that there is a race gap, as the average wages of blacks are substantially less than the other race categories. In particular, the average wage gap between white men and black men is 21%, and that between white women and black women is 9%. 2.4 Log Di¤erences* A useful approximation for the natural logarithm for small x is log (1 + x) x: (2.4) This can be derived from the in…nite series expansion of log (1 + x) : x2 x3 + 2 3 2 = x + O(x ): log (1 + x) = x x4 + 4 The symbol O(x2 ) means that the remainder is bounded by Ax2 as x ! 0 for some A < 1: A plot of log (1 + x) and the linear approximation x is shown in the following …gure. We can see that log (1 + x) and the linear approximation x are very close for jxj 0:1, and reasonably close for jxj 0:2, but the di¤erence increases with jxj. 0.4 0.2 -0.4 -0.2 0.2 -0.2 -0.4 log(1 + x) 0.4 x CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 13 Now, if y is c% greater than y; then y = (1 + c=100)y: Taking natural logarithms, log y = log y + log(1 + c=100) or c 100 where the approximation is (2.4). This shows that 100 multiplied by the di¤erence in logarithms is approximately the percentage di¤erence between y and y , and this approximation is quite good for jcj 10: log y 2.5 log y = log(1 + c=100) Conditional Expectation Function An important determinant of wage levels is education. In many empirical studies economists measure educational attainment by the number of years of schooling, and we will write this variable as education 8 . The conditional mean of log wages given gender, race, and education is a single number for each category. For example E (log(wage) j gender = man; race = white; education = 12) = 2:84 4.0 We display in Figure 2.4 the conditional means of log(wage) for white men and white women as a function of education. The plot is quite revealing. We see that the conditional mean is increasing in years of education, but at a di¤erent rate for schooling levels above and below nine years. Another striking feature of Figure 2.4 is that the gap between men and women is roughly constant for all education levels. As the variables are measured in logs this implies a constant average percentage gap between men and women regardless of educational attainment. ● 3.5 ● 3.0 ● ● ● ● 2.5 Log Dollars per Hour ● ● ● ● ● 2.0 ● ● 4 6 8 10 12 14 16 white men white women 18 20 Years of Education Figure 2.4: Mean Log Wage as a Function of Years of Education 8 Here, education is de…ned as years of schooling beyond kindergarten. A high school graduate has education=12, a college graduate has education=16, a Master’s degree has education=18, and a professional degree (medical, law or PhD) has education=20. CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 14 In many cases it is convenient to simplify the notation by writing variables using single characters, typically y; x and/or z. It is conventional in econometrics to denote the dependent variable (e.g. log(wage)) by the letter y; a conditioning variable (such as gender ) by the letter x; and multiple conditioning variables (such as race, education and gender ) by the subscripted letters x1 ; x2 ; :::; xk . Conditional expectations can be written with the generic notation E (y j x1 ; x2 ; :::; xk ) = m(x1 ; x2 ; :::; xk ): We call this the conditional expectation function (CEF). The CEF is a function of (x1 ; x2 ; :::; xk ) as it varies with the variables. For example, the conditional expectation of y = log(wage) given (x1 ; x2 ) = (gender ; race) is given by the six entries of Table 2.1. The CEF is a function of (gender ; race) as it varies across the entries. For greater compactness, we will typically write the conditioning variables as a vector in Rk : 0 1 x1 B x2 C B C x = B . C: (2.5) . @ . A xk Here we follow the convention of using lower case bold italics x to denote a vector. Given this notation, the CEF can be compactly written as E (y j x) = m (x) : The CEF E (y j x) is a random variable as it is a function of the random variable x. It is also sometimes useful to view the CEF as a function of x. In this case we can write m (u) = E (y j x = u) , which is a function of the argument u. The expression E (y j x = u) is the conditional expectation of y; given that we know that the random variable x equals the speci…c value u. However, sometimes in econometrics we take a notational shortcut and use E (y j x) to refer to this function. Hopefully, the use of E (y j x) should be apparent from the context. 2.6 Continuous Variables In the previous sections, we implicitly assumed that the conditioning variables are discrete. However, many conditioning variables are continuous. In this section, we take up this case and assume that the variables (y; x) are continuously distributed with a joint density function f (y; x): As an example, take y = log(wage) and x = experience, the number of years of labor market experience. The contours of their joint density are plotted on the left side of Figure 2.5 for the population of white men with 12 years of education. Given the joint density f (y; x) the variable x has the marginal density Z fx (x) = f (y; x)dy: R For any x such that fx (x) > 0 the conditional density of y given x is de…ned as fyjx (y j x) = f (y; x) : fx (x) (2.6) The conditional density is a slice of the joint density f (y; x) holding x …xed. We can visualize this by slicing the joint density function at a speci…c value of x parallel with the y-axis. For example, take the density contours on the left side of Figure 2.5 and slice through the contour plot at a speci…c value of experience. This gives us the conditional density of log(wage) for white men with 15 4.0 CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION Log Wage Conditional Density 3.0 2.0 2.5 Log Dollars per Hour 3.5 Exp=5 Exp=10 Exp=25 Exp=40 0 10 20 30 40 50 1.0 Labor Market Experience (Years) 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Log Dollars per Hour Figure 2.5: Left: Joint density of log(wage) and experience and conditional mean of log(wage) given experience for white men with education=12. Right: Conditional densities of log(wage) for white men with education=12. 12 years of education and this level of experience. We do this for four levels of experience (5, 10, 25, and 40 years), and plot these densities on the right side of Figure 2.5. We can see that the distribution of wages shifts to the right and becomes more di¤use as experience increases from 5 to 10 years, and from 10 to 25 years, but there is little change from 25 to 40 years experience. The CEF of y given x is the mean of the conditional density (2.6) Z m (x) = E (y j x) = yfyjx (y j x) dy: (2.7) R Intuitively, m (x) is the mean of y for the idealized subpopulation where the conditioning variables are …xed at x. This is idealized since x is continuously distributed so this subpopulation is in…nitely small. In Figure 2.5 the CEF of log(wage) given experience is plotted as the solid line. We can see that the CEF is a smooth but nonlinear function. The CEF is initially increasing in experience, ‡attens out around experience = 30, and then decreases for high levels of experience. 2.7 Law of Iterated Expectations An extremely useful tool from probability theory is the law of iterated expectations. An important special case is the known as the Simple Law. Theorem 2.7.1 Simple Law of Iterated Expectations If E jyj < 1 then for any random vector x, E (E (y j x)) = E (y) The simple law states that the expectation of the conditional expectation is the unconditional expectation. In other words, the average of the conditional averages is the unconditional average. CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION When x is discrete E (E (y j x)) = and when x is continuous 1 X j=1 E (E (y j x)) = Z 16 E (y j x j ) Pr (x = x j ) Rk E (y j x ) fx (x)dx: Going back to our investigation of average log wages for men and women, the simple law states that E (log(wage) j gender = man) Pr (gender = man) +E (log(wage) j gender = woman) Pr (gender = woman) = E (log(wage)) : Or numerically, 3:05 0:57 + 2:79 0:43 = 2:92: The general law of iterated expectations allows two sets of conditioning variables. Theorem 2.7.2 Law of Iterated Expectations If E jyj < 1 then for any random vectors x1 and x2 , E (E (y j x1 ; x2 ) j x1 ) = E (y j x1 ) Notice the way the law is applied. The inner expectation conditions on x1 and x2 , while the outer expectation conditions only on x1 : The iterated expectation yields the simple answer E (y j x1 ) ; the expectation conditional on x1 alone. Sometimes we phrase this as: “The smaller information set wins.” As an example E (log(wage) j gender = man; race = white) Pr (race = whitejgender = man) +E (log(wage) j gender = man; race = black) Pr (race = blackjgender = man) +E (log(wage) j gender = man; race = other) Pr (race = otherjgender = man) = E (log(wage) j gender = man) or numerically 3:07 0:84 + 2:86 0:08 + 3:05 0:08 = 3:05 A property of conditional expectations is that when you condition on a random vector x you can e¤ectively treat it as if it is constant. For example, E (x j x) = x and E (g (x) j x) = g (x) for any function g( ): The general property is known as the conditioning theorem. Theorem 2.7.3 Conditioning Theorem If E jg (x) yj < 1 (2.8) then E (g (x) y j x) = g (x) E (y j x) (2.9) E (g (x) y) = E (g (x) E (y j x)) (2.10) and The proofs of Theorems 2.7.1, 2.7.2 and 2.7.3 are given in Section 2.33. CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 2.8 17 CEF Error The CEF error e is de…ned as the di¤erence between y and the CEF evaluated at the random vector x: e = y m(x): By construction, this yields the formula y = m(x) + e: (2.11) In (2.11) it is useful to understand that the error e is derived from the joint distribution of (y; x); and so its properties are derived from this construction. A key property of the CEF error is that it has a conditional mean of zero. To see this, by the linearity of expectations, the de…nition m(x) = E (y j x) and the Conditioning Theorem E (e j x) = E ((y m(x)) j x) = E (y j x) = m(x) E (m(x) j x) m(x) = 0: This fact can be combined with the law of iterated expectations to show that the unconditional mean is also zero. E (e) = E (E (e j x)) = E (0) = 0: We state this and some other results formally. Theorem 2.8.1 Properties of the CEF error If E jyj < 1 then 1. E (e j x) = 0: 2. E (e) = 0: 3. If E jyjr < 1 for r 1 then E jejr < 1: 4. For any function h (x) such that E jh (x) ej < 1 then E (h (x) e) = 0: The proof of the third result is deferred to Section 2.33: The fourth result, whose proof is left to Exercise 2.3, says that e is uncorrelated with any function of the regressors. The equations y = m(x) + e E (e j x) = 0: together imply that m(x) is the CEF of y given x. It is important to understand that this is not a restriction. These equations hold true by de…nition. The condition E (e j x) = 0 is implied by the de…nition of e as the di¤erence between y and the CEF m (x) : The equation E (e j x) = 0 is sometimes called a conditional mean restriction, since the conditional mean of the error e is restricted to equal zero. The property is also sometimes called mean independence, for the conditional mean of e is 0 and thus independent of x. However, it does not imply that the distribution of e is independent of x: Sometimes the assumption “e is independent of x” is added as a convenient simpli…cation, but it is not generic feature of the conditional mean. Typically and generally, e and x are jointly dependent, even though the conditional mean of e is zero. 18 e −1.0 −0.5 0.0 0.5 1.0 CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 0 10 20 30 40 50 Labor Market Experience (Years) Figure 2.6: Joint density of CEF error e and experience for white men with education=12. As an example, the contours of the joint density of e and experience are plotted in Figure 2.6 for the same population as Figure 2.5. The error e has a conditional mean of zero for all values of experience, but the shape of the conditional distribution varies with the level of experience. As a simple example of a case where x and e are mean independent yet dependent, let e = x" where x and " are independent N(0; 1): Then conditional on x; the error e has the distribution N(0; x2 ): Thus E (e j x) = 0 and e is mean independent of x; yet e is not fully independent of x: Mean independence does not imply full independence. 2.9 Regression Variance An important measure of the dispersion about the CEF function is the unconditional variance of the CEF error e: We write this as 2 = var (e) = E (e Ee)2 = E e2 : Theorem 2.8.1.3 implies the following simple but useful result. Theorem 2.9.1 If Ey 2 < 1 then 2 <1 We can call 2 the regression variance or the variance of the regression error. The magnitude of 2 measures the amount of variation in y which is not “explained” or accounted for in the conditional mean E (y j x) : The regression variance depends on the regressors x. Consider two regressions y = E (y j x1 ) + e1 y = E (y j x1 ; x2 ) + e2 : We write the two errors distinctly as e1 and e2 as they are di¤erent – changing the conditioning information changes the conditional mean and therefore the regression error as well. In our discussion of iterated expectations, we have seen that by increasing the conditioning set, the conditional expectation reveals greater detail about the distribution of y: What is the implication for the regression error? CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 19 It turns out that there is a simple relationship. We can think of the conditional mean E (y j x) as the “explained portion”of y: The remainder e = y E (y j x) is the “unexplained portion”. The simple relationship we now derive shows that the variance of this unexplained portion decreases when we condition on more variables. This relationship is monotonic in the sense that increasing the amont of information always decreases the variance of the unexplained portion. Theorem 2.9.2 If Ey 2 < 1 then var (y) var (y E (y j x1 )) var (y E (y j x1 ; x2 )) Theorem 2.9.2 says that the variance of the di¤erence between y and its conditional mean (weakly) decreases whenever an additional variable is added to the conditioning information. The proof of Theorem 2.9.2 is given in Section 2.33. 2.10 Best Predictor Suppose that given a realized value of x, we want to create a prediction or forecast of y: We can write any predictor as a function g (x) of x. The prediction error is the realized di¤erence y g(x): A non-stochastic measure of the magnitude of the prediction error is the expectation of its square E (y g (x))2 : (2.12) We can de…ne the best predictor as the function g (x) which minimizes (2.12). What function is the best predictor? It turns out that the answer is the CEF m(x). This holds regardless of the joint distribution of (y; x): To see this, note that the mean squared error of a predictor g (x) is E (y g (x))2 = E (e + m (x) g (x))2 = Ee2 + 2E (e (m (x) = Ee2 + E (m (x) g (x))) + E (m (x) g (x))2 g (x))2 Ee2 = E (y m (x))2 where the …rst equality makes the substitution y = m(x) + e and the third equality uses Theorem 2.8.1.4. The right-hand-side after the third equality is minimized by setting g (x) = m (x), yielding the …nal inequality. The minimum is …nite under the assumption Ey 2 < 1 as shown by Theorem 2.9.1. We state this formally in the following result. Theorem 2.10.1 Conditional Mean as Best Predictor If Ey 2 < 1; then for any predictor g (x), E (y where m (x) = E (y j x). g (x))2 E (y m (x))2 CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 2.11 20 Conditional Variance While the conditional mean is a good measure of the location of a conditional distribution, it does not provide information about the spread of the distribution. A common measure of the dispersion is the conditional variance. De…nition 2.11.1 If Ey 2 < 1; the conditional variance of y given x is 2 (x) = var (y j x) E (y j x))2 j x = E (y = E e2 j x Generally, 2 (x) is a non-trivial function of x and can take any form subject to the restriction p 2 (x): that it is non-negative. The conditional standard deviation is its square root (x) = 2 2 One way to think about (x) is that it is the conditional mean of e given x. As an example of how the conditional variance depends on observables, compare the conditional log wage densities for men and women displayed in Figure 2.3. The di¤erence between the densities is not purely a location shift, but is also a di¤erence in spread. Speci…cally, we can see that the density for men’s log wages is somewhat more spread out than that for women, while the density for women’s wages is somewhat more peaked. Indeed, the conditional standard deviation for men’s wages is 3.05 and that for women is 2.81. So while men have higher average wages, they are also somewhat more dispersed. The unconditional error variance and the conditional variance are related by the law of iterated expectations 2 = E e2 = E E e2 j x = E 2 (x) : That is, the unconditional error variance is the average conditional variance. Given the conditional variance, we can de…ne a rescaled error "= e : (x) (2.13) We can calculate that since (x) is a function of x E (" j x) = E and var (" j x) = E "2 j x = E e jx (x) e2 jx 2 (x) = = 1 E (e j x) = 0 (x) 1 E e2 j x = 2 (x) 2 (x) 2 (x) = 1: Thus " has a conditional mean of zero, and a conditional variance of 1. Notice that (2.13) can be rewritten as e = (x)": and substituting this for e in the CEF equation (2.11), we …nd that y = m(x) + (x)": This is an alternative (mean-variance) representation of the CEF equation. (2.14) CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 21 Many econometric studies focus on the conditional mean m(x) and either ignore the conditional variance 2 (x); treat it as a constant 2 (x) = 2 ; or treat it as a nuisance parameter (a parameter not of primary interest). This is appropriate when the primary variation in the conditional distribution is in the mean, but can be short-sighted in other cases. Dispersion is relevant to many economic topics, including income and wealth distribution, economic inequality, and price dispersion. Conditional dispersion (variance) can be a fruitful subject for investigation. The perverse consequences of a narrow-minded focus on the mean has been parodied in a classic joke: An economist was standing with one foot in a bucket of boiling water and the other foot in a bucket of ice. When asked how he felt, he replied, “On average I feel just …ne.” Clearly, the economist in question ignored variance! 2.12 Homoskedasticity and Heteroskedasticity An important special case obtains when the conditional variance pendent of x. This is called homoskedasticity. 2 (x) is a constant and inde- De…nition 2.12.1 The error is homoskedastic if E e2 j x = does not depend on x. In the general case where 2 (x) 2 depends on x we say that the error e is heteroskedastic. De…nition 2.12.2 The error is heteroskedastic if E e2 j x = depends on x. 2 (x) It is helpful to understand that the concepts homoskedasticity and heteroskedasticity concern the conditional variance, not the unconditional variance. By de…nition, the unconditional variance 2 is a constant and independent of the regressors x. So when we talk about the variance as a function of the regressors, we are talking about the conditional variance 2 (x). Some older or introductory textbooks describe heteroskedasticity as the case where “the variance of e varies across observations”. This is a poor and confusing de…nition. It is more constructive to understand that heteroskedasticity means that the conditional variance 2 (x) depends on observables. Older textbooks also tend to describe homoskedasticity as a component of a correct regression speci…cation, and describe heteroskedasticity as an exception or deviance. This description has in‡uenced many generations of economists, but it is unfortunately backwards. The correct view is that heteroskedasticity is generic and “standard”, while homoskedasticity is unusual and exceptional. The default in empirical work should be to assume that the errors are heteroskedastic, not the converse. In apparent contradiction to the above statement, we will still frequently impose the homoskedasticity assumption when making theoretical investigations into the properties of estimation CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 22 and inference methods. The reason is that in many cases homoskedasticity greatly simpli…es the theoretical calculations, and it is therefore quite advantageous for teaching and learning. It should always be remembered, however, that homoskedasticity is never imposed because it is believed to be a correct feature of an empirical model, but rather because of its simplicity. 2.13 Regression Derivative One way to interpret the CEF m(x) = E (y j x) is in terms of how marginal changes in the regressors x imply changes in the conditional mean of the response variable y: It is typical to consider marginal changes in a single regressor, say x1 , holding the remainder …xed. When a regressor x1 is continuously distributed, we de…ne the marginal e¤ect of a change in x1 , holding the variables x2 ; :::; xk …xed, as the partial derivative of the CEF @ m(x1 ; :::; xk ): @x1 When x1 is discrete we de…ne the marginal e¤ect as a discrete di¤erence. For example, if x1 is binary, then the marginal e¤ect of x1 on the CEF is m(1; x2 ; :::; xk ) m(0; x2 ; :::; xk ): We can unify the continuous and discrete cases with the notation 8 @ > > m(x1 ; :::; xk ); if x1 is continuous < @x1 r1 m(x) = > > : m(1; x ; :::; x ) m(0; x ; :::; x ); if x is binary. 2 Collecting the k e¤ects into one k k 2 k 1 1 vector, we de…ne the regression derivative of x on y : 2 3 r1 m(x) 6 r2 m(x) 7 6 7 rm(x) = 6 7 .. 4 5 . rk m(x) @ m(x), the When all elements of x are continuous, then we have the simpli…cation rm(x) = @x vector of partial derivatives. There are two important points to remember concerning our de…nition of the regression derivative. First, the e¤ect of each variable is calculated holding the other variables constant. This is the ceteris paribus concept commonly used in economics. But in the case of a regression derivative, the conditional mean does not literally hold all else constant. It only holds constant the variables included in the conditional mean. This means that the regression derivative depends on which regressors are included. For example, in a regression of wages on education, experience, race and gender, the regression derivative with respect to education shows the marginal e¤ect of education on wages, holding constant a person’s observable characteristic experience, race and gender. But it does not hold constant a person’s unobservable characteristics (such as ability), or variables not included in the regression (such as the quality of education). Second, the regression derivative is the change in the conditional expectation of y, not the change in the actual value of y for an individual. It is tempting to think of the regression derivative as the change in the actual value of y, but this is not a correct interpretation. The regression derivative rm(x) is the change in the actual value of y only if the error e is una¤ected by the change in the regressor x. We return to a discussion of causal e¤ects in Section 2.29. CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 2.14 23 Linear CEF An important special case is when the CEF m (x) = E (y j x) is linear in x: In this case we can write the mean equation as m(x) = x1 1 + x2 2 + + xk k + : Notationally it is convenient to write this as a simple function of the vector x. An easy way to do so is to augment the regressor vector x by listing the number “1” as an element. We call this the “constant”and the corresponding coe¢ cient is called the “intercept”. Equivalently, assuming that the …nal element9 of the vector x is the intercept, then xk = 1. Thus (2.5) has been rede…ned as the k 1 vector 0 1 x1 B x2 C B C B .. C x = B . C: (2.15) B C @ xk 1 A 1 With this rede…nition, then the CEF is m(x) = x1 1 + x2 2 + + xk k 0 = x (2.16) where 0 1 1 B C = @ ... A (2.17) k is a k 1 coe¢ cient vector. This is the linear CEF model. It is also often called the linear regression model, or the regression of y on x: In the linear CEF model, the regression derivative is simply the coe¢ cient vector. That is rm(x) = : This is one of the appealing features of the linear CEF model. The coe¢ cients have simple and natural interpretations as the marginal e¤ects of changing one variable, holding the others constant. Linear CEF Model y = x0 + e E (e j x) = 0 If in addition the error is homoskedastic, we call this the homoskedastic linear CEF model. Homoskedastic Linear CEF Model y = x0 + e E (e j x) = 0 E e2 j x 9 The order doesn’t matter. It could be any element. = 2 CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 2.15 24 Linear CEF with Nonlinear E¤ects The linear CEF model of the previous section is less restrictive than it might appear, as we can include as regressors nonlinear transformations of the original variables. In this sense, the linear CEF framework is ‡exible and can capture many nonlinear e¤ects. For example, suppose we have two scalar variables x1 and x2 : The CEF could take the quadratic form m(x1 ; x2 ) = x1 1 + x2 2 + x21 3 + x22 4 + x1 x2 5 + 6 : (2.18) This equation is quadratic in the regressors (x1 ; x2 ) yet linear in the coe¢ cients ( 1 ; :::; 6 ): We will descriptively call (2.18) a quadratic CEF, and yet (2.18) is also a linear CEF in the sense of being linear in the coe¢ cients. The key is to understand that (2.18) is quadratic in the variables (x1 ; x2 ) yet linear in the coe¢ cients ( 1 ; :::; 6 ): To simplify the expression, we de…ne the transformations x3 = x21 ; x4 = x22 ; x5 = x1 x2 ; and x6 = 1; and rede…ne the regressor vector as x = (x1 ; :::; x6 ): With this rede…nition, m(x1 ; x2 ) = x0 which is linear in . For most econometric purposes (estimation and inference on ) the linearity in is all that is important. An exception is in the analysis of regression derivatives. In nonlinear equations such as (2.18), the regression derivative should be de…ned with respect to the original variables, not with respect to the transformed variables. Thus @ m(x1 ; x2 ) = @x1 @ m(x1 ; x2 ) = @x2 1 + 2x1 3 + x2 5 2 + 2x2 4 + x1 5 We see that in the model (2.18), the regression derivatives are not a simple coe¢ cient, but are functions of several coe¢ cients plus the levels of (x1; x2 ): Consequently it is di¢ cult to interpret the coe¢ cients individually. It is more useful to interpret them as a group. We typically call 5 the interaction e¤ect. Notice that it appears in both regression derivative equations, and has a symmetric interpretation in each. If 5 > 0 then the regression derivative of x1 on y is increasing in the level of x2 (and the regression derivative of x2 on y is increasing in the level of x1 ); while if 5 < 0 the reverse is true. It is worth noting that this symmetry is an arti…cial implication of the quadratic equation (2.18), and is not a general feature of nonlinear conditional means m(x1 ; x2 ). 2.16 Linear CEF with Dummy Variables When all regressors takes a …nite set of values, it turns out the CEF can be written as a linear function of regressors. This simplest example is a binary variable, which takes only two distinct values. For example, the variable gender takes only the values man and woman. Binary variables are extremely common in econometric applications, and are alternatively called dummy variables or indicator variables. Consider the simple case of a single binary regressor. In this case, the conditional mean can only take two distinct values. For example, 8 gender=man < 0 if E (y j gender) = : 1 if gender=woman CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 25 To facilitate a mathematical treatment, we typically record dummy variables with the values f0; 1g: For example 0 if gender=man x1 = (2.19) 1 if gender=woman Given this notation we can write the conditional mean as a linear function of the dummy variable x1 ; that is E (y j x1 ) = + x1 where = 0 and = 1 is equal to the 0 : In this simple regression equation the intercept conditional mean of y for the x1 = 0 subpopulation (men) and the slope is equal to the di¤erence in the conditional means between the two subpopulations. Equivalently, we could have de…ned x1 as x1 = 1 if gender=man 0 if gender=woman (2.20) In this case, the regression intercept is the mean for women (rather than for men) and the regression slope has switched signs. The two regressions are equivalent but the interpretation of the coe¢ cients has changed. Therefore it is always important to understand the precise de…nitions of the variables, and illuminating labels are helpful. For example, labelling x1 as “gender”does not help distinguish between de…nitions (2.19) and (2.20). Instead, it is better to label x1 as “women” or “female” if de…nition (2.19) is used, or as “men” or “male” if (2.20) is used. Now suppose we have two dummy variables x1 and x2 : For example, x2 = 1 if the person is married, else x2 = 0: The conditional mean given x1 and x2 takes at most four possible values: 8 if x1 = 0 and x2 = 0 (unmarried men) > > < 00 if x = 0 and x = 1 (married men) 1 2 01 E (y j x1 ; x2 ) = if x = 1 and x = 0 (unmarried women) > 1 2 > : 10 (married women) 11 if x1 = 1 and x2 = 1 In this case we can write the conditional mean as a linear function of x1 , x2 and their product x1 x2 : E (y j x1 ; x2 ) = + 1 x1 + 2 x2 + 3 x1 x2 where = 00 ; 1 = 10 00 ; 2 = 01 00 ; and 3 = 11 10 01 + 00 : We can view the coe¢ cient 1 as the e¤ect of gender on expected log wages for unmarried wages earners, the coe¢ cient 2 as the e¤ect of marriage on expected log wages for men wage earners, and the coe¢ cient 3 as the di¤erence between the e¤ects of marriage on expected log wages among women and among men. Alternatively, it can also be interpreted as the di¤erence between the e¤ects of gender on expected log wages among married and non-married wage earners. Both interpretations are equally valid. We often describe 3 as measuring the interaction between the two dummy variables, or the interaction e¤ect, and describe 3 = 0 as the case when the interaction e¤ect is zero. In this setting we can see that the CEF is linear in the three variables (x1 ; x2 ; x1 x2 ): Thus to put the model in the framework of Section 2.14, we would de…ne the regressor x3 = x1 x2 and the regressor vector as 0 1 x1 B x2 C C x=B @ x3 A : 1 So even though we started with only 2 dummy variables, the number of regressors (including the intercept) is 4. CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 26 If there are 3 dummy variables x1 ; x2 ; x3 ; then E (y j x1 ; x2 ; x3 ) takes at most 23 = 8 distinct values and can be written as the linear function E (y j x1 ; x2 ; x3 ) = + 1 x1 + 2 x2 + 3 x3 + 4 x1 x2 + 5 x1 x3 + 6 x2 x3 + 7 x1 x2 x3 which has eight regressors including the intercept. In general, if there are p dummy variables x1 ; :::; xp then the CEF E (y j x1 ; x2 ; :::; xp ) takes at most 2p distinct values, and can be written as a linear function of the 2p regressors including x1 ; x2 ; :::; xp and all cross-products. This might be excessive in practice if p is modestly large. In the next section we will discuss projection approximations which yield more parsimonious parameterizations. We started this section by saying that the conditional mean is linear whenever all regressors take only a …nite number of possible values. How can we see this? Take a categorical variable, such as race. For example, we earlier divided race into three categories. We can record categorical variables using numbers to indicate each category, for example 8 < 1 if white 2 if black x3 = : 3 if other When doing so, the values of x3 have no meaning in terms of magnitude, they simply indicate the relevant category. When the regressor is categorical the conditional mean of y given x3 takes a distinct value for each possibility: 8 < 1 if x3 = 1 if x3 = 2 E (y j x3 ) = : 2 3 if x3 = 3 This is not a linear function of x3 itself, but it can be made so by constructing dummy variables for two of the three categories. For example x4 = 1 if black 0 if not black x5 = 1 if other 0 if not other In this case, the categorical variable x3 is equivalent to the pair of dummy variables (x4 ; x5 ): The explicit relationship is 8 < 1 if x4 = 0 and x5 = 0 2 if x4 = 1 and x5 = 0 x3 = : 3 if x4 = 0 and x5 = 1 Given these transformations, we can write the conditional mean of y as a linear function of x4 and x5 E (y j x3 ) = E (y j x4 ; x5 ) = + 1 x4 + 2 x5 We can write the CEF as either E (y j x3 ) or E (y j x4 ; x5 ) (they are equivalent), but it is only linear as a function of x4 and x5 : This setting is similar to the case of two dummy variables, with the di¤erence that we have not included the interaction term x4 x5 : This is because the event fx4 = 1 and x5 = 1g is empty by construction, so x4 x5 = 0 by de…nition. CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 2.17 27 Best Linear Predictor While the conditional mean m(x) = E (y j x) is the best predictor of y among all functions of x; its functional form is typically unknown. In particular, the linear CEF model is empirically unlikely to be accurate unless x is discrete and low-dimensional so all interactions are included. Consequently in most cases it is more realistic to view the linear speci…cation (2.16) as an approximation. In this section we derive a speci…c approximation with a simple interpretation. Theorem 2.10.1 showed that the conditional mean m (x) is the best predictor in the sense that it has the lowest mean squared error among all predictors. By extension, we can de…ne an approximation to the CEF by the linear function with the lowest mean squared error among all linear predictors. For this derivation we require the following regularity condition. Assumption 2.17.1 1. Ey 2 < 1: 2. E kxk2 < 1: 3. Qxx = E (xx0 ) is positive de…nite. In Assumption 2.17.1.2 we use the notation kxk = (x0 x)1=2 to denote the Euclidean length of the vector x. The …rst two parts of Assumption 2.17.1 imply that the variables y and x have …nite means, variances, and covariances. The third part of the assumption is more technical, and its role will become apparent shortly. It is equivalent to imposing that the columns of Qxx = E (xx0 ) are linearly independent, or equivalently that the matrix Qxx is invertible. A linear predictor for y is a function of the form x0 for some 2 Rk . The mean squared prediction error is 2 S( ) = E y x0 : The best linear predictor of y given x, written P(y j x); is found by selecting the vector minimize S( ): De…nition 2.17.1 The Best Linear Predictor of y given x is P(y j x) = x0 where minimizes the mean squared prediction error S( ) = E y x0 2 : The minimizer = argmin S( ) 2Rk is called the Linear Projection Coe¢ cient. We now calculate an explicit expression for its value. (2.21) to CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION The mean squared prediction error can be written out as a quadratic function of S( ) = Ey 2 2 0 E (xy) + 0 E xx0 28 : : The quadratic structure of S( ) means that we can solve explicitly for the minimizer. The …rstorder condition for minimization (from Appendix A.9) is 0= @ S( ) = @ 2E (xy) + 2E xx0 : (2.22) Rewriting (2.22) as 2E (xy) = 2E xx0 and dividing by 2, this equation takes the form Qxy = Qxx (2.23) where Qxy = E (xy) is k 1 and Qxx = E (xx0 ) is k k. The solution is found by inverting the matrix Qxx , and is written = Qxx1 Qxy or = E xx0 1 E (xy) : (2.24) It is worth taking the time to understand the notation involved in the expression (2.24). Qxx is a E(xy) k k matrix and Qxy is a k 1 column vector. Therefore, alternative expressions such as E(xx 0) or E (xy) (E (xx0 )) 1 are incoherent and incorrect. We also can now see the role of Assumption 2.17.1.3. It is necessary in order for the solution (2.24) to exist. Otherwise, there would be multiple solutions to the equation (2.23). We now have an explicit expression for the best linear predictor: 1 P(y j x) = x0 E xx0 E (xy) : This expression is also referred to as the linear projection of y on x. The projection error is e=y x0 : (2.25) This equals the error from the regression equation when (and only when) the conditional mean is linear in x; otherwise they are distinct. Rewriting, we obtain a decomposition of y into linear predictor and error y = x0 + e: (2.26) In general we call equation (2.26) or x0 the best linear predictor of y given x; or the linear projection of y on x. Equation (2.26) is also often called the regression of y on x but this can sometimes be confusing as economists use the term regression in many contexts. (Recall that we said in Section 2.14 that the linear CEF model is also called the linear regression model.) An important property of the projection error e is E (xe) = 0: (2.27) To see this, using the de…nitions (2.25) and (2.24) and the matrix properties AA Ia = a; E (xe) = E x y = E (xy) = 0 1 = I and x0 E xx0 E xx0 1 E (xy) (2.28) CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 29 as claimed. Equation (2.27) is a set of k equations, one for each regressor. In other words, (2.27) is equivalent to E (xj e) = 0 (2.29) for j = 1; :::; k: As in (2.15), the regressor vector x typically contains a constant, e.g. xk = 1. In this case (2.29) for j = k is the same as E (e) = 0: (2.30) Thus the projection error has a mean of zero when the regressor vector contains a constant. (When x does not have a constant, (2.30) is not guaranteed. As it is desirable for e to have a zero mean, this is a good reason to always include a constant in any regression model.) It is also useful to observe that since cov(xj ; e) = E (xj e) E (xj ) E (e) ; then (2.29)-(2.30) together imply that the variables xj and e are uncorrelated. This completes the derivation of the model. We summarize some of the most important properties. Theorem 2.17.1 Properties of Linear Projection Model Under Assumption 2.17.1, 1. The moments E (xx0 ) and E (xy) exist with …nite elements. 2. The Linear Projection Coe¢ cient (2.21) exists, is unique, and equals 1 = E xx0 E (xy) : 3. The best linear predictor of y given x is P(y j x) = x0 E xx0 4. The projection error e = y x0 1 E (xy) : exists and satis…es E e2 < 1 and E (xe) = 0: 5. If x contains an constant, then E (e) = 0: 6. If E jyjr < 1 and E kxkr < 1 for r 1 then E jejr < 1: A complete proof of Theorem 2.17.1 is given in Section 2.33. It is useful to re‡ect on the generality of Theorem 2.17.1. The only restriction is Assumption 2.17.1. Thus for any random variables (y; x) with …nite variances we can de…ne a linear equation (2.26) with the properties listed in Theorem 2.17.1. Stronger assumptions (such as the linear CEF model) are not necessary. In this sense the linear model (2.26) exists quite generally. However, it is important not to misinterpret the generality of this statement. The linear equation (2.26) is de…ned as the best linear predictor. It is not necessarily a conditional mean, nor a parameter of a structural or causal economic model. CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 30 Linear Projection Model y = x0 + e: E (xe) = 0 = E xx0 1 E (xy) We illustrate projection using three log wage equations introduced in earlier sections. For our …rst example, we consider a model with the two dummy variables for gender and race similar to Table 2.1. As we learned in Section 2.16, the entries in this table can be equivalently expressed by a linear CEF. For simplicity, let’s consider the CEF of log(wage) as a function of Black and Female. E(log(wage) j Black; F emale) = 0:20Black 0:24F emale + 0:10Black F emale + 3:06: (2.31) This is a CEF as the variables are dummys and all interactions are included. Now consider a simpler model omitting the interaction e¤ect. This is the linear projection on the variables Black and F emale P(log(wage) j Black; F emale) = 0:15Black 0:23F emale + 3:06: (2.32) What is the di¤erence? The full CEF (2.31) shows that the race gap is di¤erentiated by gender: it is 20% for black men (relative to non-black men) and 10% for black women (relative to non-black women). The projection model (2.32) simpli…es this analysis, calculating an average 15% wage gap for blacks, ignoring the role of gender. Notice that this is despite the fact that the gender variable is included in (2.32). For our second example we consider the CEF of log wages as a function of years of education for white men which was illustrated in Figure 2.4 and is repeated in Figure 2.7. Superimposed on the …gure are two projections. The …rst (given by the dashed line) is the linear projection of log wages on years of education P(log(wage) j Education) = 1:5 + 0:11Education This simple equation indicates an average 11% increase in wages for every year of education. An inspection of the Figure shows that this approximation works well for education 9, but underpredicts for individuals with lower levels of education. To correct this imbalance we use a linear spline equation which allows di¤erent rates of return above and below 9 years of education: P (log(wage) j Education; (Education = 2:3 + 0:02Education + :10 (Education 9) 1 (Education > 9)) 9) 1 (Education > 9) This equation is displayed in Figure 2.7 using the solid line, and appears to …t much better. It indicates a 2% increase in mean wages for every year of education below 9, and a 12% increase in mean wages for every year of education above 9. It is still an approximation to the conditional mean but it appears to be fairly reasonable. For our third example we take the CEF of log wages as a function of years of experience for white men with 12 years of education, which was illustrated in Figure 2.5 and is repeated as the solid line in Figure 2.8. Superimposed on the …gure are two projections. The …rst (given by the dot-dashed line) is the linear projection on experience P(log(wage) j Experience) = 2:5 + 0:011Experience 31 4.0 CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION ● 3.5 ● 3.0 ● ● 2.5 ● ● ● ● ● 4 6 ● ● 2.0 Log Dollars per Hour ● 8 10 12 14 16 18 20 Years of Education Figure 2.7: Projections of log(wage) onto Education and the second (given by the dashed line) is the linear projection on experience and its square P(log(wage) j Experience) = 2:3 + 0:046Experience 0:0007Experience2 : 4.0 It is fairly clear from an examination of Figure 2.8 that the …rst linear projection is a poor approximation. It over-predicts wages for young and old workers, and under-predicts for the rest. Most importantly, it misses the strong downturn in expected wages for older wage-earners. The second projection …ts much better. We can call this equation a quadratic projection since the function is quadratic in experience: 3.0 2.0 2.5 Log Dollars per Hour 3.5 Conditional Mean Linear Projection Quadratic Projection 0 10 20 30 40 50 Labor Market Experience (Years) Figure 2.8: Linear and Quadratic Projections of log(wage) onto Experience CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 32 Invertibility and Identi…cation The linear projection coe¢ cient = (E (xx0 )) 1 E (xy) exists and is unique as long as the k k matrix Qxx = E (xx0 ) is invertible. The matrix Qxx is sometimes called the design matrix, as in experimental settings the researcher is able to control Qxx by manipulating the distribution of the regressors x: Observe that for any non-zero 2 Rk ; 0 Qxx 0 =E xx0 0 =E x 2 0 so Qxx by construction is positive semi-de…nite. The assumption that it is positive de…nite means that this is a strict inequality, E ( 0 x)2 > 0: Equivalently, there cannot exist a non-zero vector such that 0 x = 0 identically. This occurs when redundant variables are included in x: Positive semi-de…nite matrices are invertible if and only if they are positive de…nite. When Qxx is invertible then = (E (xx0 )) 1 E (xy) exists and is uniquely de…ned. In other words, in order for to be uniquely de…ned, we must exclude the degenerate situation of redundant varibles. Theorem 2.17.1 shows that the linear projection coe¢ cient is identi…ed (uniquely determined) under Assumptions 2.17.1. The key is invertibility of Qxx . Otherwise, there is no unique solution to the equation = Qxy : Qxx (2.33) When Qxx is not invertible there are multiple solutions to (2.33), all of which yield an equivalent best linear predictor x0 . In this case the coe¢ cient is not identi…ed as it does not have a unique value. Even so, the best linear predictor x0 still identi…ed. One solution is to set = E xx0 E (xy) where A denotes the generalized inverse of A (see Appendix A.5). 2.18 Linear Predictor Error Variance As in the CEF model, we de…ne the error variance as 2 = E e2 : Setting Qyy = E y 2 and Qyx = E (yx0 ) we can write 2 2 as 2 =E y x0 = Ey 2 2E yx0 = Qyy 2Qyx Qxx1 Qxy + Qyx Qxx1 Qxx Qxx1 Qxy = Qyy Qyx Qxx1 Qxy + 0 E xx0 def = Qyy x One useful feature of this formula is that it shows that Qyy x = Qyy variance of the error from the linear projection of y on x. (2.34) Qyx Qxx1 Qxy equals the CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 2.19 33 Regression Coe¢ cients Sometimes it is useful to separate the intercept from the other regressors, and write the linear projection equation in the format y = + x0 + e (2.35) where is the intercept and x does not contain a constant. Taking expectations of this equation, we …nd Ey = E + Ex0 + Ee or y where y = Ey and x = + 0 x = Ex; since E (e) = 0 from (2.30). Rearranging, we …nd = 0 x y : Subtracting this equation from (2.35) we …nd y y 0 x) = (x + e; (2.36) a linear equation between the centered variables y y and x x . (They are centered at their means, so are mean-zero random variables.) Because x x is uncorrelated with e; (2.36) is also a linear projection, thus by the formula for the linear projection model, = x ) (x E (x = var (x) 1 x) 1 0 E (x x) y y cov (x; y) a function only of the covariances10 of x and y: Theorem 2.19.1 In the linear projection model y= + x0 + e; then = 0 x y (2.37) and = var (x) 2.20 1 cov (x; y) : (2.38) Regression Sub-Vectors Let the regressors be partitioned as x= 10 x1 x2 : The covariance matrix between vectors x and z is cov (x; z) = E (x matrix of the vector x is var (x) = cov (x; x) = E (x Ex) (x Ex)0 : (2.39) Ex) (z Ez)0 : The (co)variance CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 34 We can write the projection of y on x as y = x0 + e = x01 1 + x02 2 +e (2.40) and 2: E (xe) = 0: In this section we derive formula for the sub-vectors Partition Qxx comformably with x Qxx = Q11 Q12 Q21 Q22 1 E (x1 x01 ) E (x1 x02 ) E (x2 x01 ) E (x2 x02 ) = and similarly Qxy Q1y Q2y Qxy = = E (x1 y) E (x2 y) : By the partitioned matrix inversion formula (A.4) Qxx1 = 1 Q11 Q12 Q21 Q22 def where Q11 2 = Q11 def = Q11 Q12 Q21 Q22 Q1112 Q2211 Q21 Q111 = def Q1112 Q12 Q221 Q2211 : (2.41) Q21 Q111 Q12 . Thus Q12 Q221 Q21 and Q22 1 = Q22 1 = 2 Q1112 Q12 Q221 Q2211 Q1112 Q2211 Q21 Q111 = = Q1112 Q1y Q2211 Q2y = Q1112 Q1y 2 Q2211 Q2y 1 Q1y Q2y Q12 Q221 Q2y Q21 Q111 Q1y We have shown that 2.21 1 = Q1112 Q1y 2 2 = Q2211 Q2y 1 Coe¢ cient Decomposition In the previous section we derived the formula for the coe¢ cient sub-vectors 1 and 2 : We now use these formula to give a useful interpretation of the coe¢ cients as obtaining from an iterated projection. Take equation (2.40) for the case dim(x1 ) = 1 so that 1 2 R: y = x1 1 + x02 2 + e: (2.42) Now consider the projection of x1 on x2 : x1 = x02 2 + u1 E (x2 u1 ) = 0: From (2.24) and (2.34), that E (u1 y) = E x1 2 = Q221 Q21 and Eu21 = Q11 2 = Q11 0 2 x2 y = E (x1 y) 0 2 E (x2 y) Q12 Q221 Q21 : We can also calculate = Q1y Q12 Q221 Q2y = Q1y 2 : CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION We have found that 1 = Q1112 Q1y 2 = 35 E (u1 y) Eu21 the coe¢ cient from the simple regression of y on u1 : What this means is that in the multivariate projection equation (2.42), the coe¢ cient 1 equals the projection coe¢ cient from a regression of y on u1 ; the error from a projection of x1 on the other regressors x2 : The error u1 can be thought of as the component of x1 which is not linearly explained by the other regressors. Thus the coe¢ cient 1 equals the linear e¤ect of x1 on y; after stripping out the e¤ects of the other variables. There was nothing special in the choice of the variable x1 : So this derivation applies symmetrically to all coe¢ cients in a linear projection. Each coe¢ cient equals the simple regression of y on the error from a projection of that regressor on all the other regressors. Each coe¢ cient equals the linear e¤ect of that variable on y; after linearly controlling for all the other regressors. 2.22 Omitted Variable Bias Again, let the regressors be partitioned as in (2.39). Consider the projection of y on x1 only. Perhaps this is done because the variables x2 are not observed. This is the equation y = x01 1 +u (2.43) E (x1 u) = 0 Notice that we have written the coe¢ cient on x1 as 1 rather than 1 and the error as u rather than e: This is because (2.43) is di¤erent than (2.40). Goldberger (1991) introduced the catchy labels long regression for (2.40) and short regression for (2.43) to emphasize the distinction. Typically, 1 6= 1 , except in special cases. To see this, we calculate 1 = = E x1 x01 1 E x1 x01 1 E (x1 y) E x1 x01 = 1 + E x1 x01 = 1 + 1 1 + x02 E x1 x02 2 +e 2 2 where = E x1 x01 1 E x1 x02 is the coe¢ cient matrix from a projection of x2 on x1 : Observe that 1 = 1 + = 0 or 2 = 0: Thus the short and long regressions 2 6= 1 unless have di¤erent coe¢ cients on x1 : They are the same only under one of two conditions. First, if the projection of x2 on x1 yields a set of zero coe¢ cients (they are uncorrelated), or second, if the coe¢ cient on x2 in (2.40) is zero. In general, the coe¢ cient in (2.43) is 1 rather than 1 : The di¤erence 2 between 1 and 1 is known as omitted variable bias. It is the consequence of omission of a relevant correlated variable. To avoid omitted variables bias the standard advice is to include all potentially relevant variables in estimated models. By construction, the general model will be free of such bias. Unfortunately in many cases it is not feasible to completely follow this advice as many desired variables are not observed. In this case, the possibility of omitted variables bias should be acknowledged and discussed in the course of an empirical investigation. For example, suppose y is log wages, x1 is education, and x2 is intellectual ability. It seems reasonable to suppose that education and intellectual ability are positively correlated (highly able individuals attain higher levels of education) which means > 0. It also seems reasonable to suppose that conditional on education, individuals with higher intelligence will earn higher wages CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 36 so that 2 > 0: This implies that 2 > 0 and 1 = 1 + 2 > 1 : Therefore, it seems reasonable to expect that in a regression of wages on education with ability omitted, the coe¢ cient on education is higher than in a regression where ability is included. In other words, in this context the omitted variable biases the regression coe¢ cient upwards. 2.23 Best Linear Approximation There are alternative ways we could construct a linear approximation x0 to the conditional mean m(x): In this section we show that one alternative approach turns out to yield the same answer as the best linear predictor. We start by de…ning the mean-square approximation error of x0 to m(x) as the expected squared di¤erence between x0 and the conditional mean m(x) x0 d( ) = E m(x) 2 : (2.44) The function d( ) is a measure of the deviation of x0 from m(x): If the two functions are identical then d( ) = 0; otherwise d( ) > 0: We can also view the mean-square di¤erence d( ) as a densityweighted average of the function (m(x) x0 )2 ; since Z 2 d( ) = m(x) x0 fx (x)dx Rk where fx (x) is the marginal density of x: We can then de…ne the best linear approximation to the conditional m(x) as the function x0 obtained by selecting to minimize d( ) : = argmin d( ): (2.45) 2Rk Similar to the best linear predictor we are measuring accuracy by expected squared error. The di¤erence is that the best linear predictor (2.21) selects to minimize the expected squared prediction error, while the best linear approximation (2.45) selects to minimize the expected squared approximation error. Despite the di¤erent de…nitions, it turns out that the best linear predictor and the best linear approximation are identical. By the same steps as in (2.17) plus an application of conditional expectations we can …nd that = = E xx0 1 E xx0 1 E (xm(x)) (2.46) E (xy) (2.47) (see Exercise 2.19). Thus (2.45) equals (2.21). We conclude that the de…nition (2.45) can be viewed as an alternative motivation for the linear projection coe¢ cient. 2.24 Normal Regression Suppose the variables (y; x) are jointly normally distributed. Consider the best linear predictor of y given x y = x0 + e = E xx0 1 E (xy) : Since the error e is a linear transformation of the normal vector (y; x); it follows that (e; x) is jointly normal, and since they are jointly normal and uncorrelated (since E (xe) = 0) they are also independent (see Appendix B.9). Independence implies that E (e j x) = E (e) = 0 CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 37 and E e2 j x = E e2 = 2 which are properties of a homoskedastic linear CEF. We have shown that when (y; x) are jointly normally distributed, they satisfy a normal linear CEF y = x0 + e where e N(0; 2 ) is independent of x. This is an alternative (and traditional) motivation for the linear CEF model. This motivation has limited merit in econometric applications since economic data is typically non-normal. 2.25 Regression to the Mean The term regression originated in an in‡uential paper by Francis Galton published in 1886, where he examined the joint distribution of the stature (height) of parents and children. E¤ectively, he was estimating the conditional mean of children’s height given their parent’s height. Galton discovered that this conditional mean was approximately linear with a slope of 2/3. This implies that on average a child’s height is more mediocre (average) than his or her parent’s height. Galton called this phenomenon regression to the mean, and the label regression has stuck to this day to describe most conditional relationships. One of Galton’s fundamental insights was to recognize that if the marginal distributions of y and x are the same (e.g. the heights of children and parents in a stable environment) then the regression slope in a linear projection is always less than one. To be more precise, take the simple linear projection y= +x +e (2.48) where y equals the height of the child and x equals the height of the parent. Assume that y and x have the same mean, so that y = x = : Then from (2.37) = (1 ) so we can write the linear projection (2.48) as P (y j x) = (1 ) +x : This shows that the projected height of the child is a weighted average of the population average height and the parent’s height x; with the weight equal to the regression slope : When the height distribution is stable across generations, so that var(y) = var(x); then this slope is the simple correlation of y and x: Using (2.38) = cov (x; y) = corr(x; y): var(x) By the properties of correlation (e.g. equation (B.7) in the Appendix), 1 corr(x; y) 1; with corr(x; y) = 1 only in the degenerate case y = x: Thus if we exclude degeneracy, is strictly less than 1. This means that on average a child’s height is more mediocre (closer to the population average) than the parent’s. CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 38 Sir Francis Galton Sir Francis Galton (1822-1911) of England was one of the leading …gures in late 19th century statistics. In addition to inventing the concept of regression, he is credited with introducing the concepts of correlation, the standard deviation, and the bivariate normal distribution. His work on heredity made a signi…cant intellectual advance by examing the joint distributions of observables, allowing the application of the tools of mathematical statistics to the social sciences. A common error –known as the regression fallacy –is to infer from < 1 that the population is converging, meaning that its variance is declining towards zero. This is a fallacy because we derived the implication < 1 under the assumption of constant means and variances. So certainly < 1 does not imply that the variance y is less than than the variance of x: Another way of seeing this is to examine the conditions for convergence in the context of equation (2.48). Since x and e are uncorrelated, it follows that var(y) = 2 var(x) + var(e): Then var(y) < var(x) if and only if 2 <1 var(e) var(x) which is not implied by the simple condition j j < 1: The regression fallacy arises in related empirical situations. Suppose you sort families into groups by the heights of the parents, and then plot the average heights of each subsequent generation over time. If the population is stable, the regression property implies that the plots lines will converge –children’s height will be more average than their parents. The regression fallacy is to incorrectly conclude that the population is converging. A message to be learned from this example is that such plots are misleading for inferences about convergence. The regression fallacy is subtle. It is easy for intelligent economists to succumb to its temptation. A famous example is The Triumph of Mediocrity in Business by Horace Secrist, published in 1933. In this book, Secrist carefully and with great detail documented that in a sample of department stores over 1920-1930, when he divided the stores into groups based on 1920-1921 pro…ts, and plotted the average pro…ts of these groups for the subsequent 10 years, he found clear and persuasive evidence for convergence “toward mediocrity”. Of course, there was no discovery – regression to the mean is a necessary feature of stable distributions. 2.26 Reverse Regression Galton noticed another interesting feature of the bivariate distribution. There is nothing special about a regression of y on x: We can also regress x on y: (In his heredity example this is the best linear predictor of the height of parents given the height of their children.) This regression takes the form x= +y +e : (2.49) This is sometimes called the reverse regression. In this equation, the coe¢ cients error e are de…ned by linear projection. In a stable population we …nd that = corr(x; y) = = (1 ) = ; and CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 39 which are exactly the same as in the projection of y on x! The intercept and slope have exactly the same values in the forward and reverse projections! While this algebraic discovery is quite simple, it is counter-intuitive. Instead, a common yet mistaken guess for the form of the reverse regression is to take the equation (2.48), divide through by and rewrite to …nd the equation x= +y 1 1 e (2.50) suggesting that the projection of x on y should have a slope coe¢ cient of 1= instead of ; and intercept of - = rather than : What went wrong? Equation (2.50) is perfectly valid, because it is a simple manipulation of the valid equation (2.48). The trouble is that (2.50) is not a CEF nor a linear projection. Inverting a projection (or CEF) does not yield a projection (or CEF). Instead, (2.49) is a valid projection, not (2.50). In any event, Galton’s …nding was that when the variables are standardized, the slope in both projections (y on x; and x and y) equals the correlation, and both equations exhibit regression to the mean. It is not a causal relation, but a natural feature of all joint distributions. 2.27 Limitations of the Best Linear Predictor Let’s compare the linear projection and linear CEF models. From Theorem 2.8.1.4 we know that the CEF error has the property E (xe) = 0: Thus a linear CEF is a linear projection. However, the converse is not true as the projection error does not necessarily satisfy E (e j x) = 0: Furthermore, the linear projection may be a poor approximation to the CEF. To see these points in a simple example, suppose that the true CEF is y = x+x2 and x N(0; 1): In this case the true CEF is m(x) = x + x2 and there is no error. Now consider the linear projection of y on x and an intercept, namely the model y = + x + u: Since x and x2 are uncorrelated the linear projection takes the form P (y j x) = 1 + x: This is quite di¤erent from the true CEF m(x) = x + x2 : The projection error equals e = x2 1; which is a deterministic function of x; yet is uncorrelated with x. We see in this example that a projection error need not be a CEF error, and a linear projection can be a poor approximation to the CEF. Figure 2.9: Conditional Mean and Two Linear Projections CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 40 Another defect of linear projection is that it is sensitive to the marginal distribution of the regressors when the conditional mean is non-linear. We illustrate the issue in Figure 2.9 for a constructed11 joint distribution of y and x. The solid line is the non-linear CEF of y given x: The data are divided in two –Group 1 and Group 2 –which have di¤erent marginal distributions for the regressor x; and Group 1 has a lower mean value of x than Group 2. The separate linear projections of y on x for these two groups are displayed in the Figure by the dashed lines. These two projections are distinct approximations to the CEF. A defect with linear projection is that it leads to the incorrect conclusion that the e¤ect of x on y is di¤erent for individuals in the two groups. This conclusion is incorrect because in fact there is no di¤erence in the conditional mean function. The apparant di¤erence is a by-product of a linear approximation to a non-linear mean, combined with di¤erent marginal distributions for the conditioning variables. 2.28 Random Coe¢ cient Model A model which is notationally similar to but conceptually distinct from the linear CEF model is the linear random coe¢ cient model. It takes the form y = x0 where the individual-speci…c coe¢ cient is random and independent of x. For example, if x is years of schooling and y is log wages, then is the individual-speci…c returns to schooling. If a person obtains an extra year of schooling, is the actual change in their wage. The random coe¢ cient model allows the returns to schooling to vary in the population. Some individuals might have a high return to education (a high ) and others a low return, possibly 0, or even negative. In the linear CEF model the regressor coe¢ cient equals the regression derivative –the change in the conditional mean due to a change in the regressors, = rm(x). This is not the e¤ect on a given individual, it is the e¤ect on the population average. In contrast, in the random coe¢ cient model, the random vector = rx0 is the true causal e¤ect –the change in the response variable y itself due to a change in the regressors. It is interesting, however, to discover that the linear random coe¢ cient model implies a linear CEF. To see this, let and denote the mean and covariance matrix of : = E( ) = var ( ) and then decompose the random coe¢ cient as = +u where u is distributed independently of x with mean zero and covariance matrix write E(y j x) = x0 E( j x) = x0 E( ) = x0 : Then we can so the CEF is linear in x; and the coe¢ cients equal the mean of the random coe¢ cient . We can thus write the equation as a linear CEF y = x0 + e where e = x0 u and u = (2.51) . The error is conditionally mean zero: E(e j x) = 0: 11 The x in Group 1 are N(2; 1) and those in Group 2 are N(4; 1); and the conditional distriubtion of y given x is N(m(x); 1) where m(x) = 2x x2 =6: CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 41 Furthermore var (e j x) = x0 var ( )x = x0 x so the error is conditionally heteroskedastic with its variance a quadratic function of x. Theorem 2.28.1 In the linear random coe¢ cient model y = x0 independent of x, E kxk2 < 1; and E k k2 < 1; then with E (y j x) = x0 var (y j x) = x0 x where 2.29 = E( ) and = var ( ): Causal E¤ects So far we have avoided the concept of causality, yet often the underlying goal of an econometric analysis is to uncover a causal relationship between variables. It is often of great interest to understand the causes and e¤ects of decisions, actions, and policies. For example, we may be interested in the e¤ect of class sizes on test scores, police expenditures on crime rates, climate change on economic activity, years of schooling on wages, institutional structure on growth, the e¤ectiveness of rewards on behavior, the consequences of medical procedures for health outcomes, or any variety of possible causal relationships. In each case, the goal is to understand what is the actual e¤ect on the outcome y due to a change in the input x: We are not just interested in the conditional mean or linear projection, we would like to know the actual change. The causal e¤ect is typically speci…c to an individual, and also cannot be directly observed. For example, the causal e¤ect of schooling on wages is the actual di¤erence a person would receive in wages if we could change their level of education. The causal e¤ect of a medical treatment is the actual di¤erence in an individual’s health outcome, comparing treatment versus non-treatment. In both cases the e¤ects are individual and unobservable. For example, suppose that Jennifer would have earned $10 an hour as a high-school graduate and $20 a hour as a college graduate while George would have earned $8 as a high-school graduate and $12 as a college graduate. In this example the causal e¤ect of schooling is $10 a hour for Jennifer and $4 an hour for George. Furthermore, the causal e¤ect is unobserved as we only observe the wage corresponding to the actual outcome. A variable x1 can be said to have a causal e¤ect on the response variable y if the latter changes when all other inputs are held constant. To make this precise we need a mathematical formulation. We can write a full model for the response variable y as y = h (x1 ; x2 ; u) (2.52) where x1 and x2 are the observed variables, u is an ` 1 unobserved random factor, and h is a functional relationship. This framework includes as a special case the random coe¢ cient model (2.28) studied earlier. We de…ne the causal e¤ect of x1 within this model as the change in y due to a change in x1 holding the other variables x2 and u constant. CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 42 De…nition 2.29.1 In the model (2.52) the causal e¤ ect of x1 on y is C(x1 ; x2 ; u) = r1 h (x1 ; x2 ; u) ; (2.53) the change in y due to a change in x1 ; holding x2 and u constant. To understand this concept, imagine taking a single individual. As far as our structural model is concerned, this person is described by their observables x1 and x2 ; and their unobservables u. In a wage regression the unobservables would include characteristics such as the person’s abilities, skills, work ethic, interpersonal connections, and preferences. The causal e¤ect of x1 (say, education) is the change in the wage as x1 changes, holding constant all other observables and unobservables. It may be helpful to understand that (2.53) is a de…nition, and does not necessarily describe causality in a fundamental or experimental sense. Perhaps it would be more appropriate to label (2.53) as a structural e¤ect (the e¤ect within the structural model). Sometimes it is useful to write this relationship as a potential outcome function y(x1 ) = h (x1 ; x2 ; u) where the notation implies that y(x1 ) is holding x2 and u constant. A popular example arises in the analysis of treatment e¤ects with a binary regressor x1 . Let x1 = 1 indicate treatment (e.g. a medical procedure) and x1 = 0 indicating non-treatment. In this case y(x1 ) can be written y(0) = h (0; x2 ; u) y(1) = h (1; x2 ; u) In the literature on treatment e¤ects, it is common to refer to y(0) and y(1) as the latent outcomes associated with non-treatment and treatment, respectively. That is, for a given individual, y(0) is the health outcome if there is no treatment, and y(1) is the health outcome if there is treatment. The causal e¤ect of treatment for the individual is the change in their health outcome due to treatment –the change in y as we hold both x2 and u constant: C (x2 ; u) = y(1) y(0): This is random (a function of x2 and u) as both potential outcomes y(0) and y(1) are di¤erent across individuals. In a sample, we cannot observe both outcomes from the same individual, we only observe the realized value 8 < y(0) if x1 = 0 y= : y(1) if x1 = 1 As the causal e¤ect varies across individuals and is not observable, it cannot be measured on the individual level. We therefore focus on aggregate causal e¤ects, in particular what is known as the average causal e¤ect. De…nition 2.29.2 In the model (2.52) the average causal e¤ ect of x1 on y conditional on x2 is ACE(x1 ; x2 ) = E (C(x1 ; x2 ; u) j x1 ; x2 ) Z = r1 h (x1 ; x2 ; u) f (u j x1 ; x2 )du R` where f (u j x1 ; x2 ) is the conditional density of u given x1 ; x2 . (2.54) CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 43 We can think of the average causal e¤ect ACE(x1 ; x2 ) as the average e¤ect in the general population. In our Jennifer & George schooling example given earlier, supposing that half of the population are Jennifer’s and the other half George’s, then the average causal e¤ect of college is (10 + 4)=2 = $7 an hour. What is the relationship between the average causal e¤ect ACE(x1 ; x2 ) and the regression derivative r1 m (x1 ; x2 )? Equation (2.52) implies that the CEF is m(x1 ; x2 ) = E (h (x1 ; x2 ; u) j x1 ; x2 ) Z = h (x1 ; x2 ; u) f (u j x1 ; x2 )du; R` the average causal equation, averaged over the conditional distribution of the unobserved component u. Applying the marginal e¤ect operator, the regression derivative is Z r1 h (x1 ; x2 ; u) f (u j x1 ; x2 )du r1 m(x1 ; x2 ) = ` RZ h (x1 ; x2 ; u) r1 f (ujx1 ; x2 )du + R` Z = ACE(x1 ; x2 ) + h (x1 ; x2 ; u) r1 f (u j x1 ; x2 )du: (2.55) R` In general, the average causal e¤ect is not the regression derivative. However, they equal when the second component in (2.55) is zero. This occurs when r1 f (u j x1 ; x2 ) = 0; that is, when the conditional density of u given (x1 ; x2 ) does not depend on x1 : The condition is su¢ ciently important that it has a special name in the treatment e¤ects literature. De…nition 2.29.3 Conditional Independence Assumption (CIA). Conditional on x2 ; the random variables x1 and u are statistically independent. The CIA implies f (u j x1 ; x2 ) = f (u j x2 ) does not depend on x1 ; and thus r1 f (u j x1 ; x2 ) = 0: Thus the CIA implies that r1 m(x1 ; x2 ) = ACE(x1 ; x2 ); the regression derivative equals the average causal e¤ect. Theorem 2.29.1 In the structural model (2.52), the Conditional Independence Assumption implies r1 m(x1 ; x2 ) = ACE(x1 ; x2 ) the regression derivative equals the average causal e¤ ect for x1 on y conditional on x2 . This is a fascinating result. It shows that whenever the unobservable is independent of the treatment variable (after conditioning on appropriate regressors) the regression derivative equals the average causal e¤ect. In this case, the CEF has causal economic meaning, giving strong justi…cation to estimation of the CEF. Our derivation also shows the critical role of the CIA. If CIA fails, then the equality of the regression derivative and ACE fails. CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 44 This theorem is quite general. It applies equally to the treatment-e¤ects model where x1 is binary or to more general settings where x1 is continuous. It is also helpful to understand that the CIA is weaker than full independence of u from the regressors (x1 ; x2 ): The CIA was introduced precisely as a minimal su¢ cient condition to obtain the desired result. Full independence implies the CIA and implies that each regression derivative equals that variable’s average causal e¤ect, but full independence is not necessary in order to causally interpret a subset of the regressors. 2.30 Expectation: Mathematical Details* We de…ne the mean or expectation Ey of a random variable y as follows. If y is discrete on the set f 1 ; 2 ; :::g then 1 X Ey = j Pr (y = j ) ; j=1 and if y is continuous with density f then Ey = Z 1 yf (y)dy: 1 We can unify these de…nitions by writing the expectation as the Lebesgue integral with respect to the distribution function F Z 1 Ey = ydF (y): (2.56) 1 In the event that the integral (2.56) is not …nite, separately evaluate the two integrals Z 1 I1 = ydF (y) 0 Z 0 I2 = ydF (y): (2.57) (2.58) 1 If I1 = 1 and I2 < 1 then it is typical to de…ne Ey = 1: If I1 < 1 and I2 = 1 then we de…ne Ey = 1: However, if both I1 = 1 and I2 = 1 then Ey is unde…ned. If Z 1 E jyj = jyj dF (y) = I1 + I2 < 1 1 then Ey exists and is …nite. In this case it is common to say that the mean Ey is “well-de…ned”. More generally, y has a …nite r’th moment if E jyjr < 1: (2.59) By Liapunov’s Inequality (B.20), (2.59) implies E jyjs < 1 for all s r: Thus, for example, if the fourth moment is …nite then the …rst, second and third moments are also …nite. It is common in econometric theory to assume that the variables, or certain transformations of the variables, have …nite moments of a certain order. How should we interpret this assumption? How restrictive is it? One way to visualize the importance is to consider the class of Pareto densities given by f (y) = ay a 1 ; y > 1: The parameter a of the Pareto distribution indexes the rate of decay of the tail of the density. Larger a means that the tail declines to zero more quickly. See the …gure below where we show the CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 45 Pareto density for a = 1 and a = 2: The parameter a also determines which moments are …nite. We can calculate that 8 R a 1 r a 1 > dy = if r < a < a 1 y a r r E jyj = > : 1 if r a This shows that if y is Pareto distributed with parameter a; then the r’th moment of y is …nite if and only if r < a: Higher a means higher …nite moments. Equivalently, the faster the tail of the density declines to zero, the more moments are …nite. f(y) 2.0 a=2 1.5 1.0 a=1 0.5 0.0 1 2 3 4 y Pareto Densities, a = 1 and a = 2 This connection between tail decay and …nite moments is not limited to the Pareto distribution. We can make a similar analysis using a tail bound. Suppose that y has density f (y) which satis…es the bound f (y) A jyj a 1 for some A < 1 and a > 0. Since f (y) is bounded below a scale of a Pareto density, its tail behavior is similarly bounded. This means that for r < a Z 1 Z 1 Z 1 2A r r E jyj = jyj f (y)dy f (y)dy + 2A y r a 1 dy 1 + < 1: a r 1 1 1 Thus if the tail of the density declines at the rate jyj a 1 or faster, then y has …nite moments up to (but not including) a: Broadly speaking, the restriction that y has a …nite r’th moment means that the tail of y’s density declines to zero faster than y r 1 : The faster decline of the tail means that the probability of observing an extreme value of y is a more rare event. We complete this section by adding an alternative representation of expectation in terms of the distribution function. Theorem 2.30.1 For any non-negative random variable y Z 1 Ey = Pr (y > u) du 0 Proof of Theorem 2.30.1: Let F (x) = Pr (y > x) = 1 F (x), where F (x) is the distribution function. By integration by parts Z 1 Z 1 Z 1 Z 1 1 Ey = ydF (y) = ydF (y) = [yF (y)]0 + F (y)dy = Pr (y > u) du 0 as stated. 0 0 0 CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 2.31 46 Existence and Uniqueness of the Conditional Expectation* In Sections 2.3 and 2.6 we de…ned the conditional mean when the conditioning variables x are discrete and when the variables (y; x) have a joint density. We have explored these cases because these are the situations where the conditional mean is easiest to describe and understand. However, the conditional mean exists quite generally without appealing to the properties of either discrete or continuous random variables. To justify this claim we now present a deep result from probability theory. What is says is that the conditional mean exists for all joint distributions (y; x) for which y has a …nite mean. Theorem 2.31.1 Existence of the Conditional Mean If E jyj < 1 then there exists a function m(x) such that for all measurable sets X E (1 (x 2 X ) y) = E (1 (x 2 X ) m(x)) : (2.60) The function m(x) is almost everywhere unique, in the sense that if h(x) satis…es (2.60), then there is a set S such that Pr(S) = 1 and m(x) = h(x) for x 2 S: The function m(x) is called the conditional mean and is written m(x) = E (y j x) : See, for example, Ash (1972), Theorem 6.3.3. The conditional mean m(x) de…ned by (2.60) specializes to (2.7) when (y; x) have a joint density. The usefulness of de…nition (2.60) is that Theorem 2.31.1 shows that the conditional mean m(x) exists for all …nite-mean distributions. This de…nition allows y to be discrete or continuous, for x to be scalar or vector-valued, and for the components of x to be discrete or continuously distributed. 2.32 Identi…cation* A critical and important issue in structural econometric modeling is identi…cation, meaning that a parameter is uniquely determined by the distribution of the observed variables. It is relatively straightforward in the context of the unconditional and conditional mean, but it is worthwhile to introduce and explore the concept at this point for clarity. Let F denote the distribution of the observed data, for example the distribution of the pair (y; x): Let F be a collection of distributions F: Let be a parameter of interest (for example, the mean Ey). De…nition 2.32.1 A parameter 2 R is identi…ed on F if for all F 2 F; there is a uniquely determined value of : Equivalently, is identi…ed if we can write it as a mapping = g(F ) on the set F: The restriction to the set F is important. Most parameters are identi…ed only on a strict subset of the space of all distributions. Take, for example, the mean Ey: It is uniquelyodetermined if E jyj < 1; so it is clear that n = R1 is identi…ed for the set F = F : 1 jyj dF (y) < 1 . However, is also well de…ned when it is either positive or negative in…nity. Hence, de…ning I1 and I2 as in (2.57) and (2.58), we can deduce that is identi…ed on the set F = fF : fI1 < 1g [ fI2 < 1gg : Next, consider the conditional mean. Theorem 2.31.1 demonstrates that E jyj < 1 is a su¢ cient condition for identi…cation. CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 47 Theorem 2.32.1 Identi…cation of the Conditional Mean If E jyj < 1; the conditional mean m(x) = E (y j x) is identi…ed almost everywhere. It might seem as if identi…cation is a general property for parameters, so long as we exclude degenerate cases. This is true for moments of observed data, but not necessarily for more complicated models. As a case in point, consider the context of censoring. Let y be a random variable with distribution F: Instead of observing y; we observe y de…ned by the censoring rule y = y if y if y > : That is, y is capped at the value : A common example is income surveys, where income responses are “top-coded”, meaning that incomes above the top code are recorded as equalling the top code. The observed variable y has distribution F (u) = F (u) 1 for u for u We are interested in features of the distribution F not the censored distribution F : For example, we are interested in the mean wage = E (y) : The di¢ culty is that we cannot calculate from F except in the trivial case where there is no censoring Pr (y ) = 0: Thus the mean is not generically identi…ed from the censored distribution. A typical solution to the identi…cation problem is to assume a parametric distribution. For example, let F be the set of normal distributions y N( ; 2 ): It is possible to show that the 2 parameters ( ; ) are identi…ed for all F 2 F: That is, if we know that the uncensored distribution is normal, we can uniquely determine the parameters from the censored distribution. This is often called parametric identi…cation as identi…cation is restricted to a parametric class of distributions. In modern econometrics this is generally viewed as a second-best solution, as identi…cation has been achieved only through the use of an arbitrary and unveri…able parametric assumption. A pessimistic conclusion might be that it is impossible to identi…ed parameters of interest from censored data without parametric assumptions. Interestingly, this pessimism is unwarranted. It turns out that we can identify the quantiles q of F for Pr (y ) : For example, if 20% of the distribution is censored, we can identify all quantiles for 2 (0; 0:8): This is often called nonparametric identi…cation as the parameters are identi…ed without restriction to a parametric class. What we have learned from this little exercise is that in the context of censored data, moments can only be parametrically identi…ed, while (non-censored) quantiles are nonparametrically identi…ed. Part of the message is that a study of identi…cation can help focus attention on what can be learned from the data distributions available. 2.33 Technical Proofs* Proof of Theorem 2.7.1: For convenience, assume that the variables have a joint density f (y; x). Since E (y j x) is a function of the random vector x only, to calculate its expectation we integrate with respect to the density fx (x) of x; that is Z E (E (y j x)) = E (y j x) fx (x) dx: Rk CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 48 Substituting in (2.7) and noting that fyjx (yjx) fx (x) = f (y; x) ; we …nd that the above expression equals Z Z Z Z Rk yfyjx (yjx) dy fx (x) dx = R Rk R yf (y; x) dydx = E (y) the unconditional mean of y: Proof of Theorem 2.7.2: Again assume that the variables have a joint density. It is useful to observe that f (y; x1 ; x2 ) f (x1 ; x2 ) f (yjx1 ; x2 ) f (x2 jx1 ) = = f (y; x2 jx1 ) ; (2.61) f (x1 ; x2 ) f (x1 ) the density of (y; x2 ) given x1 : Here, we have abused notation and used a single symbol f to denote the various unconditional and conditional densities to reduce notational clutter. Note that Z E (y j x1 ; x2 ) = yf (yjx1 ; x2 ) dy: (2.62) R Integrating (2.62) with respect to the conditional density of x2 given x1 , and applying (2.61) we …nd that Z E (y j x1 ; x2 ) f (x2 jx1 ) dx2 E (E (y j x1 ; x2 ) j x1 ) = Rk2 Z Z = yf (yjx1 ; x2 ) dy f (x2 jx1 ) dx2 k ZR 2 Z R yf (yjx1 ; x2 ) f (x2 jx1 ) dydx2 = k ZR 2 ZR = yf (y; x2 jx1 ) dydx2 Rk2 R = E (y j x1 ) as stated. Proof of Theorem 2.7.3: Z Z E (g (x) y j x) = g (x) yfyjx (yjx) dy = g (x) yfyjx (yjx) dy = g (x) E (y j x) R R This is (2.9). The assumption that E jg (x) yj < 1 is required for the …rst equality to be wellde…ned. Equation (2.10) follows by applying the Simple Law of Iterated Expectations to (2.9). Proof of Theorem 2.9.2: The assumption that Ey 2 < 1 implies that all the conditional expectations below exist. Set z = E(y j x1 ; x2 ). By the conditional Jensen’s inequality (B.13), (E(z j x1 ))2 E z 2 j x1 : Taking unconditional expectations, this implies E (E(y j x1 ))2 E (E(y j x1 ; x2 ))2 : Similarly, (Ey)2 E (E(y j x1 ))2 E (E(y j x1 ; x2 ))2 : (2.63) CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 49 The variables y; E(y j x1 ) and E(y j x1 ; x2 ) all have the same mean Ey; so the inequality (2.63) implies that the variances are ranked monotonically: 0 Next, for var (E(y j x1 )) var (E(y j x1 ; x2 )) : (2.64) = Ey observe that E (y E(y j x)) (E(y j x) ) = E (y E(y j x)) (E(y j x) )=0 so the decomposition y =y E(y j x) + E(y j x) satis…es var (y) = var (y E(y j x)) + var (E(y j x)) : (2.65) The monotonicity of the variances of the conditional mean (2.64) applied to the variance decomposition (2.65) implies the reverse monotonicity of the variances of the di¤erences, completing the proof. Proof of Theorem 2.8.1. Applying Minkowski’s Inequality (B.19) to e = y (E jejr )1=r = (E jy m(x)jr )1=r m(x); (E jyjr )1=r + (E jm(x)jr )1=r < 1; where the two parts on the right-hand are …nite since E jyjr < 1 by assumption and E jm(x)jr < 1 by the Conditional Expectation Inequality (B.14). The fact that (E jejr )1=r < 1 implies E jejr < 1: Proof of Theorem 2.17.1. For part 1, by the Expectation Inequality (B.15), (A.9) and Assumption 2.17.1, E xx0 E xx0 = E kxk2 < 1: Similarly, using the Expectation Inequality (B.15), the Cauchy-Schwarz Inequality (B.17) and Assumption 2.17.1, kE (xy)k 1=2 E kxyk = E kxk2 Ey 2 1=2 < 1: Thus the moments E (xy) and E (xx0 ) are …nite and well de…ned. For part 2, the coe¢ cient = (E (xx0 )) 1 E (xy) is well de…ned since (E (xx0 )) Assumption 2.17.1. Part 3 follows from De…nition 2.17.1 and part 2. For part 4, …rst note that Ee2 = E y x0 exists under 2 = Ey 2 2E yx0 = Ey 2 2E yx0 Ey 1 + 0 E xx0 E xx0 1 E (xy) 2 < 1 The …rst inequality holds because E (yx0 ) (E (xx0 )) 1 E (xy) is a quadratic form and therefore necessarily non-negative. Second, by the Expectation Inequality (B.15), the Cauchy-Schwarz Inequality (B.17) and Assumption 2.17.1, kE (xe)k E kxek = E kxk2 1=2 Ee2 1=2 < 1: It follows that the expectation E (xe) is …nite, and is zero by the calculation (2.28). CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION For part 6, Applying Minkowski’s Inequality (B.19) to e = y (E jejr )1=r = E y x0 x0 ; r 1=r (E jyjr )1=r + E x0 r 1=r (E jyjr )1=r + (E kxkr )1=r k k < 1; the …nal inequality by assumption: 50 CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 51 Exercises Exercise 2.1 Find E (E (E (y j x1 ; x2 ; x3 ) j x1 ; x2 ) j x1 ) : Exercise 2.2 If E (y j x) = a + bx; …nd E (yx) as a function of moments of x: Exercise 2.3 Prove Theorem 2.8.1.4 using the law of iterated expectations. Exercise 2.4 Suppose that the random variables y and x only take the values 0 and 1, and have the following joint probability distribution y=0 y=1 x=0 .1 .4 x=1 .2 .3 Find E (y j x) ; E y 2 j x and var (y j x) for x = 0 and x = 1: Exercise 2.5 Show that 2 (x) is the best predictor of e2 given x: (a) Write down the mean-squared error of a predictor h(x) for e2 : (b) What does it mean to be predicting e2 ? (c) Show that 2 (x) minimizes the mean-squared error and is thus the best predictor. Exercise 2.6 Use y = m(x) + e to show that var (y) = var (m(x)) + 2 Exercise 2.7 Show that the conditional variance can be written as 2 (x) = E y 2 j x (E (y j x))2 : Exercise 2.8 Suppose that y is discrete-valued, taking values only on the non-negative integers, and the conditional distribution of y given x is Poisson: Pr (y = j j x) = exp ( x0 ) (x0 )j ; j! j = 0; 1; 2; ::: Compute E (y j x) and var (y j x) : Does this justify a linear regression model of the form y = x0 + e? j Hint: If Pr (y = j) = exp( j! ) ; then Ey = and var(y) = : Exercise 2.9 Suppose you have two regressors: x1 is binary (takes values 0 and 1) and x2 is categorical with 3 categories (A; B; C): Write E (y j x1 ; x2 ) as a linear regression. Exercise 2.10 True or False. If y = x + e; x 2 R; and E (e j x) = 0; then E x2 e = 0: Exercise 2.11 True or False. If y = x + e; x 2 R; and E (xe) = 0; then E x2 e = 0: Exercise 2.12 True or False. If y = x0 + e and E (e j x) = 0; then e is independent of x: Exercise 2.13 True or False. If y = x0 + e and E(xe) = 0; then E (e j x) = 0: CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION Exercise 2.14 True or False. If y = x0 + e, E (e j x) = 0; and E e2 j x = e is independent of x: Exercise 2.15 Consider the intercept-only model y = Show that = E(y): 52 2; a constant, then + e de…ned as the best linear predictor. Exercise 2.16 Let x and y have the joint density f (x; y) = 23 x2 + y 2 on 0 x 1; 0 y 1: Compute the coe¢ cients of the best linear predictor y = + x + e: Compute the conditional mean m(x) = E (y j x) : Are the best linear predictor and conditional mean di¤erent? Exercise 2.17 Let x be a random variable with g xj ; 2 x = and x3 = 1 + 2 x2 )2 (x Show that Eg (x j m; s) = 0 if and only if m = Exercise 2.18 Suppose that 2 = Ex and and s = 2 = var(x): De…ne : 2: 0 1 1 x = @ x2 A x3 is a linear function of x2 : (a) Show that Qxx = E (xx0 ) is not invertible. (b) Use a linear transformation of x to …nd an expression for the best linear predictor of y given x. (Be explicit, do not just use the generalized inverse formula.) Exercise 2.19 Show (2.46)-(2.47), namely that for d( ) = E m(x) x0 2 then = argmin d( ) 2Rk = = E xx0 1 E xx0 1 E (xm(x)) E (xy) : Hint: To show E (xm(x)) = E (xy) use the law of iterated expectations. Chapter 3 The Algebra of Least Squares 3.1 Introduction In this chapter we introduce the popular least-squares estimator. Most of the discussion will be algebraic, with questions of distribution and inference defered to later chapters. 3.2 Random Samples In Section 2.17 we derived and discussed the best linear predictor of y given x for a pair of random variables (y; x) 2 R Rk ; and called this the linear projection model. We are now interested in estimating the parameters of this model, in particular the projection coe¢ cient = E xx0 1 E (xy) : We can estimate from observational data which includes joint measurements on the variables (y; x) : For example, supposing we are interested in estimating a wage equation, we would use a dataset with observations on wages (or weekly earnings), education, experience (or age), and demographic characteristics (gender, race, location). One possible dataset is the Current Population Survey (CPS), a survey of U.S. households which includes questions on employment, income, education, and demographic characteristics. Notationally we wish to emphasize when we are discussing observations. Typically in econometrics we denote observations by appending a subscript i which runs from 1 to n; thus the ith observation is (yi ; xi ); and n denotes the sample size. The dataset is then {(yi ; xi ); i = 1; :::; n}. From the viewpoint of empirical analysis, a dataset is a array of numbers often organized as a table, where the columns of the table correspond to distinct variables and the rows correspond to distinct observations. For empirical analysis, the dataset and observations are …xed in the sense that they are numbers presented to the researcher. For statistical analysis we need to view the dataset as random, or more precisely as a realization of a random process. For cross-sectional studies, the most common approach is to treat the individual observations as independent draws from an underlying population F: When the observations are realizations of independent and identically distributed random variables, we say that the data is a random sample. Assumption 3.2.1 The observations f(y1 ; x1 ); :::; (yi ; xi ); :::; (yn ; xn )g are a random sample. With a random sample, the ordering of the data is irrelevant. There is nothing special about any speci…c observation or ordering. You can permute the order of the observations and no information is gained or lost. 53 CHAPTER 3. THE ALGEBRA OF LEAST SQUARES 54 As most economic data sets are not literally the result of a random experiment, the random sampling framework is best viewed as an approximation rather than being literally true. The linear projection model applies to the random observations (yi ; xi ) : This means that the probability model for the observations is the same as that described in Section 2.17. We can write the model as yi = x0i + ei (3.1) where the linear proejction is de…ned as = argmin S( ); (3.2) 2Rk the minimizer of the expected squared error and has the explicit solution 1 = E xi x0i 3.3 2 x0i S( ) = E yi ; (3.3) E (xi yi ) : (3.4) Least Squares Estimator When a parameter is de…ned as the minimizer of a function as in (3.2), a standard approach to estimation is to construct an empirical analog of the function, and de…ne the estimator of the parameter as the minimizer of the empirical function. The empirical analog of the expected squared error (3.3) is the sample average squared error n Sn ( ) = 1X yi n x0i 2 (3.5) i=1 = 1 SSEn ( ) n where SSEn ( ) = n X yi x0i 2 i=1 is called the sum-of-squared-errors function. An estimator for is the minimizer of (3.5): b = argmin Sn ( ): 2Rk Alternatively, as Sn ( ) is a scale multiple of SSEn ( ); we may equivalently de…ne b as the minimizer of SSEn ( ): Hence b is commonly called the least-squares (LS) (or ordinary least squares (OLS)) estimator of . Here, as is common in econometrics, we put a hat “^”over the parameter to indicate that b is a sample estimate of : This is a helpful convention, as just by seeing the symbol b we can immediately interpret it as an estimator (because of the hat), and as an estimator of a parameter labelled . Sometimes when we want to be explicit about the estimation method, we will write b ols to signify that it is the OLS estimator. It is also common to see the notation b n ; where the subscript “n” indicates that the estimator depends on the sample size n: It is important to understand the distinction between population parameters such as and sample estimates such as b . The population parameter is a non-random feature of the population while the sample estimate b is a random feature of a random sample. is …xed, while b varies across samples. To visualize the quadratic function Sn ( ), Figure 3.1 displays an example sum-of-squared errors function SSEn ( ) for the case k = 2: The least-squares estimator b is the the pair ( b 1 ; b 2 ) minimizing this function. CHAPTER 3. THE ALGEBRA OF LEAST SQUARES 55 Figure 3.1: Sum-of-Squared Errors Function 3.4 Solving for Least Squares with One Regressor For simplicity, we start by considering the case k = 1 so that the coe¢ cient the sum of squared errors is a simple quadratic SSEn ( ) = = n X xi )2 (yi i=1 n X i=1 yi2 is a scalar. Then ! 2 n X x i yi i=1 ! + 2 n X x2i i=1 ! : The OLS estimator b minimizes this function. From elementary algebra we know that the minimizer of the quadratic function a + 2bx + cx2 is x = b=c: Thus the minimizer of SSEn ( ) is Pn b = Pi=1 xi yi : (3.6) n 2 i=1 xi The intercept-only model is the special case xi = 1: In this case we …nd Pn n yi 1X b = Pi=1 = yi = y; n n i=1 1 (3.7) i=1 the sample mean of yi : Here, as is common, we put a bar “ ” over y to indicate that the quantity is a sample mean. This calculation shows that the OLS estimator in the intercept-only model is the sample mean. 3.5 Solving for Least Squares with Multiple Regressors We now consider the case with k 1 so that the coe¢ cient To solve for b , expand the SSE function to …nd SSEn ( ) = n X i=1 yi2 2 0 n X i=1 xi yi + 0 is a vector. n X i=1 xi x0i : CHAPTER 3. THE ALGEBRA OF LEAST SQUARES 56 This is a quadratic expression in the vector argument . The …rst-order-condition for minimization of SSEn ( ) is n n X X @ b 0= SSEn ( ) = 2 xi yi + 2 xi x0i b : (3.8) @ i=1 i=1 We have written this using a single expression, but it is actually a system of k equations with k unknowns (the elements of b ). The solution for b may be found by solving the system of k equations in We can write P(3.8). n this solution compactly using matrix algebra. Inverting the k k matrix i=1 xi x0i we …nd an explicit formula for the least-squares estimator ! 1 n ! n X X b= xi x0 (3.9) xi yi : i i=1 i=1 This is the natural estimator of the best linear projection coe¢ cient de…ned in (3.2), and can also be called the linear projection estimator. We see that (3.9) simpli…es to the expression (3.6) when k = 1: The expression (3.9) is a notationally simple generalization but requires a careful attention to vector and matrix manipulations. Alternatively, equation (3.4) writes the projection coe¢ cient as an explicit function of the population moments Qxy and Qxx : Their moment estimators are the sample moments n 1X xi yi n b xy = Q The moment estimator of i=1 n 1X xi x0i : n b xx = Q i=1 replaces the population moments in (3.4) with the sample moments: b = Q b 1Q b xx xy n 1X xi x0i n i=1 ! n X 0 xi xi = = ! 1 1 i=1 n 1X x i yi n i=1 ! n X xi yi ! i=1 which is identical with (3.9). Least Squares Estimation De…nition 3.5.1 The least-squares estimator b is b = argmin Sn ( ) 2Rk where n 1X Sn ( ) = yi n x0i 2 i=1 and has the solution b= n X i=1 xi x0i ! 1 n X i=1 xi yi ! : CHAPTER 3. THE ALGEBRA OF LEAST SQUARES 57 Adrien-Marie Legendre The method of least-squares was …rst published in 1805 by the French mathematician Adrien-Marie Legendre (1752-1833). Legendre proposed leastsquares as a solution to the algebraic problem of solving a system of equations when the number of equations exceeded the number of unknowns. This was a vexing and common problem in astronomical measurement. As viewed by Legendre, (3.1) is a set of n equations with k unknowns. As the equations cannot be solved exactly, Legendre’s goal was to select to make the set of errors as small as possible. He proposed the sum of squared error criterion, and derived the algebraic solution presented above. As he noted, the …rstorder conditions (3.8) is a system of k equations with k unknowns, which can be solved by “ordinary” methods. Hence the method became known as Ordinary Least Squares and to this day we still use the abbreviation OLS to refer to Legendre’s estimation method. 3.6 Illustration We illustrate the least-squares estimator in practice with the data set used to generate the estimates from Chapter 2. This is the March 2009 Current Population Survey, which has extensive information on the U.S. population. This data set is described in more detail in Section ? For this illustration, we use the sub-sample of non-white married non-military female wages earners with 12 years potential work experience. This sub-sample has 61 observations. Let yi be log wages and xi be an intercept and years of education. Then n 1X x i yi = n i=1 and n 1X xi x0i = n 1 15:426 15:426 243 i=1 Thus b = = 3:025 47:447 1 15:426 15:426 243 0:626 0:156 1 : 3:025 47:447 : (3.10) We often write the estimated equation using the format \ log(W age) = 0:626 + 0:156 education: (3.11) An interpretation of the estimated equation is that each year of education is associated with an 16% increase in mean wages. Equation (3.11) is called a bivariate regression as there are only two variables. A multivariate regression has two or more regressors, and allows a more detailed investigation. Let’s redo the example, but now including all levels of experience. This expanded sample includes 2454 observations. Including as regressors years of experience and its square (experience 2 =100) (we divide by 100 to simplify reporting), we obtain the estimates \ log(W age) = 1:06 + 0:116 education + 0:010 experience 0:014 experience2 =100: (3.12) CHAPTER 3. THE ALGEBRA OF LEAST SQUARES 58 These estimates suggest a 12% increase in mean wages per year of education, holding experience constant. 3.7 Least Squares Residuals As a by-product of estimation, we de…ne the …tted value y^i = x0i b and the residual e^i = yi x0i b : y^i = yi (3.13) Sometimes y^i is called the predicted value, but this is a misleading label. The …tted value y^i is a function of the entire sample, including yi , and thus cannot be interpreted as a valid prediction of yi . It is thus more accurate to describe y^i as a …tted value rather than a predicted value. Note that yi = y^i + e^i and yi = x0i b + e^i : (3.14) We make a distinction between the error ei and the residual e^i : The error ei is unobservable while the residual e^i is a by-product of estimation. These two variables are frequently mislabeled, which can cause confusion. Equation (3.8) implies that n X xi e^i = 0: (3.15) i=1 To see this by a direct calculation, using (3.13) and (3.9), n X xi e^i = i=1 n X x i yi i=1 = n X xi yi i=1 = n X = i=1 = 0: n X i=1 xi yi i=1 n X x0i b n X xi x0i b xi x0i i=1 xi yi n X n X xi x0i i=1 ! 1 n X i=1 xi yi ! xi yi i=1 When xi contains a constant, an implication of (3.15) is n 1X e^i = 0: n (3.16) i=1 Thus the residuals have a sample mean of zero and the sample correlation between the regressors and the residual is zero. These are algebraic results, and hold true for all linear regression estimates. 3.8 Model in Matrix Notation For many purposes, including computation, it is convenient to write the model and statistics in matrix notation. The linear equation (2.26) is a system of n equations, one for each observation. CHAPTER 3. THE ALGEBRA OF LEAST SQUARES 59 We can stack these n equations together as y1 = x01 + e1 y2 = x02 + e2 .. . yn = x0n + en : Now de…ne 0 B B y=B @ y1 y2 .. . yn 1 C C C; A 0 B B X=B @ x01 x02 .. . x0n 1 0 C C C; A Observe that y and e are n 1 vectors, and X is an n can be compactly written in the single equation B B e=B @ e1 e2 .. . en 1 C C C: A k matrix. Then the system of n equations y = X + e: (3.17) Sample sums can be written in matrix notation. For example n X xi x0i = X 0 X i=1 n X xi yi = X 0 y: i=1 Therefore the least-squares estimator can be written as b = X 0X 1 X 0y : (3.18) The matrix version of (3.14) and estimated version of (3.17) is or equivalently the residual vector is y = Xb + e ^; e ^= y Using the residual vector, we can write (3.15) as X b: X 0e ^ = 0: (3.20) Using matrix notation we have simple expressions for most estimators. This is particularly convenient for computer programming, as most languages allow matrix notation and manipulation. Important Matrix Expressions y = X +e b = X 0X e ^ = y Xb X 0e ^ = 0: 1 X 0y CHAPTER 3. THE ALGEBRA OF LEAST SQUARES 60 Early Use of Matrices The earliest known treatment of the use of matrix methods to solve simultaneous systems is found in Chapter 8 of the Chinese text The Nine Chapters on the Mathematical Art, written by several generations of scholars from the 10th to 2nd century BCE. 3.9 Projection Matrix De…ne the matrix 1 P = X X 0X Observe that 1 P X = X X 0X X 0: X 0 X = X: This is a property of a projection matrix. More generally, for any matrix Z which can be written as Z = X for some matrix (we say that Z lies in the range space of X); then PZ = PX 1 = X X 0X X 0X =X = Z: As an important example, if we partition the matrix X into two matrices X 1 and X 2 so that X = [X 1 X 2] ; then P X 1 = X 1 . (See Exercise 3.7.) The matrix P is symmetric and idempotent1 . To see that it is symmetric, P0 = = 1 X X 0X X0 = X 0 1 0 X 0X X 0X 1 0 = X (X)0 X 0 0 X0 (X)0 X0 1 0 X0 = P: To establish that it is idempotent, the fact that P X = X implies that PP 1 = P X X 0X = X X 0X 1 X0 X0 = P: The matrix P has the property that it creates the …tted values in a least-squares regression: P y = X X 0X 1 X 0y = X b = y ^: Because of this property, P is also known as the “hat matrix”. 1 A matrix P is symmetric if P 0 = P : A matrix P is idempotent if P P = P : See Appendix A.8. CHAPTER 3. THE ALGEBRA OF LEAST SQUARES 61 A special example of a projection matrix occurs when X = 1 is an n-vector of ones. Then P 1 = 1 10 1 1 0 = 11 : n 1 10 P 1 y = 1 10 1 1 10 y Note that = 1y creates an n-vector whose elements are the sample mean y of yi : 1 The i’th diagonal element of P = X (X 0 X) X 0 is 1 hii = x0i X 0 X (3.21) xi which is called the leverage of the i’th observation. Some useful properties of the the matrix P and the leverage values hii are now summarized. Theorem 3.9.1 n X hii = tr P = k (3.22) 0 (3.23) i=1 and hii 1 To show (3.22), tr P = tr X X 0 X = tr X 0X 1 1 X0 X 0X = tr (I k ) = k: See Appendix A.4 for de…nition and properties of the trace operator. The proof of (3.23) is defered to Section 3.18. 3.10 Orthogonal Projection De…ne M where I n is the n = In P = In X X 0X 1 X0 n identity matrix. Note that M X = (I n P)X = X PX = X X = 0: Thus M and X are orthogonal. We call M an orthogonal projection matrix or an annihilator matrix due to the property that for any matrix Z in the range space of X then MZ = Z P Z = 0: CHAPTER 3. THE ALGEBRA OF LEAST SQUARES 62 For example, M X 1 = 0 for any subcomponent X 1 of X, and M P = 0 (see Exercise 3.7). The orthogonal projection matrix M has many similar properties with P , including that M is symmetric (M 0 = M ) and idempotent (M M = M ). Similarly to (3.22) we can calculate tr M = n k: (3.24) (See Exercise 3.9.) While P creates …tted values, M creates least-squares residuals: My = y Xb = b e: Py = y (3.25) As discussed in the previous section, a special example of a projection matrix occurs when X = 1 is an n-vector of ones, so that P 1 = 1 (10 1) 1 10 : Similarly, set M 1 = In P1 1 1 10 1 = In 10 : While P 1 creates a vector of sample means, M 1 creates demeaned values: M 1y = y 1y: For simplicity we will often write the right-hand-side as y y: The i’th element is yi y; the demeaned value of yi : We can also use (3.25) to write an alternative expression for the residual vector. Substituting y = X + e into e ^ = M y and using M X = 0 we …nd e ^ = M y = M (X + e) = M e which is free of dependence on the regression coe¢ cient 3.11 (3.26) . Estimation of Error Variance The error variance 2 = Ee2i is a moment, so a natural estimator is a moment estimator. If ei were observed we would estimate 2 by n 1X 2 ei : ~ = n 2 (3.27) i=1 However, this is infeasible as ei is not observed. In this case it is common to take a two-step approach to estimation. The residuals e^i are calculated in the …rst step, and then we substitute e^i for ei in expression (3.27) to obtain the feasible estimator n ^2 = 1X 2 e^i : n (3.28) i=1 In matrix notation, we can write (3.27) and (3.28) as ~2 = n 1 0 ^2 = n 1 0 ee and b eb e: Recall the expressions b e = M y = M e from (3.25) and (3.26). Applied to (3.29) we …nd ^2 = n 1 0 = n 1 0 = n 1 0 = n 1 0 e ^e ^ y MMy y My e Me (3.29) CHAPTER 3. THE ALGEBRA OF LEAST SQUARES 63 the third equality since M M = M . An interesting implication is that ~2 ^2 = n 1 0 = n 1 0 1 0 n ee e Me e Pe 0: The …nal inequality holds because P is positive semi-de…nite and e0 P e is a quadratic form. This shows that the feasible estimator ^ 2 is numerically smaller than the idealized estimator (3.27). 3.12 Analysis of Variance Another way of writing (3.25) is y = P y + My = y ^+e ^: (3.30) This decomposition is orthogonal, that is y ^0 e ^ = (P y)0 (M y) = y 0 P M y = 0: It follows that y0y = y ^0 y ^ + 2^ y0e ^+ e ^0 e ^= y ^0 y ^+e ^0 e ^ or n X yi2 = i=1 n X y^i2 i=1 + n X e^2i : i=1 Now subtracting y from both sizes of (3.30) we obtain 1y = y ^ y 1y + e ^ This decomposition is also orthogonal when X contains a constant, as (^ y 1y)0 e ^= y ^0 e ^ y10 e ^= 0 under (3.16). It follows that (y or 1y)0 (y n X i=1 (yi 1y) = (^ y 2 y) = n X i=1 (^ yi 1y)0 (^ y 2 y) + 1y) + e ^0 e ^ n X e^2i : i=1 This is commonly called the analysis-of-variance formula for least squares regression. A commonly reported statistic is the coe¢ cient of determination or R-squared: Pn Pn 2 yi y)2 e^ 2 i=1 (^ R = Pn Pn i=1 i 2 : 2 =1 y) y) i=1 (yi i=1 (yi It is often described as the fraction of the sample variance of yi which is explained by the leastsquares …t. R2 is a crude measure of regression …t. We have better measures of …t, but these require a statistical (not just algebraic) analysis and we will return to these issues later. One di¢ culty with R2 is that it increases when regressors are added to a regression (see Exercise 3.16). CHAPTER 3. THE ALGEBRA OF LEAST SQUARES 3.13 64 Regression Components Partition X = [X 1 X 2] and 1 = : 2 Then the regression model can be rewritten as y = X1 The OLS estimator of written as =( 0 1; 0 0 2) 1 + X2 2 + e: (3.31) is obtained by regression of y on X = [X 1 X 2 ] and can be y = Xb + e ^ = X1 b1 + X2 b2 + e ^: (3.32) b b We are interested in algebraic expressions for 1 and 2 : The algebra for the estimator is identical as that for the population coe¢ cients as presented in Section 2.20. b xx and Q b xy as Partition Q 2 1 3 1 0 2 3 0 X X X X b b 1 2 Q11 Q12 n 1 6 n 1 7 b xx = 4 7 5=6 Q 4 5 1 1 b b 0 0 Q21 Q22 X X1 X X2 n 2 n 2 and similarly Qxy 2 1 3 3 2 X 01 y b 1y Q 6 n 7 b xy = 4 7: 5=6 Q 4 5 1 0 b 2y Q X 2y n By the partitioned matrix inversion formula (A.4) 3 2 11 3 2 2 3 1 b 11 Q b 12 b b 12 b 1 b 1 Q b 12 Q b 1 Q Q Q Q Q 11 2 11 2 22 6 7 6 7 b 1=4 5 def Q = 4 (3.33) 5 5=4 xx 21 22 1 1 1 b b b b b b 21 Q b b Q21 Q22 Q Q Q Q Q 22 1 b 11 b 11 2 = Q where Q Thus b 12 Q b 22 1 = Q b 22 b 1Q b 21 and Q Q 22 b = = " = b 1 b 2 22 1 b 21 Q b 1Q b 12 : Q 11 ! b 1 Q 11 2 1 b b 21 Q b 1 Q22 1 Q 11 ! 1 b b Q 11 2 Q1y 2 1 b b 2y 1 Q Q 11 b 1 Q b b 1 Q 11 2 12 Q22 b 1 Q 22 1 #" b 1y Q b Q2y # 22 1 Now b 11 2 = Q b 11 Q = = b 12 Q b 1Q b 21 Q 22 1 0 1 0 X 1X 1 X X2 n n 1 1 0 X M 2X 1 n 1 1 0 X X2 n 2 1 1 0 X X1 n 2 CHAPTER 3. THE ALGEBRA OF LEAST SQUARES where X 2 X 02 X 2 M 2 = In 65 1 X 02 b 22 1 = 1 X 0 M 1 X 2 where is the orthogonal projection matrix for X 2 : Similarly Q n 2 X 1 X 01 X 1 M 1 = In 1 X 01 is the orthogonal projection matrix for X 1 : Also b 1y 2 = Q b 1y Q = b = X 0 M 2X 1 1 1 and 1 0 X X2 n 2 1 0 1 0 X 1y X X2 n n 1 1 0 X M 2y n 1 = b 2y 1 = 1 X 0 M 1 y: and Q n 2 Therefore b 12 Q b 1Q b 2y Q 22 b = X 0 M 1X 2 2 2 1 1 1 1 0 X y n 2 X 01 M 2 y (3.34) X 02 M 1 y : (3.35) These are algebraic expressions for the sub-coe¢ cient estimates from (3.32). 3.14 Residual Regression As …rst recognized by Frisch and Waugh (1933), expressions (3.34) and (3.35) can be used to show that the least-squares estimators b 1 and b 2 can be found by a two-step regression procedure. Take (3.35). Since M 1 is idempotent, M 1 = M 1 M 1 and thus b 2 X 02 M 1 X 2 = X 02 M 1 M 1 X 2 = where and 1 = f0 X f X 2 2 1 X 02 M 1 y 1 X 02 M 1 M 1 y f0 e X 2 ~1 f2 = M 1 X 2 X e ~1 = M 1 y: Thus the coe¢ cient estimate b 2 is algebraically equal to the least-squares regression of e ~1 on f X 2 : Notice that these two are y and X 2 , respectively, premultiplied by M 1 . But we know that multiplication by M 1 is equivalent to creating least-squares residuals. Therefore e ~1 is simply the f2 are the least-squares least-squares residual from a regression of y on X 1 ; and the columns of X residuals from the regressions of the columns of X 2 on X 1 : We have proven the following theorem. CHAPTER 3. THE ALGEBRA OF LEAST SQUARES 66 Theorem 3.14.1 Frisch-Waugh-Lovell In the model (3.31), the OLS estimator of 2 and the OLS residuals e ^ may be equivalently computed by either the OLS regression (3.32) or via the following algorithm: 1. Regress y on X 1 ; obtain residuals e ~1 ; f2 ; 2. Regress X 2 on X 1 ; obtain residuals X f2 ; obtain OLS estimates b 2 and residuals e 3. Regress e ~1 on X ^: In some contexts, the FWL theorem can be used to speed computation, but in most cases there is little computational advantage to using the two-step algorithm. This result is a direct analogy of the coe¢ cient representation obtained in Section 2.21. The result obtained in that section concerned the population projection coe¢ cients, the result obtained here concern the least-squares estimates. The key message is the same. In the least-squares regression (3.32), the estimated coe¢ cient b 2 numerically equals the regression of y on the regressors X 2 ; only after the regressors X 1 have been linearly projected out. Similarly, the coe¢ cient estimate b numerically equals the regression of y on the regressors X 1 ; after the regressors X 2 have been 1 linearly projected out. This result can be very insightful when intrepreting regression coe¢ cients. A common application of the FWL theorem, which you may have seen in an introductory econometrics course, is the demeaning formula for regression. Partition X = [X 1 X 2 ] where X 1 = 1 is a vector of ones and X 2 is a matrix of observed regressors. In this case, M 1 = In Observe that f2 = M 1 X 2 = X 2 X and 1 1 10 1 y ~ = M 1y = y 10 : X2 y; are the “demeaned”variables. The FWL theorem says that b 2 is the OLS estimate from a regression of yi y on x2i x2 : b = 2 n X i=1 (x2i x2 ) (x2i x2 ) 0 ! 1 n X i=1 (x2i x2 ) (yi ! y) : Thus the OLS estimator for the slope coe¢ cients is a regression with demeaned data. Ragnar Frisch Ragnar Frisch (1895-1973) was co-winner with Jan Tinbergen of the …rst Nobel Memorial Prize in Economic Sciences in 1969 for their work in developing and applying dynamic models for the analysis of economic problems. Frisch made a number of foundational contributions to modern economics beyond the Frisch-Waugh-Lovell Theorem, including formalizing consumer theory, production theory, and business cycle theory. CHAPTER 3. THE ALGEBRA OF LEAST SQUARES 3.15 67 Prediction Errors The least-squares residual e^i are not true prediction errors, as they are constructed based on the full sample including yi . A proper prediction for yi should be based on estimates constructed using only the other observations. We can do this by de…ning the leave-one-out OLS estimator of as that obtained from the sample of n 1 observations excluding the i’th observation: 0 1 10 1 X X 1 b @ 1 xj x0j A @ xj yj A ( i) = n 1 n 1 j6=i X 0( = Here, X ( i) and y ( value for yi is i) j6=i 1 i) X ( i) i) y ( i) : X( (3.36) are the data matrices omitting the i’th row. The leave-one-out predicted y~i = x0i b ( i) ; and the leave-one-out residual or prediction error is e~i = yi A convenient alternative expression for b ( b ( i) =b i) (1 y~i : (derived in Section 3.18) is hii ) 1 X 0X 1 xi e^i (3.37) where hii are the leverage values as de…ned in (3.21). Using (3.37) we can simplify the expression for the prediction error: x0i b ( e~i = yi i) x0i ^ + (1 = yi = e^i + (1 hii ) = (1 1 hii ) hii ) 1 1 x0i X 0 X 1 xi e^i hii e^i e^i : (3.38) To write this in vector notation, de…ne M = (I n diagfh11 ; ::; hnn g) = diagf(1 h11 ) 1 ; ::; (1 1 hnn ) 1 g: (3.39) Then (3.38) is equivalent to e e=M b e: (3.40) A convenient feature of this expression is that it shows that computation of the full vector of prediction errors e e is based on a simple linear operation, and does not really require n separate estimations. One use of the prediction errors is to estimate the out-of-sample mean squared error n ~2 = = 1X 2 e~i n 1 n i=1 n X (1 hii ) 2 2 e^i : (3.41) i=1 This is also known as the sample mean squared prediction error. Its square root ~ = the prediction standard error. p ~ 2 is CHAPTER 3. THE ALGEBRA OF LEAST SQUARES 3.16 68 In‡uential Observations Another use of the leave-one-out estimator is to investigate the impact of in‡uential observations, sometimes called outliers. We say that observation i is in‡uential if its omission from the sample induces a substantial change in a parameter of interest. From (3.37)-(3.38) we know that b b ( i) = (1 1 X 0X = 1 hii ) X 0X xi e~i : 1 xi e^i (3.42) By direct calculation of this quantity for each observation i; we can directly discover if a speci…c observation i is in‡uential for a coe¢ cient estimate of interest. For a general assessment, we can focus on the predicted values. The di¤erence between the full-sample and leave-one-out predicted values is y^i y~i = x0i b = x0i 0 x0i b ( XX = hii e~i i) 1 xi e~i which is a simple function of the leverage values hii and prediction errors e~i : Observation i is in‡uential for the predicted value if jhii e~i j is large, which requires that both hii and j~ ei j are large. One way to think about this is that a large leverage value hii gives the potential for observation i to be in‡uential. A large hii means that observation i is unusual in the sense that the regressor xi is far from its sample mean. We call an observation with large hii a leverage point. A leverage point is not necessarily in‡uential as the latter also requires that the prediction error e~i is large. To determine if any individual observations are in‡uential in this sense, several diagnostics have been proposed (some names include DFITS, Cook’s Distance, and Welsch Distance). Unfortunately, from a statistical perspective it is di¢ cult to recommend these diagnostics for applications as they are not based on statistical theory. Probably the most relevant measure is the change in the coe¢ cient estimates given in (3.42). The ratio of these changes to the coe¢ cient’s standard error is called its DFBETA, and is a postestimation diagnostic available in STATA. While there is no magic threshold, the concern is whether or not an individual observation meaningfully changes an estimated coe¢ cient of interest. For illustration, consider Figure 3.2 which shows a scatter plot of random variables (yi ; xi ). The 25 observations shown with the open circles are generated by xi U [1; 10] and yi N (xi ; 4): The 26’th observation shown with the …lled circle is x26 = 9; y26 = 0: (Imagine that y26 = 0 was incorrectly recorded due to a mistaken key entry.) The Figure shows both the least-squares …tted line from the full sample and that obtained after deletion of the 26’th observation from the sample. In this example we can see how the 26’th observation (the “outlier”) greatly tilts the least-squares …tted line towards the 26’th observation. In fact, the slope coe¢ cient decreases from 0.97 (which is close to the true value of 1.00) to 0.56, which is substantially reduced. Neither y26 nor x26 are unusual values relative to their marginal distributions, so this outlier would not have been detected from examination of the marginal distributions of the data. The change in the slope coe¢ cient of 0:41 is meaningful and should raise concern to an applied economist. If an observation is determined to be in‡uential, what should be done? As a common cause of in‡uential observations is data entry error, the in‡uential observations should be examined for evidence that the observation was mis-recorded. Perhaps the observation falls outside of permitted ranges, or some observables are inconsistent (for example, a person is listed as having a job but receives earnings of $0). If it is determined that an observation is incorrectly recorded, then the observation is typically deleted from the sample. This process is often called “cleaning the data”. The decisions made in this process involve an fair amount of individual judgement. When this is done it is proper empirical practice to document such choices. (It is useful to keep the source data CHAPTER 3. THE ALGEBRA OF LEAST SQUARES 69 10 ● ● leave−one−out OLS 8 ● ● ● ● 6 ● ● ● OLS ● ● ● ● ● 4 y ● ● ● ● ● 2 ● ● ● 0 ● ● ● ● ● 2 4 6 8 10 x Figure 3.2: Impact of an in‡uential observation on the least-squares estimator in its original form, a revised data …le after cleaning, and a record describing the revision process. This is especially useful when revising empirical work at a later date.) It is also possible that an observation is correctly measured, but unusual and in‡uential. In this case it is unclear how to proceed. Some researchers will try to alter the speci…cation to properly model the in‡uential observation. Other researchers will delete the observation from the sample. The motivation for this choice is to prevent the results from being skewed or determined by individual observations, but this practice is viewed skeptically by many researchers who believe it reduces the integrity of reported empirical results. 3.17 Normal Regression Model The normal regression model is the linear regression model under the restriction that the error ei is independent of xi and has the distribution N 0; 2 : We can write this as ei j xi N 0; 2 : This assumption implies N x0i ; yi j xi 2 : Normal regression is a parametric model, where likelihood methods can be used for estimation, testing, and distribution theory. The log-likelihood function for the normal regression model is ! n X 1 1 2 log L( ; 2 ) = log exp yi x0i 2 2 (2 2 )1=2 i=1 = n log (2 ) 2 n log 2 1 2 2 2 SSEn ( ): (3.43) The maximum likelihood estimator (MLE) ( b mle ; ^ 2mle ) maximizes log L( ; 2 ): Since the latter is a function of only through the sum of squared errors SSEn ( ); maximizing the likelihood is identical to minimizing SSEn ( ). Hence b mle = b ols ; CHAPTER 3. THE ALGEBRA OF LEAST SQUARES 70 the MLE for equals the OLS estimator: Due to this equivalence, the least squares estimator b ols is often called the MLE. We can also …nd the MLE for 2 : Plugging b into the log-likelihood we obtain log L b mle ; Maximization with respect to 2 2 = n log (2 ) 2 n log 2 2 yields the …rst-order condition @ log L b mle ; ^ 2 = @ 2 Solving for ^ 2 yields the MLE for n 1 2 + 2^ 2 ^2 SSEn ( b mle ) : 2 2 b 2 SSEn ( mle ) 2 = 0: n ^ 2mle = SSEn ( b mle ) 1X 2 = e^i n n i=1 which is the same as the moment estimator (3.28). Plugging the estimates into (3.43) we obtain the maximized log-likelihood log L b mle ; ^ 2mle = n (log (2 ) + 1) 2 n log ^ 2mle : 2 (3.44) The log-likelihood (or the negative log-likelihood) is typically reported as a measure of …t. It may seem surprising that the MLE b mle is numerically equal to the OLS estimator, despite emerging from quite di¤erent motivations. It is not completely accidental. The least-squares estimator minimizes a particular sample loss function – the sum of squared error criterion – and most loss functions are equivalent to the likelihood of a speci…c parametric distribution, in this case the normal regression model. In this sense it is not surprising that the least-squares estimator can be motivated as either the minimizer of a sample loss function or as the maximizer of a likelihood function. Carl Friedrich Gauss The mathematician Carl Friedrich Gauss (1777-1855) proposed the normal regression model, and derived the least squares estimator as the maximum likelihood estimator for this model. He claimed to have discovered the method in 1795 at the age of eighteen, but did not publish the result until 1809. Interest in Gauss’s approach was reinforced by Laplace’s simultaneous discovery of the central limit theorem, which provided a justi…cation for viewing random disturbances as approximately normal. 3.18 Technical Proofs* 1 Proof of Theorem 3.9.1, equation (3.23): First, hii = x0i (X 0 X) xi 0 since it is a 0 quadratic form and X X > 0: Next, since hii is the i’th diagonal element of the projection matrix 1 P = X (X 0 X) X; then hii = s0 P s CHAPTER 3. THE ALGEBRA OF LEAST SQUARES where 71 1 0 B .. C B . C B C C s=B B 1 C B .. C @ . A 0 0 is a unit vector with a 1 in the i’th place (and zeros elsewhere). By the spectral decomposition of the idempotent matrix P (see equation (A.5)) Ik 0 0 0 P = B0 B where B 0 B = I n . Thus letting b = Bs denote the i’th column of B, and partitioning b0 = b01 then Ik 0 0 0 hii = s0 B 0 Ik 0 0 0 = b01 b02 Bs b1 = b01 b1 b0 b = 1 the …nal equality since b is the i’th column of B and B 0 B = I n : We have shown that hii establishing (3.23). 1; Proof of Equation (3.37). The Sherman–Morrison formula (A.3) from Appendix A.5 states that for nonsingular A and vector b A 1 bb0 1 =A b0 A + 1 1 1 b 1 A bb0 A 1 : This implies X 0X xi x0i 1 1 = X 0X + (1 1 hii ) 1 X 0X xi x0i X 0 X 1 and thus b ( i) = = X 0X xi x0i X 0X 1 + (1 = b = b = b 1 X 0y hii ) 1 X 0X X 0y xi yi X 0X 1 0 XX 1 1 xi yi xi x0i xi yi + (1 X 0X hii ) (1 hii ) 1 X 0X 1 (1 hii ) 1 X 0X 1 1 X 0y X 0X xi (1 hii ) yi xi e^i the third equality making the substitutions b = (X 0 X) remainder collecting terms. 1 1 1 x i yi xi x0i b hii yi x0i b + hii yi X 0 y and hii = x0i (X 0 X) 1 xi ; and the CHAPTER 3. THE ALGEBRA OF LEAST SQUARES 72 Exercises Exercise 3.1 Let y be a random variable with g y; ; 2 = Ey and 2 y = )2 (y 2 = var(y): De…ne : Let (^ ; ^ 2 ) be the values such that g n (^ ; ^ 2 ) = 0 where g n (m; s) = n ^ and ^ 2 are the sample mean and variance. 1 Pn i=1 g (yi ; m; s) : Show that Exercise 3.2 Consider the OLS regression of the n 1 vector y on the n k matrix X. Consider an alternative set of regressors Z = XC; where C is a k k non-singular matrix. Thus, each column of Z is a mixture of some of the columns of X: Compare the OLS estimates and residuals from the regression of y on X to the OLS estimates from the regression of y on Z: Exercise 3.3 Using matrix algebra, show X 0 e ^ = 0: Exercise 3.4 Let e ^ be the OLS residual from a regression of y on X = [X 1 X 2 ]. Find X 02 e ^: Exercise 3.5 Let e ^ be the OLS residual from a regression of y on X: Find the OLS coe¢ cient from a regression of e ^ on X: Exercise 3.6 Let y ^ = X(X 0 X) 1 X 0 y: Find the OLS coe¢ cient from a regression of y ^ on X: Exercise 3.7 Show that if X = [X 1 X 2 ] then P X 1 = X 1 and M X 1 = 0: Exercise 3.8 Show that M is idempotent: M M = M : Exercise 3.9 Show that tr M = n k: Exercise 3.10 Show that if X = [X 1 X 2 ] and X 01 X 2 = 0 then P = P 1 + P 2 . Exercise 3.11 Show that when X contains a constant, 1 Pn y^i = y: n i=1 Exercise 3.12 A dummy variable takes on only the values 0 and 1. It is used for categorical data, such as an individual’s gender. Let d1 and d2 be vectors of 1’s and 0’s, with the i0 th element of d1 equaling 1 and that of d2 equaling 0 if the person is a man, and the reverse if the person is a woman. Suppose that there are n1 men and n2 women in the sample. Consider …tting the following three equations by OLS y = + d1 y = d1 y = 1 1 + d2 + d2 2 2 +e +e + d1 + e (3.45) (3.46) (3.47) Can all three equations (3.45), (3.46), and (3.47) be estimated by OLS? Explain if not. (a) Compare regressions (3.46) and (3.47). Is one more general than the other? Explain the relationship between the parameters in (3.46) and (3.47). (b) Compute 0d 1 and 0d ; 2 where is an n 1 is a vector of ones. (c) Letting = ( 1 2 )0 ; write equation (3.46) as y = X + e: Consider the assumption E(xi ei ) = 0. Is there any content to this assumption in this setting? CHAPTER 3. THE ALGEBRA OF LEAST SQUARES 73 Exercise 3.13 Let d1 and d2 be de…ned as in the previous exercise. (a) In the OLS regression y = d1 ^ 1 + d2 ^ 2 + u ^; show that ^ 1 is the sample mean of the dependent variable among the men of the sample (y 1 ), and that ^ 2 is the sample mean among the women (y 2 ). (b) Let X (n k) be an additional matrix of regressions. Describe in words the transformations y X where x1 and x2 are the k = y d1 y 1 d2 y 2 d1 x01 = X d2 x02 1 means of the regressors for men and women, respectively. (c) Compare e from the OLS regresion with b from the OLS regression y =X e +e ~ y = d1 ^ 1 + d2 ^ 2 + X b + e ^: 1 Exercise 3.14 Let b n = (X 0n X n ) X 0n y n denote the OLS estimate when y n is n 1 and X n is n k. A new observation (yn+1 ; xn+1 ) becomes available. Prove that the OLS estimate computed using this additional observation is b n+1 = bn + 1 1+ 1 x0n+1 (X 0n X n ) xn+1 X 0n X n 1 xn+1 yn+1 x0n+1 b n : Exercise 3.15 Prove that R2 is the square of the sample correlation between y and y ^: Exercise 3.16 Consider two least-squares regressions y = X1 e1 + e ~ and ^: y = X1 b1 + X2 b2 + e Let R12 and R22 be the R-squared from the two regressions. Show that R22 (explain) when there is equality R22 = R12 ? Exercise 3.17 Show that ~ 2 R12 : Is there a case ^ 2 : Is equality possible? Exercise 3.18 For which observations will b ( i) = b? Exercise 3.19 The data …le cps85.dat contains a random sample of 528 individuals from the 1985 Current Population Survey by the U.S. Census Bureau. The …le contains observations on nine variables, listed in the …le cps85.pdf. V1 V2 V3 V4 V5 V6 V7 V8 V9 = = = = = = = = = education (in years) region of residence (coded 1 if South, 0 otherwise) (coded 1 if nonwhite and non-Hispanic, 0 otherwise) (coded 1 if Hispanic, 0 otherwise) gender (coded 1 if female, 0 otherwise) marital status (coded 1 if married, 0 otherwise) potential labor market experience (in years) union status (coded 1 if in union job, 0 otherwise) hourly wage (in dollars) CHAPTER 3. THE ALGEBRA OF LEAST SQUARES 74 Estimate a regression of wage yi on education x1i , experience x2i , and experienced-squared x3i = x22i (and a constant). Report the OLS estimates. Let e^i be the OLS residual and y^i the predicted value from the regression. Numerically calculate the following: (a) (b) (c) (d) (e) (f) (g) Pn ^i i=1 e Pn ^i i=1 x1i e Pn ^i i=1 x2i e Pn 2 ^ i i=1 x1i e Pn 2 ^ i i=1 x2i e Pn ^i e^i i=1 y Pn ^2i i=1 e (h) R2 Are these calculations consistent with the theoretical properties of OLS? Explain. Exercise 3.20 Using the data from the previous problem, re-estimate the slope on education using the residual regression approach. Regress yi on (1; x2i ; x22i ), regress x1i on (1; x2i ; x22i ), and regress the residuals on the residuals. Report the estimate from this regression. Does it equal the value from the …rst OLS regression? Explain. In the second-stage residual regression, (the regression of the residuals on the residuals), calculate the equation R2 and sum of squared errors. Do they equal the values from the initial OLS regression? Explain. Chapter 4 Least Squares Regression 4.1 Introduction In this chapter we investigate some …nite-sample properties of least-squares applied to a random sample in the the linear regression model. In particular, we calculate the …nite-sample mean and covariance matrix and propose standard errors for the coe¢ cient estimates 4.2 Sample Mean To start with the simplest setting, we …rst consider the intercept-only model yi = + ei E (ei ) = 0: which is equivalent to the regression model with k = 1 and xi = 1: In the intercept model, = E (yi ) is the mean of yi : (See Exercise 2.15.) The least-squares estimator b = y equals the sample mean as shown in (3.7). We now calculate the mean and variance of the estimator y. Since the sample mean is a linear function of the observations, its expectation is simple to calculate ! n n 1X 1X Ey = E yi = Eyi = : n n i=1 i=1 This shows that the expected value of least-squares estimator (the sample mean) equals the projection coe¢ cient (the population mean). An estimator with the property that its expectation equals the parameter it is estimating is called unbiased. De…nition 4.2.1 An estimator b for is unbiased if Eb = . We next calculate the variance of the estimator y: Making the substitution yi = n y = 1X ei : n i=1 75 + ei we …nd CHAPTER 4. LEAST SQUARES REGRESSION 76 Then )2 var (y) = E (y n 1X ei n = E i=1 = 1 n2 = 1 n2 = 1 n n X n X !0 1 n X @1 ej A n j=1 E (ei ej ) i=1 j=1 n X 2 i=1 2 : The second-to-last inequality is because E (ei ej ) = 2 for i = j yet E (ei ej ) = 0 for i 6= j due to independence. We have shown that var (y) = n1 2 . This is the familiar formula for the variance of the sample mean. 4.3 Linear Regression Model We now consider the linear regression model. Throughout the remainder of this chapter we maintain the following. Assumption 4.3.1 Linear Regression Model The observations (yi ; xi ) come from a random sample and satisfy the linear regression equation yi = x0i + ei E (ei j xi ) = 0: (4.1) (4.2) The variables have …nite second moments Eyi2 < 1; E kxi k2 < 1; and an invertible design matrix Qxx = E xi x0i > 0: We will consider both the general case of heteroskedastic regression, where the conditional variance E e2i j xi = 2 (xi ) = 2i is unrestricted, and the specialized case of homoskedastic regression, where the conditional variance is constant. In the latter case we add the following assumption. CHAPTER 4. LEAST SQUARES REGRESSION 77 Assumption 4.3.2 Homoskedastic Linear Regression Model In addition to Assumption 4.3.1, E e2i j xi = 2 (xi ) = 2 (4.3) is independent of xi : 4.4 Mean of Least-Squares Estimator In this section we show that the OLS estimator is unbiased in the linear regression model. This calculation can be done using either summation notation or matrix notation. We will use both. First take summation notation. Observe that under (4.1)-(4.2) E (yi j X) = E (yi j xi ) = x0i : (4.4) The …rst equality states that the conditional expectation of yi given fx1 ; :::; xn g only depends on xi ; since the observations are independent across i: The second equality is the assumption of a linear conditional mean. Using de…nition (3.9), the conditioning theorem, the linearity of expectations, (4.4), and properties of the matrix inverse, 0 1 ! 1 n ! n X X E bjX = E@ xi x0i xi yi j X A i=1 = n X xi x0i i=1 = n X xi x0i i=1 = n X xi x0i i=1 = n X i=1 = xi x0i i=1 ! ! ! ! : 1 E n X i=1 1 n X i=1 1 n X i=1 1 n X xi yi ! jX ! E (xi yi j X) xi E (yi j X) xi x0i i=1 Now let’s show the same result using matrix notation. (4.4) implies 0 1 0 1 .. .. . B C B 0. C C=B x C=X : E (y j X) E (y j X) = B i @ A @ i A .. .. . . Similarly 0 .. . 1 0 .. . 1 B C B C C B C E (e j X) = B @ E (ei j X) A = @ E (ei j xi ) A = 0: .. .. . . (4.5) (4.6) CHAPTER 4. LEAST SQUARES REGRESSION 78 Using de…nition (3.18), the conditioning theorem, the linearity of expectations, (4.5), and the properties of the matrix inverse, E bjX 1 X 0X = E = X 0X 1 = X 0X 1 = : X 0y j X X 0 E (y j X) X 0X At the risk of belaboring the derivation, another way to calculate the same result is as follows. Insert y = X + e into the formula (3.18) for b to obtain b = = = X 0X 1 X 0 (X + e) X 0X 1 X 0X + X 0X + X 0X 1 1 X 0e X 0 e: (4.7) This is a useful linear decomposition of the estimator b into the true parameter 1 component (X 0 X) X 0 e: Once again, we can calculate that E b jX = E = 1 X 0X X 0X = 0: 1 and the stochastic X 0e j X X 0 E (e j X) Regardless of the method, we have shown that E b j X = expectations, we …nd that E b =E E bjX = : : Applying the law of iterated We have shown the following theorem. Theorem 4.4.1 Mean of Least-Squares Estimator In the linear regression model (Assumption 4.3.1) and E bjX = (4.8) E( b ) = : (4.9) Equation (4.9) says that the estimator b is unbiased for , meaning that the distribution of b is centered at . Equation (4.8) says that the estimator is conditionally unbiased, which is a stronger result. It says that b is unbiased for any realization of the regressor matrix X. 4.5 Variance of Least Squares Estimator In this section we calculate the conditional variance of the OLS estimator. For any r 1 random vector Z de…ne the r r covariance matrix var(Z) = E (Z = EZZ 0 EZ) (Z EZ)0 (EZ) (EZ)0 CHAPTER 4. LEAST SQUARES REGRESSION 79 and for any pair (Z; X) de…ne the conditional covariance matrix var(Z j X) = E (Z E (Z j X))0 j X : E (Z j X)) (Z The conditional covariance matrix of the n 1 regression error e is the n n matrix D = E ee0 j X : The i’th diagonal element of D is E e2i j X = E e2i j xi = 2 i while the ij 0 th o¤-diagonal element of D is E (ei ej j X) = E (ei j xi ) E (ej j xj ) = 0: where the …rst equality uses independence of the observations (Assumption 1.5.1) and the second is (4.2). Thus D is a diagonal matrix with i’th diagonal element 2i : 0 2 1 0 0 1 2 B 0 0 C 2 B C D = diag 21 ; :::; 2n = B . (4.10) .. . . .. C : @ .. . . . A 0 2 n 0 In the special case of the linear homoskedastic regression model (4.3), then E e2i j xi = 2 i D = In 2 2 = and we have the simpli…cation : In general, however, D need not necessarily take this simpli…ed form. For any matrix n r matrix A = A(X), var(A0 y j X) = var(A0 e j X) = A0 DA: In particular, we can write b = A0 y where A = X (X 0 X) 1 var( b j X) = A0 DA = X 0 X It is useful to note that 0 X DX = n X 1 (4.11) and thus X 0 DX X 0 X xi x0i 1 : 2 i; i=1 0 a weighted version of X X. Rather than working with the variance of the unscaled estimator b ; it will be useful to work p : We calculate that with the conditional variance of the scaled estimator n b V def b = var p n b = n var( b j X) = n X 0X = 1 0 XX n 1 1 X 0 DX jX X 0X 1 0 X DX n 1 1 0 XX n 1 : CHAPTER 4. LEAST SQUARES REGRESSION 80 This rescaling might seem rather odd, but it will help provide continuity between the …nite-sample treatment of this chapter and the asymptotic treatment of later chapters. As we will see in the next chapter, var( b j X) vanishes as n tends to in…nity, yet V b converges to a constant matrix. In the special case of the linear homoskedastic regression model, D = I n 2 , so X 0 DX = 0 X X 2 ; and the variance matrix simpli…es to V b 1 1 0 XX n = 2 : It may be worth observing that without rescaling, the variance can be written as var b j X = X 0 X 1 X 0 DX X 0X 1 and under conditional homoskedasticity var b j X = X 0 X 1 2 : Theorem 4.5.1 Variance of Least-Squares Estimator In the linear regression model (Assumption 4.3.1) V b = var = p n b 1 0 XX n jX 1 1 0 X DX n 1 0 XX n 1 (4.12) where D is de…ned in (4.10). In the homoskedastic linear regression model (Assumption 4.3.2) V 4.6 b = 1 0 XX n 1 2 : Gauss-Markov Theorem Now consider the class of estimators of can be written as which are linear functions of the vector y; and thus e = A0 y where A is an n k function of X. As noted before, the least-squares estimator is the special case obtained by setting A = X(X 0 X) 1 : What is the best choice of A? The Gauss-Markov theorem, which we now present, says that the least-squares estimator is the best choice among linear unbiased estimators when the errors are homoskedastic, in the sense that the least-squares estimator has the smallest variance among all unbiased linear estimators. To see this, since E (y j X) = X ; then for any linear estimator e = A0 y we have E e j X = A0 E (y j X) = A0 X ; so e is unbiased if (and only if) A0 X = I k : Furthermore, we saw in (4.11) that var e j X = var A0 y j X = A0 DA = A0 A 2 CHAPTER 4. LEAST SQUARES REGRESSION 81 the last equality using the homoskedasticity assumption D = I n 2 . The “best” unbiased linear estimator is obtained by …nding the matrix A0 satisfying A00 X = I k such that A00 A0 is minimized in the positive de…nite sense, in that for any other matrix A satisfying A0 X = I k ; then A0 A A00 A0 is positive semi-de…nite. Theorem 4.6.1 Gauss-Markov 1. In the homoskedastic linear regression model (Assumption 4.3.2), the best (minimum-variance) unbiased linear estimator is the leastsquares estimator b = X 0X 1 X 0y 2. In the linear regression model (Assumption 4.3.1), the best unbiased linear estimator is e = X 0D 1 X 1 X 0D 1 (4.13) y The …rst part of the Gauss-Markov theorem is a limited e¢ ciency justi…cation for the leastsquares estimator. The justi…cation is limited because the class of models is restricted to homoskedastic linear regression and the class of potential estimators is restricted to linear unbiased estimators. This latter restriction is particularly unsatisfactory as the theorem leaves open the possibility that a non-linear or biased estimator could have lower mean squared error than the least-squares estimator. The second part of the theorem shows that in the (heteroskedastic) linear regression model, within the class of linear unbiased estimators the best estimator is not least-squares but is (4.13). This is called the Generalized Least Squares (GLS) estimator. The GLS estimator is infeasible as the matrix D is unknown. This result does not suggest a practical alternative to least-squares. We return to the issue of feasible implementation of GLS in Section 9.1. We give a proof of the …rst part of the theorem below, and leave the proof of the second part for Exercise 4.3. Proof of Theorem 4.6.1.1. Let A be any n k function of X such that A0 X = I k : The variance 1 2 of the least-squares estimator is (X 0 X) and that of A0 y is A0 A 2 : It is su¢ cient to show 1 1 0 0 that the di¤erence A A (X X) is positive semi-de…nite. Set C = A X (X 0 X) : Note that X 0 C = 0: Then we calculate that A0 A X 0X 1 = C + X X 0X 1 0 = C 0C + C 0X X 0X + X 0X 1 C + X X 0X 1 + X 0X 1 1 X 0X X 0X X 0X 1 X 0C 1 = C 0C The matrix C 0 C is positive semi-de…nite (see Appendix A.8) as required. X 0X 1 CHAPTER 4. LEAST SQUARES REGRESSION 4.7 82 Residuals What are some properties of the residuals e^i = yi x0i b and prediction errors e~i = yi at least in the context of the linear regression model? Recall from (3.26) that we can write the residuals in vector notation as x0i b ( i) , e ^ = Me where M = I n X (X 0 X) conditional expectation 1 X 0 is the orthogonal projection matrix. Using the properties of E (^ e j X) = E (M e j X) = M E (e j X) = 0 and var (^ e j X) = var (M e j X) = M var (e j X) M = M DM (4.14) where D is de…ned in (4.10). We can simplify this expression under the assumption of conditional homoskedasticity E e2i j xi = 2 : In this case (4.14) simplies to var (^ e j X) = M 2 : In particular, for a single observation i; we obtain var (^ ei j X) = E e^2i j X = (1 hii ) 2 (4.15) since the diagonal elements of M are 1 hii as de…ned in (3.21). Thus the residuals e^i are heteroskedastic even if the errors ei are homoskedastic. Similarly, recall from (3.40) that the prediction errors e~i = (1 hii ) 1 e^i can be written in vector notation as e ~= M e ^ where M is a diagonal matrix with ith diagonal element (1 hii ) 1 : Thus e ~ = M M e: We can calculate that E (~ e j X) = M M E (e j X) = 0 and var (~ e j X) = M M var (e j X) M M = M M DM M which simpli…es under homoskedasticity to var (~ e j X) = M M M M = M MM 2 2 : The variance of the i’th prediction error is then var (~ ei j X) = E e~2i j X = (1 hii ) 1 (1 = (1 hii ) 1 2 hii ) (1 hii ) 1 2 : A residual with constant conditional variance can be obtained by rescaling. The standardized residuals are ei = (1 hii ) 1=2 e^i ; (4.16) and in vector notation e = (e1 ; :::; en )0 = M 1=2 M e: CHAPTER 4. LEAST SQUARES REGRESSION 83 From our above calculations, under homoskedasticity, var (e j X) = M 1=2 MM 1=2 2 and var (ei j X) = E e2i j X = 2 (4.17) and thus these standardized residuals have the same bias and variance as the original errors when the latter are homoskedastic. 4.8 Estimation of Error Variance The error variance 2 = Ee2i can be a parameter of interest, even in a heteroskedastic regression or a projection model. 2 measures the variation in the “unexplained” part of the regression. Its method of moments estimator (MME) is the sample average of the squared residuals: n 1X 2 ^ = e^i n 2 i=1 and equals the MLE in the normal regression model (3.28). In the linear regression model we can calculate the mean of ^ 2 : From (3.26), the properties of projection matrices and the trace operator, observe that ^2 = 1 1 1 1 1 0 e ^e ^ = e0 M M e = e0 M e = tr e0 M e = tr M ee0 : n n n n n Then E ^2 j X 1 tr E M ee0 j X n 1 tr M E ee0 j X n 1 tr (M D) : n = = = Adding the assumption of conditional homoskedasticity E e2i j xi = (4.18) simpli…es to E ^2 j X = = (4.18) 2; so that D = I n 2; then 1 tr M 2 n k 2 n ; n the …nal equality by (3.24). This calculation shows that ^ 2 is biased towards zero. The order of the bias depends on k=n, the ratio of the number of estimated coe¢ cients to the sample size. Another way to see this is to use (4.15). Note that n 2 E ^ jX = 1X E e^2i j X n i=1 = n 1X (1 n hii ) 2 i=1 = n k n 2 (4.19) CHAPTER 4. LEAST SQUARES REGRESSION 84 the line equality using Theorem 3.9.1. Since the bias takes a scale form, a classic method to obtain an unbiased estimator is by rescaling the estimator. De…ne n 1 X 2 2 s = e^i : (4.20) n k i=1 By the above calculation, E s2 j X = so E s2 = 2 (4.21) 2 and the estimator s2 is unbiased for 2 : Consequently, s2 is known as the “bias-corrected estimator” for 2 and in empirical practice s2 is the most widely used estimator for 2 : Interestingly, this is not the only method to construct an unbiased estimator for 2 . An estimator constructed with the standardized residuals ei from (4.16) is n 2 n 1X 2 1X = ei = (1 n n i=1 hii ) 1 2 e^i : (4.22) i=1 You can show (see Exercise 4.6) that E 2 jX = 2 (4.23) and thus 2 is unbiased for 2 (in the homoskedastic linear regression model). When k=n is small (typically, this occurs when n is large), the estimators ^ 2 ; s2 and 2 are likely to be close. However, if not then s2 and 2 are generally preferred to ^ 2 : Consequently it is best to use one of the bias-corrected variance estimators in applications. 4.9 Mean-Square Forecast Error A major purpose of estimated regressions is to predict out-of-sample values. Consider an outof-sample observation (yn+1 ; xn+1 ) where xn+1 will be observed but not yn+1 . Given the coe¢ cient estimate b the standard point estimate of E (yn+1 j xn+1 ) = x0n+1 is y~n+1 = x0n+1 b : The forecast error is the di¤erence between the actual value yn+1 and the point forecast, e~n+1 = yn+1 y~n+1 : The mean-squared forecast error (MSFE) is M SF En = E~ e2n+1 : In the linear regression model, e~n+1 = en+1 M SF En = Ee2n+1 x0n+1 b ; so 2E en+1 x0n+1 b +E x0n+1 b b (4.24) xn+1 : The …rst term in (4.24) is 2 : The second term in (4.24) is zero since en+1 x0n+1 is independent of b and both are mean zero. The third term in (4.24) is b tr E xn+1 x0n+1 E b = tr E xn+1 x0n+1 E b = = = 1 tr E xn+1 x0n+1 EV n 1 E tr xn+1 x0n+1 V b n 1 E x0n+1 V b xn+1 n b b (4.25) CHAPTER 4. LEAST SQUARES REGRESSION 85 where we use the fact that xn+1 is independent of b and use the de…nition V Thus 1 M SF En = 2 + E x0n+1 V b xn+1 : n Under conditional homoskedasticity, this simpli…es to 2 M SF En = 1 1 + E x0n+1 X 0 X b = var p n b jX : : xn+1 A simple estimator for the MSFE is the averaging of squared prediction errors (3.41) n 1X 2 ~ = e~i n 2 i=1 where e~i = yi x0i b ( i) = e^i (1 1: hii ) Indeed, we can calculate that E~ 2 = E~ e2i = 2 2 x0i b ( = E ei i) + E x0i b ( b i) By the same calculations as in (4.25) we …nd E~ 2 = 2 + 1 n 1 ( i) E x0i V b xi = M SF En xi 1: This is the MSFE based on a sample of size n 1; rather than size n: The di¤erence arises because the in-sample prediction errors e~i for i n are calculated using an e¤ective sample size of n 1, while the out-of sample prediction error e~n+1 is calculated from a sample with the full n observations. Unless n is very small we should expect M SF En 1 (the MSFE based on n 1 observations) to be close to M SF En (the MSFE based on n observations). Thus ~ 2 is a reasonable estimator for M SF En Theorem 4.9.1 MSFE In the linear regression model (Assumption 4.3.1) M SF En = E~ e2n+1 = 2 + 1 E x0n+1 V b xn+1 n p b j X : Furthermore, ~ 2 de…ned in (3.41) is where V b = var n an unbiased estimator of M SF En 1 : E~ 2 = M SF En 4.10 1 Covariance Matrix Estimation Under Homoskedasticity For inference, we need an estimate of the covariance matrix V b of the least-squares estimator. In this section we consider the homoskedastic regression model (Assumption 4.3.2). CHAPTER 4. LEAST SQUARES REGRESSION 86 Under homoskedasticity, the covariance matrix takes the relatively simple form V b 1 1 0 XX n = 2 which is known up to the unknown scale 2 . In Section 4.8 we discussed three estimators of The most commonly used choice is s2 ; leading to the classic covariance matrix estimator 1 0 XX n 0 Vb b = 2: 1 s2 : (4.26) 0 Since s2 is conditionally unbiased for 2 , it is simple to calculate that Vb b is conditionally unbiased for V b under the assumption of homoskedasticity: 0 E Vb b j X 1 0 XX n 1 1 0 XX n = V b: 1 = = E s2 j X 2 This estimator was the dominant covariance matrix estimator in applied econometrics at time, and is still the default in most regression packages. If the estimator (4.26) is used, but the regression error is heteroskedastic, it is possible for 1 1 0 1 0 1 0 XX X DX XX to be quite biased for the correct covariance matrix V b = n n n For example, suppose k = 1 and 2i = x2i with Exi = 0: The ratio of the true variance of least-squares estimator to the expectation of the variance estimator is V b 0 E Vb b j X one 0 Vb b 1 : the 1 Pn 4 i=1 xi Ex4i ' = n P = : 1 n 2 2 2 Ex 2 x i n i=1 i (Notice that we use the fact that 2i = x2i implies 2 = E 2i = Ex2i :) The constant is the standardized forth moment (or kurtosis) of the regressor xi ; and can be any number greater than one. For example, if xi N 0; 2 then = 3; so the true variance is three times larger than the 2 homoskedastic estimator. But can be much larger. Suppose, for example, that xi 1: In 1 this case = 15; so that the true variance is …fteen times larger than the homoskedastic estimator. While this is an extreme and constructed example, the point is that the classic covariance matrix estimator (4.26) may be quite biased when the homoskedasticity assumption fails. 4.11 Covariance Matrix Estimation Under Heteroskedasticity In the previous section we showed that that the classic covariance matrix estimator can be highly biased if homoskedasticity fails. In this section we show how to contruct covariance matrix estimators which do not require homoskedasticity. Recall that the general form for the covariance matrix is V b = 1 0 XX n 1 1 0 X DX n 1 0 XX n 1 : CHAPTER 4. LEAST SQUARES REGRESSION 87 This depends on the unknown matrix D which we can write as 2 1 ; :::; D = diag 2 n = E ee0 j X = E (D 0 j X) : where D 0 = diag e21 ; :::; e2n : Thus D 0 is a conditionally unbiased estimator for D: Therefore, if the squared errors e2i were observable, we could construct the unbiased estimator ideal Vb b = = 1 0 XX n 1 1 0 XX n 1 1 0 1 0 X D0 X XX n n ! n 1X 1 0 0 2 xi xi ei XX n n 1 1 : i=1 Indeed, E ideal Vb b jX 1 = 1 0 XX n 1 0 XX n 1 = 1 0 XX n 1 = = V ! n 1X 1 0 XX xi x0i E e2i j X n n i=1 ! n 1 1X 1 0 0 2 xi xi i XX n n 1 i=1 1 0 X DX n 1 1 0 XX n b ideal verifying that Vb b is unbiased for V b ideal Since the errors are unobserved, Vb b is not a feasible estimator. To construct a feasible estimator we can replace the errors with the least-squares residuals e^i ; the prediction errors e~i or the standardized residuals ei ; e.g. e2i b = diag e^2 ; :::; e^2 ; D 1 n 2 e D = diag e~1 ; :::; e~2n ; D = diag e21 ; :::; e2n : Substituting these matrices into the formula for V Vb b Ve b 1 0 XX n 1 = 1 0 XX n 1 = 1 0 XX n 1 = 1 0 XX n 1 = 1 0 XX n 1 = b (4.27) we obtain the estimators 1 1 0b 1 0 X DX XX n n ! n 1X 1 0 xi x0i e^2i XX n n 1 ; (4.28) i=1 1 1 0e 1 0 X DX XX n n ! n 1X 1 0 xi x0i e~2i XX n n i=1 ! n X 1 2 (1 hii ) xi x0i e^2i n i=1 1 1 0 XX n 1 ; CHAPTER 4. LEAST SQUARES REGRESSION 88 and V b 1 0 XX n 1 = 1 = 1 0 XX n 1 0 XX n 1 = The estimators Vb b ; Ve b ; and V 1 1 0 1 0 X DX XX n n ! n 1X 1 0 xi x0i e2i XX n n i=1 ! n 1X (1 hii ) 1 xi x0i e^2i n i=1 1 1 1 0 XX n : are often called robust, heteroskedasticity-consistent, or heteroskedasticity-robust covariance matrix estimators. The estimator Vb b was …rst developed by Eicker (1963), and introduced to econometrics by White (1980), and is sometimes called the Eicker-White or White covariance matrix estimator1 . The estimator Ve b was introduced by Andrews (1991) based on the principle of leave-one-out cross-validation, and the estimator V b was introduced by Horn, Horn and Duncan (1975) as a reduced-bias covariance matrix estimator. Since (1 hii ) 2 > (1 hii ) 1 > 1 it is straightforward to show that b Vb b < V b < Ve b (4.29) (See Exercise 4.7). The inequality A < B when applied to matrices means that the matrix B A is positive de…nite. In general, the bias of the estimators Vb b ; Ve b and V b ; is quite complicated, but they greatly simplify under the assumption of homoskedasticity (4.3). For example, using (4.15), ! n 1 1 X 1 1 1 0 0 0 2 E Vb b j X = XX XX xi xi E e^i j X n n n i=1 ! n 1 1 1 0 1X 1 0 0 2 = XX XX xi xi (1 hii ) n n n i=1 ! n 1 1 1 X 1 1 0 1 1 0 2 2 = XX X 0X xi x0i hii XX n n n n i=1 1 0 XX n = V b: 1 2 This calculation shows that Vb b is biased towards zero. Similarly, (again under homoskedasticity) we can calculate that Ve b is biased away from zero, speci…cally 1 1 0 2 XX (4.30) E Ve b j X n while the estimator V b is unbiased E V (See Exercise 4.8.) 1 b jX = 1 0 XX n 1 2 : Often, this estimator is rescaled by multiplying by the ad hoc bias adjustment corrected error variance estimator. (4.31) n n k in analogy to the bias- CHAPTER 4. LEAST SQUARES REGRESSION 89 It might seem rather odd to compare the bias of heteroskedasticity-robust estimators under the assumption of homoskedasticity, but it does give us a baseline for comparison. 0 We have introduced four covariance matrix estimators, Vb b ; Vb b ; Ve b ; and V b : Which should 0 you use? The classic estimator Vb b is typically a poor choice, as it is only valid under the unlikely homoskedasticity restriction. For this reason it is not typically used in contemporary econometric research. Of the three robust estimators, Vb b is the most commonly used, as it is the most straightforward and familiar. However, Ve b and (in particular) V b are preferred based on their 0 improved bias. Unfortunately, standard regression packages set the classic estimator Vb b as the default. As Ve b and V b are simple to implement, this should not be a barrier. For example, in STATA, V b is implemented by selecting “Robust”standard errors and selecting the bias correction option “1=(1 h)” or using the vce(hc2) option. 4.12 Standard Errors A variance estimator such as n 1 Vb b is an estimate of the variance of the distribution of b . A more easily interpretable measure of spread is its square root – the standard deviation. This is so important when discussing the distribution of parameter estimates, we have a special name for estimates of their standard deviation. De…nition 4.12.1 A standard error s( b ) for a real-valued estimator b is an estimate of the standard deviation of the distribution of b : When is a vector with estimate b and covariance matrix estimate n for individual elements are the square roots of the diagonal elements of n r rh i 1=2 1 Vb b : s( ^ j ) = n Vb ^ = n j 1V b 1V b b: b, standard errors That is, jj As we discussed in the previous section, there are multiple possible covariance matrix estimators, so standard errors are not unique. It is therefore important to understand what formula and method is used by an author when studying their work. It is also important to understand that a particular standard error may be relevant under one set of model assumptions, but not under another set of assumptions. To illustrate the computation of the covariance matrix estimate and standard errors, we return to the log wage regression (3.11) of Section 3.6. We calculate that s2 = 0:215 and 0:208 3:200 3:200 49:961 b = : Therefore the homoskedastic and White covariance matrix estimates are 0 Vb b = and Vb b = = 1 15:426 15:426 243 1 15:426 15:426 243 7:092 0:445 0:445 0:029 1 : 1 0:215 = 10:387 0:659 0:208 3:200 3:200 49:961 0:659 0:043 1 15:426 15:426 243 1 CHAPTER 4. LEAST SQUARES REGRESSION 90 The standard errors are the square roots For example, p of the diagonal elements of these matrices. p the White standard error for b 0 is 7:092=61 = 0:341 and that for ^ 1 is :029=61 = 0:022: A conventional format to write the estimated equation with standard errors is \ log(W age) = 0:626 + 0:156 Education: (0:022) (0:341) Alternatively, standard errors could be calculated using Ve b or V b : We report the four possible standard errors in the following table q q q q 0 n 1 Vb b n 1 Vb b n 1 Ve b n 1V b Intercept Education 0.412 0.026 0.341 0.022 0.361 0.023 0.351 0.022 The homoskedastic standard errors are noticably di¤erent than the others, but the three robust standard errors are quite close to one another. 4.13 Measures of Fit As we described in the previous chapter, a commonly reported measure of regression …t is the regression R2 de…ned as Pn 2 e^ ^2 2 R = 1 Pn i=1 i 2 = 1 : ^ 2y y) i=1 (yi P where ^ 2y = n 1 ni=1 (yi y)2 : R2 can be viewed as an estimator of the population parameter 2 = var (x0i ) =1 var(yi ) 2 2 y : However, ^ 2 and ^ 2y are biased estimators. Theil (1961) proposed replacing these by the unbiP ased versions s2 and ~ 2y = (n 1) 1 ni=1 (yi y)2 yielding what is known as R-bar-squared or adjusted R-squared: P (n 1) ni=1 e^2i s2 2 : R =1 =1 P ~ 2y (n k) ni=1 (yi y)2 2 While R is an improvement on R2 ; a much better improvement is Pn 2 e~ ~2 2 e R = 1 Pn i=1 i 2 = 1 ^ 2y y) i=1 (yi where e~i are the prediction errors (3.38) and ~ 2 is the MSPE from (3.41). As described in Section e2 is a good (4.9), ~ 2 is a good estimator of the out-of-sample mean-squared forecast error, so R estimator of the percentage of the forecast variance which is explained by the regression forecast. e2 is a good measure of …t. In this sense, R 2 e2 ; is that R2 One problem with R2 ; which is partially corrected by R and fully corrected by R necessarily increases when regressors are added to a regression model. This occurs because R2 is a negative function of the sum of squared residuals which cannot increase when a regressor is added. 2 e2 are non-monotonic in the number of regressors. R e2 can even be negative, In contrast, R and R which occurs when an estimated model predicts worse than a constant-only model. In the statistical literature the MSPE ~ 2 is known as the leave-one-out cross validation criterion, and is popular for model comparison and selection, especially in high-dimensional (none2 or ~ 2 to compare and select models. Models with parametric) contexts. It is equivalent to use R CHAPTER 4. LEAST SQUARES REGRESSION 91 e2 (or low ~ 2 ) are better models in terms of expected out of sample squared error. In contrast, high R 2 R cannot be used for model selection, as it necessarily increases when regressors are added to a 2 regression model. R is also an inappropriate choice for model selection (it tends to select models with too many parameters), though a justi…cation of this assertion requires a study of the theory of model selection. e2 and/or ~ 2 in regression analysis, In summary, it is recommended to calculate and report R 2 and omit R2 and R : Henri Theil 2 Henri Theil (1924-2000) of Holland invented R and two-stage least squares, both of which are routinely seen in applied econometrics. He also wrote an early and in‡uential advanced textbook on econometrics (Theil, 1971). 4.14 Empirical Example We again return to our wage equation, but use an extended sample of non-military wage earners with at least 12 years of education. For regressors we include years of education, potential work experience, experience squared, and dummy variable indicators for the following: female, female union member, male union member, female married, male married, hispanic, and non-white. The available sample is 46,943 so the parameter estimates are quite precise and reported in Table 5.1. Table 5.1 displays the parameter estimates in a standard tabular format. The table clearly states the estimation method (OLS), the dependent variable (log(Wage)), and the regressors are clearly labeled. Both parameter estimates and standard errors are reported for all coe¢ cient. In addition to the coe¢ cient estimates, the table also reports the estimated error standard deviation and the sample size: These are useful summary measures of …t which aid readers. Table 5.1 OLS Estimates of Linear Equation for Log(Wage) ^ s( ^ ) 0.021 0.001 0.001 0.002 0.009 0.020 0.020 0.008 0.008 0.008 0.007 Intercept 0.915 Education 0.118 Experience 0.034 Experience2 =100 -0.057 Female -0.129 Female Union Member 0.022 Male Union Member 0.095 Married Female 0.016 Married Male 0.180 Hispanic -0.110 Non-White -0.075 ^ 0.5659 Sample Size 46,943 Note: Standard errors are heteroskedasticity-consistent CHAPTER 4. LEAST SQUARES REGRESSION 92 As a general rule, it is advisable to always report standard errors along with parameter estimates. This allows readers to assess the precision of the parameter estimates, and as we will discuss in later chapters, form con…dence intervals and t-tests for individual coe¢ cients if desired. The results in Table 5.1 con…rm our earlier …ndings that the return to a year of education is approximately 12%, the return to experience is concave, that women earn approximately 13% less then men, and non-whites earn about 7% less than whites. In addition, we see that there are wage premiums for being a member of a labor union or being married, but the premiums appear to be much larger for men than for women. 4.15 Multicollinearity 1 If X 0 X is singular, then (X 0 X) and b are not de…ned. This situation is called strict multicollinearity, as the columns of X are linearly dependent, i.e., there is some 6= 0 such that X = 0: Most commonly, this arises when sets of regressors are included which are identically related. For example, if X includes both the logs of two prices and the log of the relative prices, log(p1 ); log(p2 ) and log(p1 =p2 ); for then X 0 X will necessarily be singular. When this happens, the applied researcher quickly discovers the error as the statistical software will be unable to construct (X 0 X) 1 : Since the error is discovered quickly, this is rarely a problem for applied econometric practice. The more relevant situation is near multicollinearity, which is often called “multicollinearity” for brevity. This is the situation when the X 0 X matrix is near singular, when the columns of X are close to linearly dependent. This de…nition is not precise, because we have not said what it means for a matrix to be “near singular”. This is one di¢ culty with the de…nition and interpretation of multicollinearity. One potential complication of near singularity of matrices is that the numerical reliability of the calculations may be reduced. In practice this is rarely an important concern, except when the number of regressors is very large. A more relevant implication of near multicollinearity is that individual coe¢ cient estimates will be imprecise. We can see this most simply in a homoskedastic linear regression model with two regressors yi = x1i 1 + x2i 2 + ei ; and 1 0 XX= n In this case var b j X = 2 n 1 1 1 1 1 : 2 = n (1 1 2) 1 : The correlation indexes collinearity, since as approaches 1 the matrix becomes singular. We can see the e¤ect of collinearity on precision by observing that the variance of a coe¢ cient esti1 2 mate 2 n 1 approaches in…nity as approaches 1. Thus the more “collinear” are the regressors, the worse the precision of the individual coe¢ cient estimates. What is happening is that when the regressors are highly dependent, it is statistically di¢ cult to disentangle the impact of 1 from that of 2 : As a consequence, the precision of individual estimates are reduced. The imprecision, however, will be re‡ected by large standard errors, so there is no distortion in inference. Some earlier textbooks overemphasized a concern about multicollinearity. A very amusing parody of these texts appeared in Chapter 23.3 of Goldberger’s A Course in Econometrics (1991), which is reprinted below. To understand his basic point, you should notice how the estimation 1 2 variance 2 n 1 depends equally and symmetrically on the correlation and the sample size n. CHAPTER 4. LEAST SQUARES REGRESSION Arthur S. Goldberger Art Goldberger (1930-2009) was one of the most distinguished members of the Department of Economics at the University of Wisconsin. His PhD thesis developed an early macroeconometric forecasting model (known as the Klein-Goldberger model) but most of his career focused on microeconometric issues. He was the leading pioneer of what has been called the Wisconsin Tradition of empirical work – a combination of formal econometric theory with a careful critical analysis of empirical work. Goldberger wrote a series of highly regarded and in‡uential graduate econometric textbooks, including including Econometric Theory (1964), Topics in Regression Analysis (1968), and A Course in Econometrics (1991). 93 CHAPTER 4. LEAST SQUARES REGRESSION Micronumerosity Arthur S. Goldberger A Course in Econometrics (1991), Chapter 23.3 Econometrics texts devote many pages to the problem of multicollinearity in multiple regression, but they say little about the closely analogous problem of small sample size in estimating a univariate mean. Perhaps that imbalance is attributable to the lack of an exotic polysyllabic name for “small sample size.” If so, we can remove that impediment by introducing the term micronumerosity. Suppose an econometrician set out to write a chapter about small sample size in sampling from a univariate population. Judging from what is now written about multicollinearity, the chapter might look like this: 1. Micronumerosity The extreme case, “exact micronumerosity,”arises when n = 0; in which case the sample estimate of is not unique. (Technically, there is a violation of the rank condition n > 0 : the matrix 0 is singular.) The extreme case is easy enough to recognize. “Near micronumerosity” is more subtle, and yet very serious. It arises when the rank condition n > 0 is barely satis…ed. Near micronumerosity is very prevalent in empirical economics. 2. Consequences of micronumerosity The consequences of micronumerosity are serious. Precision of estimation is reduced. There are two aspects of this reduction: estimates of may have large errors, and not only that, but Vy will be large. Investigators will sometimes be led to accept the hypothesis = 0 because y=^ y is small, even though the true situation may be not that = 0 but simply that the sample data have not enabled us to pick up. The estimate of will be very sensitive to sample data, and the addition of a few more observations can sometimes produce drastic shifts in the sample mean. The true may be su¢ ciently large for the null hypothesis = 0 to be 2 rejected, even though Vy = =n is large because of micronumerosity. But if the true is small (although nonzero) the hypothesis = 0 may mistakenly be accepted. 94 CHAPTER 4. LEAST SQUARES REGRESSION 95 3. Testing for micronumerosity Tests for the presence of micronumerosity require the judicious use of various …ngers. Some researchers prefer a single …nger, others use their toes, still others let their thumbs rule. A generally reliable guide may be obtained by counting the number of observations. Most of the time in econometric analysis, when n is close to zero, it is also far from in…nity. Several test procedures develop critical values n ; such that micronumerosity is a problem only if n is smaller than n : But those procedures are questionable. 4. Remedies for micronumerosity If micronumerosity proves serious in the sense that the estimate of has an unsatisfactorily low degree of precision, we are in the statistical position of not being able to make bricks without straw. The remedy lies essentially in the acquisition, if possible, of larger samples from the same population. But more data are no remedy for micronumerosity if the additional data are simply “more of the same.”So obtaining lots of small samples from the same population will not help. 4.16 Normal Regression Model In the special case of the normal linear regression model introduced in Section 3.17, we can derive exact sampling distributions for the least-squares estimator, residuals, and variance estimator. In particular, under the normality assumption ei j xi N 0; 2 then we have the multivariate implication e j X N 0; I n 2 : That is, the error vector e is independent of X and is normally distributed. Since linear functions of normals are also normal, this implies that conditional on X b 1 e ^ (X 0 X) X 0 M = e 2 (X 0 X) 1 N 0; 0 1 0 2M where M = I n X (X 0 X) X 0 : Since uncorrelated normal variables are independent, it follows that b is independent of any function of the OLS residuals including the estimated error variance s2 or ^ 2 or prediction errors e ~: The spectral decomposition (see equation (A.5)) of M yields M =H In 0 k 0 0 H0 CHAPTER 4. LEAST SQUARES REGRESSION where H 0 H = I n : Let u = 1H 0e n^ 2 2 N (0; H 0 H) = = = = 96 N (0; I n ) : Then k) s2 (n 2 1 2 1 2 1 2 e ^0 e ^ e0 M e In 0 e0 H = u0 In 0 k 0 0 0 0 k H 0e u 2 n k; a chi-square distribution with n k degrees of freedom. Furthermore, if standard errors are calculated using the homoskedastic formula (4.26) h i 1 N 0; 2 (X 0 X) b b jj N (0; 1) j j j j rh = rh tn k = q 2 q i i 2 1 1 s( ^ j ) n k 0 0 2 s (X X) (X X) n k n k n k jj a t distribution with n jj k degrees of freedom. Theorem 4.16.1 Normal Regression In the linear regression model (Assumption 4.3.1) if ei is independent of xi and distributed N 0; 2 then b N 0; n^ 2 ^ 2 j = j s( ^ j ) (n k)s2 2 tn 2 (X 0 X) 1 2 n k k These are the exact …nite-sample distributions of the least-squares estimator and variance estimators, and are the basis for traditional inference in linear regression. While elegant, the di¢ culty in applying Theorem 4.16.1 is that the normality assumption is too restrictive to be empirical plausible, and therefore inference based on Theorem 4.16.1 has no guarantee of accuracy. We develop an alternative inference theory based on large sample (asymptotic) approximations in the following chapter. William Gosset William S. Gosset (1876-1937) of England is most famous for his derivation of the student’s t distribution, published in the paper “The probable error of a mean” in 1908. At the time, Gosset worked at Guiness Brewery, which prohibited its employees from publishing in order to prevent the possible loss of trade secrets. To circumvent this barrier, Gosset published under the pseudonym “Student”. Consequently, this famous distribution is known as the student’s t rather than Gosset’s t! CHAPTER 4. LEAST SQUARES REGRESSION 97 Exercises Exercise 4.1 Explain the di¤erence between 1 n Pn 0 i=1 xi xi and E (xi x0i ) : Exercise 4.2 True or False. If yi =P xi + ei , xi 2 R; E(ei j xi ) = 0; and e^i is the OLS residual from the regression of yi on xi ; then ni=1 x2i e^i = 0: Exercise 4.3 Prove Theorem 4.6.1.2. Exercise 4.4 In a linear model y = X + e; with E(e j X) = 0; 2 var (e j X) = a known function of X , the GLS estimator is the residual vector is e ^= y e = X0 1 1 X X0 X e ; and an estimate of s2 = 1 1 2 is e ^0 1 e ^: X X0 1 X n k y ; (a) Find E e j X : (b) Find var e j X : (c) Prove that e ^ = M 1 e; where M 1 = I (d) Prove that M 01 1 M1 = 1 1 X X0 1 X 1 1 X0 X0 1 1 : : (e) Find E s2 j X : (f) Is s2 a reasonable estimator for 2? Exercise 4.5 Let (yi ; xi ) be a random sample with E(y j X) = X : Consider the Weighted Least Squares (WLS) estimator of e = X 0W X 1 X 0W y where W = diag (w1 ; :::; wn ) and wi = xji2 , where xji is one of the xi : (a) In which contexts would e be a good estimator? (b) Using your intuition, in which situations would you expect that e would perform better than OLS? Exercise 4.6 Show (4.23) in the homoskedastic regression model. Exercise 4.7 Prove (4.29). Exercise 4.8 Show (4.30) and (4.31) in the homoskedastic regression model. Chapter 5 An Introduction to Large Sample Asymptotics 5.1 Introduction In Chapter 4 we derived the mean and variance of the least-squares estimator in the context of the linear regression model, but this is not a complete description of the sampling distribution, nor su¢ cient for inference (con…dence intervals and hypothesis testing) on the unknown parameters. Furthermore, the theory does not apply in the context of the linear projection model, which is more relevant for empirical applications. Figure 5.1: Sampling Density of ^ To illustrate the situation with an example, let yi and xi be drawn from the joint density f (x; y) = 1 exp 2 xy 1 (log y 2 log x)2 exp 1 (log x)2 2 and let ^ be the slope coe¢ cient estimate from a least-squares regression of yi on xi and a constant. Using simulation methods, the density function of ^ was computed and plotted in Figure 5.1 for sample sizes of n = 25; n = 100 and n = 800: The vertical line marks the true projection coe¢ cient. 98 CHAPTER 5. AN INTRODUCTION TO LARGE SAMPLE ASYMPTOTICS 99 From the …gure we can see that the density functions are dispersed and highly non-normal. As the sample size increases the density becomes more concentrated about the population coe¢ cient. Is there a simple way to characterize the sampling distribution of ^ ? In principle the sampling distribution of ^ is a function of the joint distribution of (yi ; xi ) and the sample size n; but in practice this function is extremely complicated so it is not feasible to analytically calculate the exact distribution of ^ except in very special cases. Therefore we typically rely on approximation methods. The most widely used and versatile method is asymptotic theory, which approximates sampling distributions by taking the limit of the …nite sample distribution as the sample size n tends to in…nity. It is important to understand that this is an approximation technique, as the asymptotic distributions are used to assess the …nite sample distributions of our estimators in actual practical samples. The primary tools of asymptotic theory are the weak law of large numbers (WLLN), central limit theorem (CLT), and continuous mapping theorem (CMT). With these tools we can approximate the sampling distributions of most econometric estimators. In this chapter we provide a consise summary. It will be useful for most students to review this material, even if most is familiar. 5.2 Asymptotic Limits “Asymptotic analysis”is a method of approximation obtained by taking a suitable limit. There is more than one method to take limits, but the most common method in statistics and econometrics is to approximate sampling distributions by taking the limit as the sample size tends to positive in…nity, written “as n ! 1.” It is not meant to be interpreted literally, but rather as an approximating device. The …rst building block for asymptotic analysis is the concept of a limit of a sequence. De…nition 5.2.1 A sequence an has the limit a; written an ! a as n ! 1; or alternatively as limn!1 an = a; if for all > 0 there is some n < 1 such that for all n n ; jan aj : In words, an has the limit a if the sequence gets closer and closer to a as n gets larger. If a sequence has a limit, that limit is unique (a sequence cannot have two distinct limits). If an has the limit a; we also say that an converges to a as n ! 1: Not all sequences have limits. For example, the sequence f1; 2; 1; 2; 1; 2; :::g does not have a limit. It is therefore sometimes useful to have a more general de…nition of limits which always exist, and these are the limit superior and limit inferior of sequence De…nition 5.2.2 lim inf n!1 an = limn!1 inf m De…nition 5.2.3 lim supn!1 an = limn!1 supm n an n an The limit inferior and limit superior always exist, and equal when the limit exists. In the example given earlier, the limit inferior of f1; 2; 1; 2; 1; 2; :::g is 1, and the limit superior is 2. CHAPTER 5. AN INTRODUCTION TO LARGE SAMPLE ASYMPTOTICS 5.3 100 Convergence in Probability A sequence of numbers may converge to a limit, but what about a sequence of random variables? 1 Pn For example, consider a sample mean y = yi based on an random sample of n observations. n i=1 As n increases, the distribution of y changes. In what sense can we describe the “limit” of y: In what sense does it converge? Since y is a random variable, we cannot directly apply the deterministic concept of a sequence of numbers. Instead, we require a de…nition of convergence which is appropriate for random variables. There are more than one such de…nition, but the most commonly used is called convergence in probability. De…nition 5.3.1 A random variable zn 2 R converges in probability p to z as n ! 1; denoted zn ! z; or alternatively plimn!1 zn = z, if for all > 0; lim Pr (jzn zj ) = 1: (5.1) n!1 We call z the probability limit (or plim) of zn . The de…nition looks quite abstract, but it formalizes the concept of the sequence of random variables concentrating about a point. The event fjzn zj g occurs when zn is within of the point z: Pr (jzn zj ) is the probability of this event – that zn is within of the point z. Equation (5.1) states that this probability approaches 1 as the sample size n increases. The de…nition of convergence in probability requires that this holds for any : So for any small interval about z the distribution of zn concentrates within this interval for large n: You may notice that the de…nition concerns the distribution of the random variables zn , not their realizations. Furthermore, notice that the de…nition uses to the concept of a conventional (deterministic) limit, but the latter is applied to a sequence of probabilities, not directly to the random variables zn or their realizations. p When zn ! z we call z the probability limit (or plim) of zn . Two comments about the notation are worth mentioning. First, it is conventional to write the p convergence symbol as ! where the “p” above the arrow indicates that the convergence is “in probability”. You should try and adhere to this notation, and not simply write zn ! z. Second, it is also important to include the phrase “as n ! 1”to be speci…c about how the limit is obtained. It is common to confuse convergence in probability with convergence in expectation: Ezn ! Ez: (5.2) They are related but distinct concepts. Neither (5.1) nor (5.2) implies the other. To see the distinction it might be helpful to think through a stylized example. Consider a discrete random variable zn which takes the value 0 with probability 1 n 1 and the value an 6= 0 with probability n 1 , or Pr (zn = 0) = 1 Pr (zn = an ) = 1 n (5.3) 1 : n In this example the probability distribution of zn concentrates at zero as n increases, regardless of p the sequence an : You can check that zn ! 0 as n ! 1: In this example we can also calculate that the expectation of zn is an Ezn = : n CHAPTER 5. AN INTRODUCTION TO LARGE SAMPLE ASYMPTOTICS 101 Despite the fact that zn converges in probability to zero, its expectation will not decrease to zero unless an =n ! 0: If an diverges to in…nity at a rate equal to n (or faster) then Ezn will not converge p to zero. For example, if an = n; then Ezn = 1 for all n; even though zn ! 0: This example might seem a bit arti…cial, but the point is that the concepts of convergence in probability and convergence in expectation are distinct, so it is important not to confuse one with the other. Another common source of confusion with the notation surrounding probability limits is that p the expression to the right of the arrow \ !” must be free of dependence on the sample size n: p Thus expressions of the form “zn ! cn ” are notationally meaningless and should not be used. 5.4 Weak Law of Large Numbers In large samples we expect parameter estimates to be close to the population values. For example, in Section 4.2 we saw that the sample mean y is unbiased for = Ey and has variance 2 =n: As n gets large its variance decreases and thus the distribution of y concentrates about the population mean : It turns out that this implies that the sample mean converges in probability to the population mean. When y has a …nite variance there is a fairly straightforward proof by applying Chebyshev’s inequality. Theorem 5.4.1 Chebyshev’s Inequality. For any random variable zn and constant > 0 Pr (jzn var(zn ) Ezn j > ) 2 : Chebyshev’s inequality is terri…cally important in asymptotic theory. While it’s proof is a technical exercise in probability theory, it is quite simple so we discuss it forthwith. Let Fn (u) denote the distribution of zn Ezn : Then Z 2 2 dFn (u): = Pr (jzn Ezn j > ) = Pr (zn Ezn ) > fu2 > 2 g The integral is over the event u2 > Z fu2 > 2 g dFn (u) Z fu2 > 2 g 2 , so that the inequality 1 u2 2 dFn (u) Z u2 2 dFn (u) = u2 2 holds throughout. Thus Ezn )2 E (zn 2 which establishes the desired inequality. Applied to the sample mean y, Chebyshev’s inequality shows that for any = var(zn ) 2 ; >0 2 Pr (jy Eyj > ) n 2: For …xed 2 and ; the bound on the right-hand-side shrinks to zero as n ! 1: Thus the probability that y is within of Ey = approaches 1 as n gets large, or lim Pr (jy n!1 j ) = 1: This means that y converges in probability to as n ! 1: This result is called the weak law of large numbers. Our derivation assumed that y has a …nite variance, but all that is necessary is for y to have a …nite mean. CHAPTER 5. AN INTRODUCTION TO LARGE SAMPLE ASYMPTOTICS 102 Theorem 5.4.2 Weak Law of Large Numbers (WLLN) If yi are independent and identically distributed and E jyj < 1; then as n ! 1, n 1X p y= yi ! E(y): n i=1 The proof of Theorem 5.4.2 is presented in Section 5.14. The WLLN shows that the estimator y converges in probability to the true population mean . In general, an estimator which converges in probability to the population value is called consistent. De…nition 5.4.1 An estimator ^ of a parameter as n ! 1: p is consistent if ^ ! Consistency is a good property for an estimator to possess. It means that for any given data distribution; there is a sample size n su¢ ciently large such that the estimator ^ will be arbitrarily close to the true value with high probability. Unfortunately it does not mean that ^ will actually be close to in a given …nite sample, but it is a minimal property for an estimator to be considered a “good” estimator. Theorem 5.4.3 If yi are independent and identically distributed and E jyj < 1; then b = y is consistent for the population mean : 5.5 Almost Sure Convergence and the Strong Law* Convergence in probability is sometimes called weak convergence. A related concept is almost sure convergence, also known as strong convergence. (In probability theory the term “almost sure” means “with probability equal to one”. An event which is random but occurs with probability equal to one is said to be almost sure.) De…nition 5.5.1 A random variable zn 2 R converges almost surely a:s: to z as n ! 1; denoted zn ! z; if for every > 0 Pr lim jzn n!1 zj = 1: (5.4) The convergence (5.4) is stronger than (5.1) because it computes the probability of a limit rather than the limit of a probability. Almost sure convergence is stronger than convergence in p a:s: probability in the sense that zn ! z implies zn ! z. In the example (5.3) of Section 5.3, the sequence zn converges in probability to zero for any sequence an ; but this is not su¢ cient for zn to converge almost surely. In order for zn to converge to zero almost surely, it is necessary that an ! 0. In the random sampling context the sample mean can be shown to converge almost surely to the population mean. This is called the strong law of large numbers. CHAPTER 5. AN INTRODUCTION TO LARGE SAMPLE ASYMPTOTICS 103 Theorem 5.5.1 Strong Law of Large Numbers (SLLN) If yi are independent and identically distributed and E jyj < 1; then as n ! 1, n 1 X a:s: y= yi ! E(y): n i=1 The proof of the SLLN is technically quite advanced so is not presented here. For a proof see Billingsley (1995, Section 22) or Ash (1972, Theorem 7.2.5). The WLLN is su¢ cient for most purposes in econometrics, so we will not use the SLLN in this text. 5.6 Vector-Valued Moments Our preceding discussion focused on the case where y is real-valued (a scalar), but nothing important changes if we generalize to the case where y 2 Rm is a vector. To …x notation, the elements of y are 0 1 y1 B y2 C B C y = B . C: @ .. A ym The population mean of y is just the vector of marginal means 0 1 E (y1 ) B E (y2 ) C B C = E(y) = B C: .. @ A . E (ym ) When working with random vectors y it is convenient to measure their magnitude by their Euclidean length, which is Euclidean norm kyk = y12 + 2 + ym 1=2 : In vector notation we have kyk2 = y 0 y: It turns out that it is equivalent to describe …niteness of moments in terms of the Euclidean norm of a vector or all individual components. Theorem 5.6.1 For y 2 Rm ; E kyk < 1 if and only if E jyj j < 1 for j = 1; :::; m: The m m variance matrix of y is V = var (y) = E (y ) (y )0 : V is often called a variance-covariance matrix. You can show that the elements of V are …nite if E kyk2 < 1: CHAPTER 5. AN INTRODUCTION TO LARGE SAMPLE ASYMPTOTICS 104 A random sample fy 1 ; :::; y n g consists of n observations of independent and identically draws from the distribution of y: (Each draw is an m-vector.) The vector sample mean 0 1 y1 n B y C 1X B 2 C y= yi = B . C n @ .. A i=1 ym is the vector of sample means of the individual variables. Convergence in probability of a vector can be de…ned as convergence in probability of all elep p ments in the vector. Thus y ! if and only if y j ! j for j = 1; :::; m: Since the latter holds if E jyj j < 1 for j = 1; :::; m; or equivalently E kyk < 1; we can state this formally as follows. Theorem 5.6.2 Weak Law of Large Numbers (WLLN) for random vectors If y i are independent and identically distributed and E kyk < 1; then as n ! 1, n 1X p y= y i ! E(y): n i=1 5.7 Convergence in Distribution The WLLN is a useful …rst step, but does not give an approximation to the distribution of an estimator. A large-sample or asymptotic approximation can be obtained using the concept of convergence in distribution. De…nition 5.7.1 Let z n be a random vector with distribution Fn (u) = Pr (z n u) : We say that z n converges in distribution to z as n ! 1, d denoted z n ! z; if for all u at which F (u) = Pr (z Fn (u) ! F (u) as n ! 1: u) is continuous, d When z n ! z, it is common to refer to z as the asymptotic distribution or limit distribution of z n . When the limit distribution z is degenerate (that is, Pr (z = c) = 1 for some c) we can write p d the convergence as z n ! c, which is equivalent to convergence in probability, z n ! c. The typical path to establishing convergence in distribution is through the central limit theorem (CLT), which states that a standardized sample average converges in distribution to a normal random vector. Theorem 5.7.1 Lindeberg–Lévy Central Limit Theorem (CLT). If y i are independent and identically distributed and E kyk2 < 1; then as n!1 n p 1 X d n (y )= p (y i ) ! N (0; V ) n i=1 where = Ey and V = E (y ) (y )0 : CHAPTER 5. AN INTRODUCTION TO LARGE SAMPLE ASYMPTOTICS 105 p The standardized sum z n = n (y n ) has mean zero and variance V . What the CLT adds is that the variable z n is also approximately normally distributed, and that the normal approximation improves as n increases. The CLT is one of the most powerful and mysterious results in statistical theory. It shows that the simple process of averaging induces normality. The …rst version of the CLT (for the number of heads resulting from many tosses of a fair coin) was established by the French mathematician Abraham de Moivre in an article published in 1733. This was extended to cover an approximation to the binomial distribution in 1812 by Pierre-Simon Laplace in his book Théorie Analytique des Probabilités, and the most general statements are credited to articles by the Russian mathematician Aleksandr Lyapunov (1901) and the Finnish mathematician Jarl Waldemar Lindeberg (1920, 1922). The above statement is known as the classic (or Lindeberg-Lévy) CLT due to contributions by Lindeberg (1920) and the French mathematician Paul Pierre Lévy. A more general version which does not require the restriction to identical distributions was provided by Lindeberg (1922). Theorem 5.7.2 Lindeberg Central Limit Theorem (CLT). Suppose that yi are independent but not necessarily identically distributed …nite Pn with 2 2 : If for 2 = means i = Eyi and variances 2i = E (yi ) : Set i n i=1 i all " > 0 n 1 X 2 lim 2 E (yi " n) = 0 (5.5) i ) 1 (jyi ij n!1 then n i=1 n 1 X (yi i) n i=1 d ! N (0; 1) Equation (5.5) is known as Lindeberg’s condition. A standard method to verify (5.5) is via Lyapunov’s condition: For some > 0 lim n!1 1 2+ n n X 2+ i) E (yi = 0: (5.6) i=1 It is easy to verify that (5.6) implies (5.5), and (5.6) is often easy to verify. For example, if 3 supi E (yi < 1 and inf i 2i c > 0 then i) n 1 X 3 n i=1 E (yi 3 i) n (nc)3=2 !0 so (5.6) is satis…ed. 5.8 Higher Moments Often we want to estimate a parameter which is the expected value of a transformation of a random vector y. That is, can be written as = Eh (y) for some function h : Rm ! Rk : For example, the second moment of y is Ey 2 ; the k’th is Ey k ; the moment generating function is E exp (ty) ; and the distribution function is E1 fy xg : CHAPTER 5. AN INTRODUCTION TO LARGE SAMPLE ASYMPTOTICS 106 Estimating parameters of this form …ts into our previous analysis by de…ning the random variable z = h (y) for then = Ez is just a simple moment of z. This suggests the moment estimator n n 1X 1X b= zi = h (y i ) : n n i=1 i=1 Pn k is n 1 k For example, the moment estimator of Ey i=1 yi ; that of the moment P Pngenerating function n 1 1 is n xg i=1 exp (tyi ) ; and for the distribution function the estimator is n i=1 1 fyi Since b is a sample average, and transformations of iid variables are also iid, the asymptotic results of the previous sections immediately apply. Theorem 5.8.1 If y i are independent and identically distributed, = P Eh (y) ; and E kh (y)k < 1; then for b = n1 ni=1 h (y i ) ; as n ! 1, p b ! : Theorem 5.8.2 If y i are independent and P identically distributed, Eh (y) ; and E kh (y)k2 < 1; then for b = n1 ni=1 h (y i ) ; as n ! 1; p where V = E (h (y) n (b ) (h (y) = d ) ! N (0; V ) )0 : Theorems 5.8.1 and 5.8.2 show that the estimate b is consistent for and asymptotically normally distributed, so long as the stated moment conditions hold. A word of caution. Theorems 5.8.1 and 5.8.2 give the impression that it is possible to estimate any moment of y: Technically this is the case, so long as that moment is …nite. What is hidden by the notation, however, is that estimates of highPorder momnets can be quite imprecise. For example, consider the sample 8th moment b8 = n1 ni=1 yi8 ; and suppose for simplicity that y is N(0; 1): Then we can calculate1 that var (b8 ) = n 1 645; 015; which is huge, even for large n! In general, higher-order moments are challenging to estimate because their variance depends upon even higher moments which can be quite large in some cases. 5.9 Functions of Moments We now expand our investigation and consider estimation of parameters which can be written as a continuous function of = Eh (y). That is, the parameter of interest can be written as = g ( ) = g (Eh (y)) (5.7) for some functions g : Rk ! R` and h : Rm ! Rk : As one example, the geometric mean of wages w is = exp (E (log (w))) : 1 By the formula for the variance of a mean var (b8 ) = n 2; 027; 025 and Ey 8 = 7!! = 105 where k!! = k(k 1) 1 Ey 16 (5.8) Ey 8 2 is the double factorial. : Since y is N(0; 1); Ey 16 = 15!! = CHAPTER 5. AN INTRODUCTION TO LARGE SAMPLE ASYMPTOTICS 107 This is (5.7) with g(u) = exp (u) and h(w) = log(w): A simple yet common example is the variance 2 = E (w Ew)2 = Ew2 (Ew)2 : This is (5.7) with w w2 h(w) = and g( 1; 2) = 2 1: 2 Similarly, the skewness of the wage distribution is sk = E (w 3=2 Ew)2 E (w This is (5.7) with Ew)3 : 0 1 w h(w) = @ w2 A w3 and g( 1; 2; 3) = 3 3 2 1+2 1 : 2 3=2 1 3 2 (5.9) The parameter = g ( ) is not a population moment, so it does not have a direct moment estimator. Instead, it is common to use a plug-in estimate formed by replacing the unknown with its point estimate b and then “plugging” this into the expression for . The …rst step is n 1X b= h (y i ) n i=1 and the second step is b = g (b ) : Again, the hat “^” indicates that b is a sample estimate of : For example, the plug-in estimate of the geometric mean of the wage distribution from (5.8) is b = exp(b) with n b= The plug-in estimate of the variance is 1X log (wagei ) : n i=1 n ^ 2 = 1X 2 wi n i=1 = n 1X (wi n i=1 n 1X wi n i=1 w)2 : !2 CHAPTER 5. AN INTRODUCTION TO LARGE SAMPLE ASYMPTOTICS 108 and that for the skewness is c = sk = where 3b2 b1 + 2b31 b3 3=2 b2 b21 1 Pn w)3 i=1 (wi n 3=2 1 Pn w)2 i=1 (wi n n 1X j bj = wi : n i=1 A useful property is that continuous functions are limit-preserving. p Theorem 5.9.1 Continuous Mapping Theorem (CMT). If z n ! c p as n ! 1 and g ( ) is continuous at c; then g(z n ) ! g(c) as n ! 1. The proof of Theorem 5.9.1 is given in Section 5.14. p For example, if zn ! c as n ! 1 then p zn + a ! c + a p azn ! ac p zn2 ! c2 as the functions g (u) = u + a; g (u) = au; and g (u) = u2 are continuous. Also a zn p ! a c if c 6= 0: The condition c 6= 0 is important as the function g(u) = a=u is not continuous at u = 0: If yP = Eh (y) ; and E kh (y)k < 1; then for i are independent and identically distributed, b = n1 ni=1 h (y i ) ; as n ! 1, p b ! : Theorem 5.9.2 If y i are independent and identically distributed, = g (Eh (y)) ; E kh (y)k < 1; and g (u) is continuous at u = , then for b = g 1 Pn h (y ) ; as n ! 1; b p! : i i=1 n To apply Theorem 5.9.2 it is necessary to check if the function g is continuous at . In our …rst example g(u) = exp (u) is continuous everywhere. It therefore follows from Theorem 5.6.2 and p Theorem 5.9.2 that if E jlog (wage)j < 1 then as n ! 1; b ! : In the example of the variance, g is continuous for all . Thus if Ew2 < 1 then as n ! 1; 2 p b ! 2: 2 > 0; In our third example g de…ned in (5.9) is continuous for all such that var(w) = 2 1 3 which holds unless w has a degenerate distribution. Thus if E jwj < 1 and var(w) > 0 then as c p! sk: n ! 1; sk CHAPTER 5. AN INTRODUCTION TO LARGE SAMPLE ASYMPTOTICS 5.10 109 Delta Method In this section we introduce two tools –an extended version of the CMT and the Delta Method –which allow us to calculate the asymptotic distribution of the parameter estimate b . We …rst present an extended version of the continuous mapping theorem which allows convergence in distribution. Theorem 5.10.1 Continuous Mapping Theorem d If z n ! z as n ! 1 and g : Rm ! Rk has the set of discontinuity points d Dg such that Pr (z 2 Dg ) = 0; then g(z n ) ! g(z) as n ! 1. For a proof of Theorem 5.10.1 see Theorem 2.3 of van der Vaart (1998). It was …rst proved by Mann and Wald (1943) and is therefore sometimes referred to as the Mann-Wald Theorem. Theorem 5.10.1 allows the function g to be discontinuous only if the probability at being at a discontinuity point is zero. For example, the function g(u) = u 1 is discontinuous at u = 0; but if d d zn ! z N (0; 1) then Pr (z = 0) = 0 so zn 1 ! z 1 : A special case of the Continuous Mapping Theorem is known as Slutsky’s Theorem. Theorem 5.10.2 Slutsky’s Theorem p d If zn ! z and cn ! c as n ! 1, then d 1. zn + cn ! z + c d 2. zn cn ! zc 3. zn cn d ! z if c 6= 0 c Even though Slutsky’s Theorem is a special case of the CMT, it is a useful statement as it focuses on the most common applications –addition, multiplication, and division. Despite the fact that the plug-in estimator b is a function of b for which we have an asymptotic distribution, Theorem 5.10.1 does not directly give us an asymptotic distribution for b : This is p because b = g (b ) is written as a function of b , not of the standardized sequence n (b ): We need an intermediate step – a …rst order Taylor series expansion. This step is so critical to statistical theory that it has its own name –The Delta Method. Theorem 5.10.3 Delta Method: p d If n (b ) ! ; where g(u) is continuously di¤ erentiable in a neighborhood of then as n ! 1 p where G(u) = as n ! 1 @ 0 @u g(u) p n (g (b ) d g( )) ! G0 and G = G( ): In particular, if n (g (b ) d g( )) ! N 0; G0 V G : (5.10) N (0; V ) then (5.11) CHAPTER 5. AN INTRODUCTION TO LARGE SAMPLE ASYMPTOTICS 110 The Delta Method allows us to complete our derivation of the asymptotic distribution of the estimator b of . Now by combining Theorems 5.8.2 and 5.10.3 we can …nd the asymptotic distribution of the plug-in estimator b . Theorem 5.10.4 If y i are independent and identically distributed, = @ Eh (y), = g ( ) ; E kh (y)k2 < 1; and G (u) = g (u)0 is continuous @u P in a neighborhood of , then for b = g n1 ni=1 h (y i ) ; as n ! 1 p where V = E (h (y) d n b ! N 0; G0 V G ) (h (y) )0 and G = G ( ) : Theorem 5.9.2 established the consistency of b for , and Theorem 5.10.4 established its asymptotic normality. It is instructive to compare the conditions required for these results. Consistency required that h (y) have a …nite mean, while asymptotic normality requires that this variable have a …nite variance. Consistency required that g(u) be continuous, while asymptotic normality required that g(u) be continuously di¤erentiable, the latter a stronger smoothness condition. 5.11 Stochastic Order Symbols It is convenient to have simple symbols for random variables and vectors which converge in probability to zero or are stochastically bounded. In this section we introduce some of the most commonly found notation. It might be useful to review the common notation for non-random convergence and boundedness. Let xn and an ; n = 1; 2; :::; be a non-random sequences. The notation xn = o(1) (pronounced “small oh-one”) is equivalent to an ! 0 as n ! 1. The notation xn = o(an ) is equivalent to an 1 xn ! 0 as n ! 1: The notation xn = O(1) (pronounced “big oh-one”) means that xn is bounded uniformly in n : there exists an M < 1 such that jxn j M for all n: The notation xn = O(an ) is equivalent to an 1 xn = O(1): We now introduce similar concepts for sequences of random variables. Let zn and an ; n = 1; 2; ::: be sequences of random variables. (In most applications, an is non-random.) The notation zn = op (1) p (“small oh-P-one”) means that zn ! 0 as n ! 1: We also write zn = op (an ) CHAPTER 5. AN INTRODUCTION TO LARGE SAMPLE ASYMPTOTICS if an 1 zn = op (1): For example, for any consistent estimator b for b= 111 we can write + op (1): Similarly, the notation zn = Op (1) (“big oh-P-one”) means that zn is bounded in probability. Precisely, for any " > 0 there is a constant M" < 1 such that lim sup Pr (jzn j > M" ) ": n!1 Furthermore, we write zn = Op (an ) if an 1 zn = Op (1): Op (1) is weaker than op (1) in the sense that zn = op (1) implies zn = Op (1) but not the reverse. However, if zn = Op (an ) then zn = op (bn ) for any bn such that an =bn ! 0: d If a random vector converges in distribution z n ! z (for example, if z N (0; V )) then z n = Op (1): It follows that for estimators b which satisfy the convergence of Theorem 5.10.4 then we can write b = + Op (n 1=2 ): Another useful observation is that a random sequence with a bounded moment is stochastically bounded. Theorem 5.11.1 If z n is a random vector which satis…es E kz n k = O (an ) for some sequence an and > 0; then z n = Op (a1= n ): 1= Similarly, E kz n k = o (an ) implies z n = op (an ): This can be shown using Markov’s inequality (B.21). The assumptions imply that there is some M 1= M < 1 such that E kz n k M an for all n: For any " set B = : Then " Pr an 1= kz n k > B = Pr kz n k > M an " " E kz n k M an " as required. There are many simple rules for manipulating op (1) and Op (1) sequences which can be deduced from the continuous mapping theorem or Slutsky’s Theorem. For example, op (1) + op (1) = op (1) op (1) + Op (1) = Op (1) Op (1) + Op (1) = Op (1) op (1)op (1) = op (1) op (1)Op (1) = op (1) Op (1)Op (1) = Op (1) CHAPTER 5. AN INTRODUCTION TO LARGE SAMPLE ASYMPTOTICS 5.12 112 Uniform Stochastic Bounds* For some applications it can be useful to obtain the stochastic order of the random variable max jyi j : 1 i n This is the magnitude of the largest observation in the sample fy1 ; :::; yn g: If the support of the distribution of yi is unbounded, then as the sample size n increases, the largest observation will also tend to increase. It turns out that there is a simple characterization. Theorem 5.12.1 If y i are independent and identically distributed: If E jyjr < 1; then as n ! 1 n p 1=r max jyi j ! 0: 1 i n (5.12) If E exp(ty) < 1 for all t < 1; then 1 (log n) p max jyi j ! 0: 1 i n (5.13) The proof of Theorem 5.12.1 is presented in Section 5.14. Equivalently, (5.12) can be written as max jyi j = op (n1=r ) (5.14) max jyi j = op (log n): (5.15) 1 i n and (5.13) as 1 i n Equation (5.12) says that if y has r …nite moments, then the largest observation will diverge at a rate slower than n1=r . As r increases this rate decreases. Equation (5.13) shows that if we strengthen this to y having all …nite moments and a …nite moment generating function (for example, if y is normally distributed) then the largest observation will diverge slower than log n: Thus the higher the moments, the slower the rate of divergence. To simplify the notation, we write (5.14) as yi = op (n1=r ) uniformly in 1 i n, and similarly (5.15) as yi = op (log n); uniformly in 1 i n: It is important to understand when the Op or op symbols are applied to subscript i random variables whether the convergence is pointwise in i, or is uniform in i in the sense of (5.14)-(5.15). Theorem 5.12.1 applies to random vectors. If E kykr < 1 then max ky i k = op (n1=r ); 1 i n and if E exp(t0 y) < 1 for all ktk < 1 then (log n) 1 p max ky i k ! 0: 1 i n (5.16) CHAPTER 5. AN INTRODUCTION TO LARGE SAMPLE ASYMPTOTICS 5.13 113 Semiparametric E¢ ciency In this section we argue that the sample mean b and plug-in estimator b = g (b ) are e¢ cient estimators of the parameters and . Our demonstration is based on the rich but technically challenging theory of semiparametric e¢ ciency bounds. An excellent accessible review has been provided by Newey (1990). We will also appeal to the asymptotic theory of maximum likelihood estimation (see Section B.11). We start by examining the sample mean b ; for the asymptotic e¢ ciency of b will follow from that of b : Recall, we know that if E kyk2 < 1 then the sample mean has the asymptotic distribution p d n (b ) ! N (0; V ) : We want to know if b is the best feasible estimator, or if there is another estimator with a smaller asymptotic variance. While it seems intuitively unlikely that another estimator could have a smaller asymptotic variance, how do we know that this is not the case? When we ask if b is the best estimator, we need to be clear about the class of models –the class of permissible distributions. For estimation of the mean of the distribution of y the broadest conceivable class is L1 = fF : E kyk < 1g : This class is too broad n for our current purposes, as n o b is not asymptotically N (0; V ) for all F 2 L1 : A more realistic choice is L2 = F : E kyk2 < 1 – the class of …nite-variance distributions. When we seek an e¢ cient estimator of the mean in the class of models L2 what we are seeking is the best estimator, given that all we know is that F 2 L2 : To show that the answer is not immediately obvious, it might be helpful to review a setting where the sample mean is ine¢ p cient. Suppose that y 2 R has the double exponential den1=2 sity f (y j ) = 2 exp jy j 2 : Since var (y) = 1 we see that the sample mean satp d ) ! N (0; 1). In this model the maximum likelihood estimator (MLE) ~ for is…es n (^ is the sample median. Recall from the theory of maximum likelhood that the MLE satis…es p p d 1 n (~ ) ! N 0; ES 2 where S = @@ log f (y j ) = 2 sgn (y ) is the score. We can p d calculate that ES 2 = 2 and thus conclude that n (~ ) ! N (0; 1=2) : The asymptotic variance of the MLE is one-half that of the sample mean. Thus when the true density is known to be double exponential the sample mean is ine¢ cient. But the estimator which achieves this improved e¢ ciency –the sample median –is not generically consistent for the population mean. It is inconsistent if the density is asymmetric or skewed. So the improvement comes at a great cost. Another way of looking at this p is that the sample median is e¢ cient in the class of densities f (y j ) = 2 1=2 exp jy j 2 but unless it is known that this is the correct distribution class this knowledge is not very useful. The relevant question is whether or not the sample mean is e¢ cient when the form of the distribution is unknown. We call this setting semiparametric as the parameter of interest (the mean) is …nite dimensional while the remaining features of the distribution are unspeci…ed. In the semiparametric context an estimator is called semiparametrically e¢ cient if it has the smallest asymptotic variance among all semiparametric estimators. The mathematical trick is to reduce the semiparametric model to a set of parametric “submodels”. The Cramer-Rao variance bound can be found for each parametric submodel. The variance bound for the semiparametric model (the union of the submodels) is then de…ned as the supremum of the individual variance bounds. R Formally, suppose that the true density of y is the unknown function f (y) with mean = Ey = yf (y)dy: A parametric submodel for f (y) is a density f (y j ) which is a smooth function of a parameter , and there is a true value 0 such that f (y j 0 ) = f (y): The index indicates the submodels. The equality f (y j 0 ) = f (y) means that the submodel class passes through the true density, so the submodel is a true model. The class of submodels and R parameter 0 depend on the true density f: In the submodel f (y j ) ; the mean is ( ) = yf (y j ) dy which varies with the parameter . Let 2 @ be the class of all submodels for f: CHAPTER 5. AN INTRODUCTION TO LARGE SAMPLE ASYMPTOTICS 114 Since each submodel is parametric we can calculate the e¢ ciency bound for estimation of within this submodel. Speci…cally, given the density f (y j ) its likelihood score is S = @ log f (y j @ 0) ; 1 so the Cramer-Rao lower bound for estimation of is ES S 0 : De…ning M = @@ ( by Theorem B.11.5 the Cramer-Rao lower bound for estimation of within the submodel 0 0) 0; is 1 V = M ES S 0 M . As V is the e¢ ciency bound for the submodel class f (y j ) ; no estimator can have an asymptotic variance smaller than V for any density f (y j ) in the submodel class, including the true density f . This is true for all submodels : Thus the asymptotic variance of any semiparametric estimator cannot be smaller than V for any conceivable submodel. Taking the supremum of the Cramer-Rao bounds lower from all conceivable submodels we de…ne2 V = sup V : 2@ The asymptotic variance of any semiparametric estimator cannot be smaller than V , since it cannot be smaller than any individual V : We call V the semiparametric asymptotic variance bound or semiparametric e¢ ciency bound for estimation of , as it is a lower bound on the asymptotic variance for any semiparametric estimator. If the asymptotic variance of a speci…c semiparametric estimator equals the bound V we say that the estimator is semiparametrically e¢ cient. For many statistical problems it is quite challenging to calculate the semiparametric variance bound. However, in some cases there is a simple method to …nd the solution. Suppose that we can …nd a submodel 0 whose Cramer-Rao lower bound satis…es V 0 = V where V is the asymptotic variance of a known semiparametric estimator. In this case, we can deduce that V = V 0 = V . Otherwise there would exist another submodel 1 whose Cramer-Rao lower bound satis…es V 0 < V 1 but this would imply V < V 1 which contradicts the Cramer-Rao Theorem. We now …nd this submodel for the sample mean b : Our goal is to …nd a parametric submodel whose Cramer-Rao bound for is V : This can be done by creating a tilted version of the true density. Consider the parametric submodel 0 V where f (y) is the true density and = Ey: Note that Z Z f (y j ) dy = f (y)dy + 0 V 1 f (y j ) = f (y) 1 + 1 Z (y ) f (y) (y (5.17) ) dy = 1 and for all close to zero f (y j ) 0: Thus f (y j ) is a valid density function. It is a parametric submodel since f (y j 0 ) = f (y) when 0 = 0: This parametric submodel has the mean Z ( ) = yf (y j ) dy Z Z = yf (y)dy + f (y)y (y )0 V 1 dy = + which is a smooth function of : Since @ @ log f (y j ) = log 1 + @ @ 2 0 V 1 (y ) = V 1 (y ) 0 1 1 + V (y ) It is not obvious that this supremum exists, as V is a matrix so there is not a unique ordering of matrices. However, in many cases (including the ones we study) the supremum exists and is unique. CHAPTER 5. AN INTRODUCTION TO LARGE SAMPLE ASYMPTOTICS it follows that the score function for S = is @ log f (y j @ 0) =V By Theorem B.11.3 the Cramer-Rao lower bound for E S S0 1 115 = V 1 E (y 1 (y ): (5.18) is ) (y )0 V 1 1 =V: (5.19) The Cramer-Rao lower bound for ( ) = + is also V , and this equals the asymptotic variance of the moment estimator b : This was what we set out to show. In summary, we have shown that in the submodel (5.17) the Cramer-Rao lower bound for estimation of is V which equals the asymptotic variance of the sample mean. This establishes the following result. Proposition 5.13.1 In the class of distributions F 2 L2 ; the semiparametric variance bound for estimation of is V = var(yi ); and the sample mean b is a semiparametrically e¢ cient estimator of the population mean . We call this result a proposition rather than a theorem as we have not attended to the regularity conditions. It is a simple matter to extend this result to the plug-in estimator b = g (b ). We know from Theorem 5.10.4 that if E kyk2 < 1 and g (u) is continuously di¤erentiable at u = then the plugp d in estimator has the asymptotic distribution n b ! N (0; G0 V G) : We therefore consider the class of distributions n o L2 (g) = F : E kyk2 < 1; g (u) is continuously di¤erentiable at u = Ey : For example, if = 1 = 2 where 1 = Ey1 and 2 = Ey2 then L2 (g) = F : Ey12 < 1; Ey22 < 1; and Ey2 6= 0 : For any submodel the Cramer-Rao lower bound for estimation of = g ( ) is G0 V G by Theorem B.11.5. For the submodel (5.17) this bound is G0 V G which equals the asymptotic variance of b from Theorem 5.10.4. Thus b is semiparametrically e¢ cient. Proposition 5.13.2 In the class of distributions F 2 L2 (g) the semiparametric variance bound for estimation of = g ( ) is G0 V G; and the plug-in estimator b = g (b ) is a semiparametrically e¢ cient estimator of . The result in Proposition 5.13.2 is quite general. Smooth functions of sample moments are e¢ cient estimators for their population counterparts. This is a very powerful result, as most econometric estimators can be written (or approximated) as smooth functions of sample means. CHAPTER 5. AN INTRODUCTION TO LARGE SAMPLE ASYMPTOTICS 5.14 116 Technical Proofs* In this section we provide proofs of some of the more technical points in the chapter. These proofs may only be of interest to more mathematically inclined. Proof of Theorem 5.4.2: Without loss of generality, we can assume E(yi ) = 0 by recentering yi on its expectation. We need to show that for all > 0 and > 0 there is some N < 1 so that for all n N; Pr (jyj > ) : Fix and : Set " = =3: Pick C < 1 large enough so that E (jyi j 1 (jyi j > C)) " (5.20) (where 1 ( ) is the indicator function) which is possible since E jyi j < 1: De…ne the random variables wi = yi 1 (jyi j C) E (yi 1 (jyi j zi = yi 1 (jyi j > C) C)) E (yi 1 (jyi j > C)) so that y =w+z and E jyj E jwj + E jzj : (5.21) We now show that sum of the expectations on the right-hand-side can be bounded below 3": First, by the Triangle Inequality (A.12) and the Expectation Inequality (B.15), E jzi j = E jyi 1 (jyi j > C) E (yi 1 (jyi j > C))j E jyi 1 (jyi j > C)j + jE (yi 1 (jyi j > C))j 2E jyi 1 (jyi j > C)j 2"; (5.22) and thus by the Triangle Inequality (A.12) and (5.22) n E jzj = E n 1X zi n 1X E jzi j n i=1 2": (5.23) i=1 Second, by a similar argument jwi j = jyi 1 (jyi j jyi 1 (jyi j 2 jyi 1 (jyi j C) E (yi 1 (jyi j C)j + jE (yi 1 (jyi j C))j C))j C)j 2C (5.24) where the …nal inequality is (5.20). Then by Jensen’s Inequality (B.12), the fact that the wi are iid and mean zero, and (5.24), (E jwj)2 the …nal inequality holding for n together show that E jwj2 = 4C 2 ="2 = 36C 2 = E jyj as desired. Ewi2 4C 2 = n n 3"2 2 2 . "2 (5.25) Equations (5.21), (5.23) and (5.25) (5.26) CHAPTER 5. AN INTRODUCTION TO LARGE SAMPLE ASYMPTOTICS 117 Finally, by Markov’s Inequality (B.21) and (5.26), E jyj Pr (jyj > ) 3" = ; the …nal equality by the de…nition of ": We have shown that for any n 36C 2 = 2 2 ; Pr (jyj > ) ; as needed. > 0 and > 0 then for all Proof of Theorem 5.6.1: By Loève’s cr Inequality (A.19) 0 11=2 m m X X kyk = @ yj2 A jyj j : j=1 j=1 Thus if E jyj j < 1 for j = 1; :::; m; then E kyk m X j=1 E jyj j < 1: For the reverse inequality, the Euclidean norm of a vector is larger than the length of any individual component, so for any j; jyj j kyk : Thus, if E kyk < 1; then E jyj j < 1 for j = 1; :::; m: Proof of Theorem 5.7.1: The moment bound Ey 0i y i < 1 is su¢ cient to guarantee that the elements of and V are well de…ned and …nite. Without loss of generality, it is su¢ cient to consider the case = 0: p Our proof method is to calculate the characteristic function of ny n and show that it converges pointwise to the characteristic function of N (0; V ) : By Lévy’s Continuity Theorem (see Van der p Vaart (2008) Theorem 2.13) this is su¢ cient to established that ny n converges in distribution to N (0; V ) : For 2 Rm ; let C ( ) = E exp i 0 y i denote the characteristic function of y i and set c ( ) = log C( ). Since y i has two …nite moments the …rst and second derivatives of C( ) are continuous in : They are @ C( ) = iE y i exp i 0 y i @ @2 @ @ When evaluated at 0 C( ) = i2 E y i y 0i exp i 0 y i : =0 C(0) = 1 @ C(0) = iE (y i ) = 0 @ @2 @ @ 0 C(0) = E y i y 0i = V: Furthermore, c ( ) = c ( ) = so when evaluated at @ @ c( ) = C( ) 1 C( ) @ @ 2 @2 1 @ c( ) = C( ) C( ) @ @ 0 @ @ 0 =0 c(0) = 0 c (0) = 0 c (0) = V: C( ) 2 @ @ C( ) C( ) @ @ 0 CHAPTER 5. AN INTRODUCTION TO LARGE SAMPLE ASYMPTOTICS By a second-order Taylor series expansion of c( ) about c( ) = c(0) + c (0)0 + 1 2 0 c ( 118 = 0; ) = 1 2 0 c ( ) (5.27) where lies on the line segment joining 0 and : p p We now compute Cn ( ) = E exp i 0 ny n ; the characteristic function of ny n : By the properties of the exponential function, the independence of the y i ; the de…nition of c( ) and (5.27) ! n 1 X 0 log Cn ( ) = log E exp i p yi n i=1 1 exp i p n 0 yi 1 E exp i p n 0 yi 1 log E exp i p n 0 = log E = log n Y i=1 n Y i=1 = n X i=1 p = nc = 1 2 0 c yi n ( n) p where n lies on the line segment joining 0 and = n: Since c ( n ) ! c (0) = V: We thus …nd that as n ! 1; log Cn ( ) ! 1 2 0 1 2 0 and Cn ( ) ! exp n ! 0 and c ( ) is continuous, V V which is the characteristic function of the N (0; V ) distribution. This completes the proof. Proof of Theorem 5.9.1: Since g is continuous at c; for all " > 0 we can …nd a > 0 such that if kz n ck < then kg (z n ) g (c)k ": Recall that A B implies Pr(A) Pr(B): Thus p Pr (kg (z n ) g (c)k ") Pr (kz n ck < ) ! 1 as n ! 1 by the assumption that z n ! c: p Hence g(z n ) ! g(c) as n ! 1. Proof of Theorem 5.10.3: By a vector Taylor series expansion, for each element of g; gj ( where nj n) = gj ( ) + gj ( lies on the line segment between It follows that ajn = gj ( p jn ) n (g ( gj n) n and jn ) ( n ) and therefore converges in probability to : p ! 0: Stacking across elements of g; we …nd g( )) = (G + an )0 p n( d ) ! G0 : n p d The convergence is by Theorem 5.10.1, as G + an ! G; n ( continuous. This establishes (5.10) When N (0; V ) ; the right-hand-side of (5.28) equals d n G0 = G0 N (0; V ) = N 0; G0 V G (5.28) ) ! ; and their product is CHAPTER 5. AN INTRODUCTION TO LARGE SAMPLE ASYMPTOTICS 119 establishing (5.11). Proof of Theorem 5.12.1: First consider (5.12). Take any > 0: The event max1 i n jyi j > n1=r Sn 1=r 1=r ; which is the same as the event means that at least i j exceeds n i=1 jyi j > n Sn one ofr the jy r or equivalently i=1 fjyi j > ng : Since the probability of the union of events is smaller than the sum of the probabilities, ! n [ r 1=r r Pr n max jyi j > = Pr fjyi j > ng 1 i n i=1 n X i=1 Pr (jyi jr > n r ) n 1 X E (jyi jr 1 (jyi jr > n r )) n r i=1 1 r r E (jyi j = 1 (jyi jr > n r )) where the second inequality is the strong form of Markov’s inequality (Theorem B.22) and the …nal equality is since the yi are iid. Since E jyjr < 1 this …nal expectation converges to zero as n ! 1: This is because Z r E jyi j = jyjr dF (y) < 1 implies r r E (jyi j 1 (jyi j > c)) = as c ! 1: This establishes (5.12). Now consider (5.13). Take any Pr (log n) 1 r jyj >c jyjr dF (y) ! 0 (5.29) > 0 and set t = 1= : By a similar calculation max jyi j > 1 i n Z = Pr n [ i=1 n X i=1 ! fexp jtyi j > exp (t log n)g Pr (exp jtyi j > n) E (exp jtyj 1 (exp jtyj > n)) where the second line uses exp (t log n) = exp (log n) = n: The assumption E exp(ty) < 1 means E (exp jtyj 1 (exp jtyj > n)) ! 0 as n ! 1 by the same argument as in (5.29). This establishes (5.13). Chapter 6 Asymptotic Theory for Least Squares 6.1 Introduction It turns out that the asymptotic theory of least-squares estimation applies equally to the projection model and the linear CEF model, and therefore the results in this chapter will be stated for the broader projection model described in Section 2.17. Recall that the model is yi = x0i + ei for i = 1; :::; n; where the linear proejction is = E xi x0i 1 E (xi yi ) : Many of the results of this section hold under random sampling (Assumption 1.5.1) and …nite second moments (Assumption 2.17.1). We restate this conditions here for clarity. Assumption 6.1.1 1. The observations (yi ; xi ); i = 1; :::; n; are independent and identically distributed 2. Ey 2 < 1: 3. E kxk2 < 1: 4. Qxx = E (xx0 ) is positive de…nite. Some of the results will require a strengthening to …nite fourth moments. Assumption 6.1.2 In addition to Assumption 6.1.1, Eyi4 < 1 and E kxi k4 < 1: 120 CHAPTER 6. ASYMPTOTIC THEORY FOR LEAST SQUARES 6.2 121 Consistency of Least-Squares Estimation In this section we use the weak law of large numbers (WLLN, Theorem 5.4.2 and Theorem 5.6.2) and continuous mapping theorem (CMT, Theorem 5.9.1) to show that the least-squares estimator b is consistent for the projection coe¢ cient : This derivation is based on three key components. First, the OLS estimator can be written as a continuous function of a set of sample moments. Second, the WLLN shows that sample moments converge in probability to population moments. And third, the CMT states that continuous functions preserve convergence in probability. We now explain each step in brief and then in greater detail. First, observe that the OLS estimator ! ! 1 n n X X 1 1 b= b 1Q b xi x0i xi yi = Q xx xy n n i=1 i=1 b xx = 1 Pn xi x0 and Q b xy = 1 Pn xi yi : is a function of the sample moments Q i i=1 i=1 n n Second, by an application of the WLLN these sample moments converge in probability to the population moments. Speci…cally, the fact that (yi ; xi ) are mutually independent and identically distributed implies that any function of (yi ; xi ) is iid, including xi x0i and xi yi : These variables also have …nite expectations by Theorem 2.17.1.1. Under these conditions, the WLLN (Theorem 5.6.2) implies that as n ! 1; n X p b xx = 1 Q (6.1) xi x0i ! E xi x0i = Qxx n i=1 and n X p b xy = 1 xi yi ! E (xi yi ) = Qxy : Q n (6.2) i=1 Third, the CMT ( Theorem 5.9.1) allows us to combine these equations to show that b converges in probability to : Speci…cally, as n ! 1; b=Q b 1Q b xx xy p ! Qxx1 Qxy = : (6.3) p We have shown that b ! , as n ! 1: In words, the OLS estimator converges in probability to the projection coe¢ cient vector as the sample size n gets large. To fully understand the application of the CMT we walk through it in detail. We can write b=g Q b xx ; Q b xy where g (A; b) = A 1 b is a function of A and b: The function g (A; b) is a continuous function of A and b at all values of the arguments such that A 1 exists. Assumption 2.17.1 implies that Qxx1 exists and thus g (A; b) is continuous at A = Qxx : This justi…es the application of the CMT in (6.3). For a slightly di¤erent demonstration of (6.3), recall that (4.7) implies that where b b 1Q b =Q xx xe n X b xe = 1 Q xi ei : n i=1 (6.4) CHAPTER 6. ASYMPTOTIC THEORY FOR LEAST SQUARES 122 The WLLN and (2.27) imply b xe p! E (xi ei ) = 0: Q Therefore (6.5) b 1Q b =Q xx xe b p ! Qxx1 0 =0 p which is the same as b ! . Theorem 6.2.1 Consistency of Least-Squares b xx p! Qxx ; Q b xy p! Qxy ; Q b 1 Under Assumption 6.1.1, Q xx p p b b Q ! 0; and ! as n ! 1: p ! Qxx1 ; xe Theorem 6.2.1 states that the OLS estimator b converges in probability to as n increases, and thus b is consistent for . In the stochastic order notation, Theorem 6.2.1 can be equivalently written as b = + op (1): (6.6) To illustrate the e¤ect of sample size on the least-squares estimator consider the least-squares regression ln(W agei ) = 0 + 1 Educationi + 2 Experiencei + 2 3 Experiencei + ei : We use the sample of 30,833 white men from the March 2009 CPS. Randomly sorting the observations, and sequentially estimating the model by least-squares, starting with the …rst 40 observations, and continuing until the full sample is used, the sequence of estimates are displayed in Figure 6.1. You can see how the least-squares estimate changes with the sample size, but as the number of observations increases it settles down to the full-sample estimate ^ 1 = 0:114: 6.3 Asymptotic Normality We started this chapter discussing the need for an approximation to the distribution of the OLS estimator b : In Section 6.2 we showed that b converges in probability to . Consistency is a good …rst step, but in itself does not describe the distribution of the estimator. In this section we derive an approximation typically called the asymptotic distribution. The derivation starts by writing the estimator as a function of sample moments. One of the moments must be written as a sum of zero-mean random vectors and normalized so that the central limit theorem can be applied. The steps are as follows. p Take equation (6.4) and multiply it by n: This yields the expression ! 1 ! n n X X p 1 1 0 p n b = xi xi xi ei : (6.7) n n i=1 i=1 p This shows that the normalized and centered estimator n b is a function of the sample Pn 1 Pn 1 0 average n i=1 xi xi and the normalized sample average pn i=1 xi ei : Furthermore, the latter has mean zero so the central limit theorem (CLT, Theorem 5.7.1) applies. 123 0.12 0.11 0.08 0.09 0.10 OLS Estimation 0.13 0.14 0.15 CHAPTER 6. ASYMPTOTIC THEORY FOR LEAST SQUARES 5000 10000 15000 20000 Number of Observations Figure 6.1: The least-squares estimator ^ 1 as a function of sample size n The product xi ei is iid (since the observations are iid) and mean zero (since E (xi ei ) = 0): De…ne the k k covariance matrix = E xi x0i e2i : (6.8) We require the elements of to be …nite, written k k < 1: By the Expectation Inequality (B.15), E xi x0i e2i = E kxi ei k2 = E kxi k2 e2i or equivalently that E kxi ei k2 < 1: Usingkxi ei k2 = kxi k2 e2i and the Cauchy-Schwarz Inequality (B.17), k k E xi x0i e2i = E kxi ei k2 = E kxi k2 e2i E kxi k4 1=2 Ee4i 1=2 (6.9) which is …nite if xi and ei have …nite fourth moments. As ei is a linear combination of yi and xi ; it is su¢ cient that the observables have …nite fourth moments (Theorem 2.17.1.6). We can then apply the CLT (Theorem 5.7.1). Theorem 6.3.1 Under Assumption 6.1.2, k k and E kxi ei k2 < 1 (6.10) n 1 X d p xi ei ! N (0; ) n i=1 as n ! 1: Putting together (6.1), (6.7), and (6.11), p n b d ! Qxx1 N (0; ) = N 0; Qxx1 Qxx1 (6.11) CHAPTER 6. ASYMPTOTIC THEORY FOR LEAST SQUARES 124 as n ! 1; where the …nal equality follows from the property that linear combinations of normal vectors are also normal (Theorem B.9.1). We have derived the asymptotic normal approximation to the distribution of the least-squares estimator. Theorem 6.3.2 Asymptotic Normality of Least-Squares Estimator Under Assumption 6.1.2, as n ! 1 p where d n b ! N (0; V ) = Qxx1 Qxx1 ; V Qxx = E (xi x0i ) ; and (6.12) = E xi x0i e2i : In the stochastic order notation, Theorem 6.3.2 implies that b= + Op (n 1=2 ) (6.13) which is stronger than (6.6). p The matrix V = avar( b ) is the variance of the asymptotic distribution of n b : Consequently, V is often referred to as the asymptotic covariance matrix of b : The expression V = Qxx1 Qxx1 is called a sandwich form. It might be worth noticing that there is a di¤erence between the variance of the asymptotic distribution given in (6.12) and the …nite-sample conditional variance in the CEF model as given in (4.12): V While V and V b b = 1 0 XX n 1 1 0 X DX n 1 0 XX n 1 : are di¤erent, the two are close if n is large. Indeed, as n ! 1 V There is a special case where jection Error when and V p b !V : simplify. We say that ei is a Homoskedastic Pro- cov(xi x0i ; e2i ) = 0: (6.14) Condition (6.14) holds in the homoskedastic linear regression model, but is somewhat broader. Under (6.14) the asymptotic variance formulas simplify as = E xi x0i E e2i = Qxx V = Qxx1 Qxx1 = Qxx1 2 2 V (6.15) 0 (6.16) In (6.16) we de…ne V 0 = Qxx1 2 whether (6.14) is true or false. When (6.14) is true then V = V 0 ; otherwise V 6= V 0 : We call V 0 the homoskedastic asymptotic covariance matrix. Theorem 6.3.2 states that the sampling distribution of the least-squares estimator, after rescaling, is approximately normal when the sample size n is su¢ ciently large. This holds true for all joint distributions of (yi ; xi ) which satisfy the conditions of Assumption 6.1.2, and is therefore broadly applicable. Consequently, asymptotic normality is routinely used to approximate the …nite sample p : distribution of n b CHAPTER 6. ASYMPTOTIC THEORY FOR LEAST SQUARES 125 Figure 6.2: Density of Normalized OLS estimator with Double Pareto Error A di¢ culty is that for any …xed n the sampling distribution of b can be arbitrarily far from the normal distribution. In Figure 5.1 we have already seen a simple example where the least-squares estimate is quite asymmetric and non-normal even for reasonably large sample sizes. The normal approximation improves as n increases, but how large should n be in order for the approximation to be useful? Unfortunately, there is no simple answer to this reasonable question. The trouble is that no matter how large is the sample size, the normal approximation is arbitrarily poor for some data distribution satisfying the assumptions. We illustrate this problem using a simulation. Let yi = 1 xi + 2 + ei where xi is N (0; 1) ; and ei is independent of xi with the Double Pareto 1 density f (e) = 2 jej ; jej 1: If > 2 the error ei has zero mean and variance =( 2): As approaches 2, however, q its variance diverges to in…nity. In this context the normalized leastsquares slope estimator n 2 ^ 1 > 2. 1 has the N(0; 1) asymptotic distibution for any q In Figure 6.2 we display the …nite sample densities of the normalized estimator n 2 ^ 1 1 ; setting n = 100 and varying the parameter . For = 3:0 the density is very close to the N(0; 1) density. As diminishes the density changes signi…cantly, concentrating most of the probability mass around zero. Another example is shown in Figure 6.3. Here the model is yi = + ei where uki ei = E u2k i E uki E uki 2 1=2 (6.17) p setting n = 100; for k = 1; 4, and ui N(0; 1): We show the sampling distribution of n b 6 and 8. As k increases, the sampling distribution becomes highly skewed and non-normal. The lesson from Figures 6.2 and 6.3 is that the N(0; 1) asymptotic approximation is never guaranteed to be accurate. 6.4 Joint Distribution Theorem 6.3.2 gives the joint asymptotic distribution of the coe¢ cient estimates. We can use the result to study the covariance between the coe¢ cient estimates. For example, suppose k = 2 CHAPTER 6. ASYMPTOTIC THEORY FOR LEAST SQUARES 126 Figure 6.3: Density of Normalized OLS estimator with error process (6.17) and write the estimates as ( ^ 1 ; ^ 2 ): For simplicity suppose that the regressors are mean zero. Then we can write 2 1 Qxx = 1 2 2 2 1 2 where 21 and 22 are the variances of x1i and x2i ; and is their correlation. If the error is homoskedastic, then the asymptotic variance matrix for ( ^ 1 ; ^ 2 ) is V 0 = Qxx1 2 : By the formula for inversion of a 2 2 matrix, Qxx1 = 2 2 1 2 2 (1 1 2 2) 1 2 1 2 2 1 : Thus if x1i and x2i are positively correlated ( > 0) then ^ 1 and ^ 2 are negatively correlated (and vice-versa). For illustration, Figure 6.4 displays the probability contours of the joint asymptotic distribution 2 2 2 = 1 and = 0:5: The coe¢ cient estimates are negatively ^ of ^ 1 1 and 2 2 when 1 = 2 = correlated since the regressors are positively correlated. This means that if ^ 1 is unusually negative, it is likely that ^ 2 is unusually positive, or conversely. It is also unlikely that we will observe both ^ and ^ unusually large and of the same sign. 1 2 This …nding that the correlation of the regressors is of opposite sign of the correlation of the coef…cient estimates is sensitive to the assumption of homoskedasticity. If the errors are heteroskedastic then this relationship is not guaranteed. This can be seen through a simple constructed example. Suppose that x1i and x2i only take the values f 1; +1g; symmetrically, with Pr (x1i = x2i = 1) = Pr (x1i = x2i = 1) = 3=8; and Pr (x1i = 1; x2i = 1) = Pr (x1i = 1; x2i = 1) = 1=8: You can check that the regressors are mean zero, unit variance and correlation 0.5, which is identical with the setting displayed in Figure 6.4. Now suppose that the error is heteroskedastic. Speci…cally, suppose that E e2i j x1i = x2i = 5 1 and E e2i j x1i 6= x2i = : You can check that E e2i = 1; E x21i e2i = E x22i e2i = 1 and 4 4 CHAPTER 6. ASYMPTOTIC THEORY FOR LEAST SQUARES 127 Figure 6.4: Contours of Joint Distribution of ( ^ 1 ; ^ 2 ); homoskedastic case 7 E x1i x2i e2i = : Therefore 8 V = Qxx1 Qxx1 2 32 1 9 6 1 6 1 2 7 = 4 1 54 7 16 1 2 8 2 3 1 46 1 4 7 = 4 1 5: 3 1 4 32 7 6 1 8 7 54 1 1 2 3 1 2 7 5 1 Thus the coe¢ cient estimates ^ 1 and ^ 2 are positively correlated (their correlation is 1=4:) The joint probability contours of their asymptotic distribution is displayed in Figure 6.5. We can see how the two estimates are positively associated. What we found through this example is that in the presence of heteroskedasticity there is no simple relationship between the correlation of the regressors and the correlation of the parameter estimates. We can extend the above analysis to study the covariance between coe¢ cient sub-vectors. For example, partitioning x0i = (x01i ; x02i ) and 0 = 01 ; 02 ; we can write the general model as yi = x01i 1 + x02i 2 + ei 0 0 0 and the coe¢ cient estimates as b = b 1 ; b 2 : Make the partitions Qxx = Q11 Q12 Q21 Q22 ; = 11 12 21 22 From (2.41) Qxx1 = Q1112 Q2211 Q21 Q111 Q1112 Q12 Q221 Q2211 : (6.18) CHAPTER 6. ASYMPTOTIC THEORY FOR LEAST SQUARES 128 Figure 6.5: Contours of Joint Distribution of ^ 1 and ^ 2 ; heteroskedastic case where Q11 2 = Q11 moskedastic, Q12 Q221 Q21 and Q22 1 = Q22 cov b 1 ; b 2 = 2 Q21 Q111 Q12 . Thus when the error is ho- Q1112 Q12 Q221 which is a matrix generalization of the two-regressor case. In the general case, you can show that (Exercise 6.5) V = Q12 Q221 21 V 11 V 12 V 21 V 22 (6.19) where V 11 = Q1112 V 21 = V 22 = Q2211 Q2211 11 21 22 Q21 Q111 Q21 Q111 11 12 + Q12 Q221 1 12 Q22 Q21 1 22 Q22 Q21 1 21 Q11 Q12 + + Q21 Q111 Q21 Q111 1 22 Q22 Q21 1 12 Q22 Q21 1 11 Q11 Q12 Q1112 (6.20) Q1112 Q2211 (6.21) (6.22) Unfortunately, these expressions are not easily interpretable. 6.5 Consistency of Error Variance Estimators 2 1 Pn ^2i and s2 = Using the methods of Section 6.2 we can show that the estimators ^ = i=1 e n P n 1 ^2i are consistent for 2 : i=1 e n k The trick is to write the residual e^i as equal to the error ei plus a deviation term e^i = yi x0i b = ei + x0i = ei x0i b x0i b : Thus the squared residual equals the squared error plus a deviation e^2i = e2i 2ei x0i b + b 0 xi x0i b : (6.23) CHAPTER 6. ASYMPTOTIC THEORY FOR LEAST SQUARES 129 So when we take the average of the squared residuals we obtain the average of the squared errors, plus two terms which are (hopefully) asymptotically negligable. ! ! n n n X 0 1X 2 1X 1 2 0 0 b b ^ = ei 2 ei xi xi xi + b : (6.24) n n n i=1 i=1 i=1 Indeed, the WLLN shows that n 1X 2 p ei ! n 1 n i=1 n X i=1 2 p ei x0i ! E ei x0i = 0 n 1X p xi x0i ! E xi x0i = Qxx n i=1 p and Theorem 6.2.1 shows that b ! . Hence (6.24) converges in probability to Finally, since n=(n k) ! 1 as n ! 1; it follows that s2 = n n p k ^2 ! 2 2; as desired. : Thus both estimators are consistent. Theorem 6.5.1 Under Assumption 6.1.1, ^ 2 n ! 1: 6.6 p ! 2 and s2 p ! 2 as Homoskedastic Covariance Matrix Estimation Theorem 6.3.2 describe the asymptotic covariance matrix of the least-squares estimators b . For asymptotic inference (con…dence intervals and tests) we need a consistent estimate of its covariance matrix. In this section we start with the simpli…ed problem of estimating V 0 = Qxx1 2 ; the asymptotic variance of b under conditional homoskedasticity: As we described in Section 4.10, 0 b 1 s2 where Q b xx and s2 are de…ned in (6.1) and (4.20). the conventional estimator is Vb b = Q xx We now show that this estimator is consistent for V 0 : Since the estimator is the product of two moment estimates, the method is to show consistency of each moment estimator, and then apply the continuous mapping theorem to the product. b 1 p! Q 1 ; and Theorem 6.5.1 established s2 p! 2 : It Theorem 6.2.1 established that Q xx xx follows by the CMT that 0 b 1 s2 p! Q 1 2 = V 0 Vb b = Q xx xx 0 so that Vb b is consistent for V 0 ; as desired. 0 p Theorem 6.6.1 Under Assumption 6.1.1, Vb b ! V 0 as n ! 1: It is instructive to notice that Theorem 6.6.1 does not require the assumption of homoskedastic0 ity. That is, Vb b is consistent for V 0 regardless if the regression is homoskedastic or heteroskedastic. 0 However, V 0 = V = avar( b ) only under homoskedasticity. Thus in the general case, Vb b is consistent for a well-de…ned but non-useful object. CHAPTER 6. ASYMPTOTIC THEORY FOR LEAST SQUARES 6.7 130 Heteroskedastic Covariance Matrix Estimation Theorems 6.3.2 established that the asymptotic variance of b is V = Qxx1 Qxx1 : We now consider estimation of this covariance matrix without imposing homoskedasticity. The standard approach is to use a plug-in estimator which replace the unknowns with sample moments. The moment estimator for is n X b = 1 xi x0i e^2i ; (6.25) n i=1 leading to the plug-in covariance matrix estimator Vb b 1 bQ b 1: =Q xx xx (6.26) You can check that this is identical to the White covariance matrix estimator Vb b introduced in (4.28). Here we write the estimator as Vb to indicate that it is an estimate of V = avar( b ): We will use both Vb b and Vb to indicate (6.26). b 1 p! Q 1 ; so we just need to verify the consistency of b . The As shown in Theorem 6.2.1, Q xx xx key is to write the replace the squared residual e^2i with the squared error e2i ; and then show that the di¤erence is asymptotically negligible. Speci…cally, observe that n b = 1X xi x0i e^2i n i=1 = n n 1X 1X 0 2 xi xi ei + xi x0i e^2i n n i=1 e2i : (6.27) i=1 The …rst term is an average of the iid random variables xi x0i e2i ; and therefore by the WLLN converges in probability to its expectation, namely, n 1X p xi x0i e2i ! E xi x0i e2i = n : i=1 Technically, this requires that has …nite elements, which was shown in (6.10). So to establish that b is consistent for it remains to show that n 1X p xi x0i e^2i e2i ! 0: n (6.28) i=1 There are multiple ways to do this. A reasonable straightforward yet slightly tedious derivation is to start by applying the Triangle Inequality (A.12) n 1X xi x0i e^2i n n 1X xi x0i e^2i n e2i i=1 = 1 n i=1 n X i=1 kxi k2 e^2i e2i e2i : (6.29) Then recalling the expression for the squared residual (6.23), apply the Triangle Inequality and then the Schwarz Inequality (A.10) twice e^2i e2i 2 ei x0i b = 2 jei j x0i b 2 jei j kxi k b + b + 0 b xi x0i b + kxi k2 b 0 2 xi 2 : (6.30) CHAPTER 6. ASYMPTOTIC THEORY FOR LEAST SQUARES Combining (6.29) and (6.30), we …nd n 1X xi x0i e^2i n e2i 2 i=1 ! n 1X b kxi k3 jei j n 131 n + i=1 = op (1): 1X kxi k4 n i=1 ! b 2 (6.31) p The expression is op (1) because b ! 0; and both averages in parenthesis are averages of random variables with …nite mean under Assumption ??. Indeed, by Hölder’s Inequality (B.16) E kxi k3 jei j E kxi k3 3=4 4=3 Ee4i 1=4 = E kxi k4 3=4 Ee4i 1=4 <1 We have established (6.28), as desired. Theorem 6.7.1 Under Assumption 6.1.2, as n ! 1; b p Vb b ! V : 6.8 p ! and Alternative Covariance Matrix Estimators* In Section 4.11 we also introduced the alternative heteroskedasticity-robust covariance matrix estimators Ve b ; and V b which take the form (6.26) but with b replaced by n X e = 1 (1 n hii ) 2 xi x0i e^2i hii ) 1 xi x0i e^2i ; i=1 and n 1X = (1 n i=1 p respectively. To show that these estimators also consistent for V ; given b ! , it is su¢ cient b converge in probability to zero as n ! 1: to show that the di¤erences e b and The trick is to use the fact that the leverage values are asymptotically negligible: max hii = op (1): 1 i n (See Theorem 6.20.1 in Section 6.20).) Then using the Triangle Inequality and (6.32) n b 1X xi x0i e^2i (1 hii ) 1 1 n i=1 ! n 1X hii kxi k2 e^2i max 1 i n 1 n hii i=1 = op (1): Similarly, n e b 1X xi x0i e^2i (1 hii ) 2 1 n i=1 ! n 1X 2hii h2ii 2 2 max kxi k e^i 1 i n (1 n hii )2 i=1 = op (1): (6.32) CHAPTER 6. ASYMPTOTIC THEORY FOR LEAST SQUARES Theorem 6.8.1 Under Assumption 6.1.2, as n ! 1; e p p !V : Ve ! V ; and V 6.9 132 p ! ; p ! ; Functions of Parameters Sometimes we are interested in a transformation of the coe¢ cient vector = ( 1 ; :::; k ): For example, we may be interested in a single coe¢ cient j ; or a ratio j = l : In these cases we can write the transformation as a function of the coe¢ cients, e.g. = h( ) for some function h : Rk ! Rq . The estimate of is b = h( b ): p By the continuous mapping theorem (Theorem 5.9.1) and the fact b ! b is consistent for . we can deduce that Theorem 6.9.1 Under Assumption 6.1.1, if h( ) is continuous at the p true value of ; then as n ! 1; b ! : Furthermore, by the Delta Method (Theorem 5.10.3) we know that b is asymptotically normal. Assumption 6.9.1 h( ) : Rk ! Rq is continuously di¤ erentiable at the true value of and H = @@ h( )0 has rank q: Theorem 6.9.2 Asymptotic Distribution of Functions of Parameters Under Assumptions 6.1.2 and 6.9.1, as n ! 1; p where d n b V ! N (0; V ) = H0 V H : (6.33) (6.34) In many cases, the function h( ) is linear: h( ) = R0 for some k q matrix R: In this case, H = R. In particular, if R is a “selector matrix” R= I 0 (6.35) CHAPTER 6. ASYMPTOTIC THEORY FOR LEAST SQUARES then we can conformably partition V =( = 0 1; I 0 0 0 2) so that R0 I 0 V = 1 for 133 =( 0 1; 0 0 2) : Then = V 11 ; the upper-left sub-matrix of V 11 given in (6.20). In this case (6.33) states that p d n b1 ! N (0; V 11 ) : 1 That is, subsets of b are approximately normal with variances given by the comformable subcomponents of V . To illustrate the case of a nonlinear transformation, take the example = j = l for 6= l: Then 0 1 0 B C .. B C . B C B 1= l C B C B C .. H =B (6.36) C . B C 2 B C j= l C B B C .. @ A . 0 so = V jj = V 2 l + V ll 2 4 j= l 2V jl 3 j= l where V ab denotes the ab’th element of V : For inference we need an estimate of the asymptotic variance matrix V = H 0 V H , and for this it is typical to use a plug-in estimator. The natural estimator of H is the derivative evaluated at the point estimates c = @ h( b )0 : H (6.37) @ The derivative in (6.37) may be calculated analytically or numerically. By analytically, we mean working out for the formula for the derivative and replacing the unknowns by point estimates. For example, if = j = l ; then @@ h( ) is (6.36). However in some cases the function h( ) may be extremely complicated and a formula for the analytic derivative may not be easily available. In this case calculation by numerical di¤erentiation may be preferable. Let l = (0 1 0)0 be the c is unit vector with the “1” in the l’th place. Then the jl’th element of a numerical derivative H for some small ": The estimate of V b c jl = hj ( + l ") H " is hj ( b ) c 0 Vb H c : Vb = H (6.38) 0 Alternatively, Vb b ; Ve or V may be used in place of Vb : Given (6.37), (6.38) is simple to calculate using matrix operations. As the primary justi…cation for Vb is the asymptotic approximation (6.33), Vb is often called an asymptotic covariance matrix estimator. p The estimator Vb is consistent for V under the conditions of Theorem 6.9.2 since Vb b ! V by Theorem 6.7.1, and c = @ h( b )0 p! @ h( )0 = H H @ @ p since b ! and the function @ h( )0 is continuous. @ CHAPTER 6. ASYMPTOTIC THEORY FOR LEAST SQUARES 134 Theorem 6.9.3 Under Assumptions 6.1.2 and 6.9.1, as n ! 1; p Vb 6.10 !V : Asymptotic Standard Errors As described in Section 4.12, a standard error is an estimate of the standard deviation of the p distribution of an estimator. Thus if Vb is an estimate of the asymptotic covariance of n b ; then n 1 Vb is an estimate of the variance of b ; and standard errors are the square roots of the diagonal elements of this matrix. These take the form rh q i 1=2 1 ^ b s( j ) = n V j = n Vb : jj When the justi…cation for Vb is based on asymptotic theory we call s( ^ j ) an asymptotic standard error for ^ j : Standard errors for b are constructed similarly. Supposing that q = 1 (so h( ) is real-valued), then the asymptotic standard error for ^ is the square root of n 1 Vb ; that is, q q 1=2 c 0 b c 1=2 ^ b V =n H V H : s( ) = n When calculating and reporting coe¢ cients estimates b or estimates b which are transformations of the original coe¢ cient estimates, it is good practice to report standard errors for each reported estimate. This helps users of the work assess the estimation precision. 6.11 t statistic Let = h( ) : Rk ! R be any parameter of interest (for example, could be a single element of ), b its estimate and s(b) its asymptotic standard error. Consider the statistic tn ( ) = b s(b) : (6.39) Di¤erent writers have called (6.39) a t-statistic, a t-ratio, a z-statistic or a studentized statistic, sometimes using the di¤erent labels to distinguish between …nite-sample and asymptotic inference. As the statistics themselves are always (6.39) we won’t make such as distinction, and will simply refer to tn ( ) as a t-statistic or a t-ratio. We also often suppress the parameter dependence, writing it as tn : The t-statistic is a simple function of the estimate, its standard error, and the parameter. p p d By Theorem 6.9.2, n b ! N (0; V ) and Vb^ ! V : Thus tn ( ) = b s(b) p b n q = Vbb N (0; V ) p V = Z N (0; 1) : d ! CHAPTER 6. ASYMPTOTIC THEORY FOR LEAST SQUARES 135 The last equality is by the property that linear scales of normal distributions are normal. Thus the asymptotic distribution of the t-ratio tn ( ) is the standard normal. Since this distribution does not depend on the parameters, we say that tn ( ) is asymptotically pivotal. In special cases (such as the normal regression model, see Section 3.17), the statistic tn has an exact t distribution, and is therefore exactly free of unknowns. In this case, we say that tn is exactly pivotal. In general, however, pivotal statistics are unavailable and we must rely on asymptotically pivotal statistics. As we will see in the next section, it is also useful to consider the distribution of the absolute d d t-ratio jtn ( )j : Since tn ( ) ! Z, the continuous mapping theorem yields jtn ( )j ! jZj : Letting (u) = Pr (Z u) denote the standard normal distribution function, we can calculate that the distribution function of jZj is Pr (jZj u) = Pr ( u = Pr (Z = (u) = 2 (u) := (u) Z u) u) Pr (Z < u) ( u) 1 Theorem 6.11.1 Under Assumptions 6.1.2 and 6.9.1, tn ( ) d N (0; 1) and jtn ( )j ! jZj (6.40) d ! Z The asymptotic normality of Theorem 6.11.1 is used to justify con…dence intervals and tests for the parameters. 6.12 Con…dence Intervals The OLS estimate b is a point estimate for , meaning that b is a single value in Rk . A broader concept is a set estimate Cn which is a collection of values in Rk : When the parameter is real-valued then it is common to focus on intervals Cn = [Ln ; Un ] and which is called an interval estimate for . The goal of an interval estimate Cn is typically to contain the true value, e.g. 2 Cn ; with high probability, yet without being too big. The interval estimate Cn is a function of the data and hence is random. The coverage probability of the interval Cn = [Ln ; Un ] is Pr ( 2 Cn ): The randomness comes from Cn as the parameter is treated as …xed. Interval estimates Cn are typically called con…dence intervals as the goal is typically to set the coverage probability to equal a pre-speci…ed target, typically 90% or 95%. Cn is called a (1 )% con…dence interval if inf Pr ( 2 Cn ) = 1 : There is not a unique method to construct con…dence intervals. For example, a simple (yet silly) interval is R with probability 1 Cn = b with probability By construction, if b has a continuous distribution, Pr( 2 Cn ) = 1 ; so this con…dence interval has perfect coverage, but Cn is uninformative about and is therefore not useful. When we have an asymptotically normal parameter estimate b with standard error s(b); the standard con…dence interval for takes the form h i Cn = b c s(b); b + c s(b) (6.41) CHAPTER 6. ASYMPTOTIC THEORY FOR LEAST SQUARES 136 where c > 0 is a pre-speci…ed constant. This con…dence interval is symmetric about the point estimate b; and its length is proportional to the standard error s(b): Equivalently, Cn is the set of parameter values for such that the t-statistic tn ( ) is smaller (in absolute value) than c; that is ( ) b Cn = f : jtn ( )j cg = : c c : s(b) The coverage probability of this con…dence interval is Pr ( 2 Cn ) = Pr (jtn ( )j c) which is generally unknown. We can approximate the coverage probability by taking the asymptotic limit as n ! 1: Since jtn ( )j is asymptotically jZj (Theorem 6.11.1), it follows that as n ! 1 that c) = (c) Pr ( 2 Cn ) ! Pr (jZj where (u) is given in (6.40). We call this the asymptotic coverage probability. Since the tratio is asymptotically pivotal, the asymptotic coverage probability is independent of the parameter ; and is only a function of c: As we mentioned before, an ideal con…dence interval has a pre-speci…ed probability coverage 1 ; typically 90% or 95%. This means selecting the constant c so that (c) = 1 : E¤ectively, this makes c a function of ; and can be backed out of a normal distribution table. For example, = 0:05 (a 95% interval) implies c = 1:96 and = 0:1 (a 90% interval) implies c = 1:645: Rounding 1.96 to 2, we obtain the most commonly used con…dence interval in applied econometric practice h i Cn = b 2s(b); b + 2s(b) : (6.42) This is a useful rule-of thumb. This asymptotic 95% con…dence interval Cn is simple to compute and can be roughly calculated from tables of coe¢ cient estimates and standard errors. (Technically, it is an asymptotic 95.4% interval, due to the substitution of 2.0 for 1.96, but this distinction is meaningless.) Theorem 6.12.1 Under Assumptions 6.1.2 and 6.9.1, for Cn de…ned in (6.41), Pr ( 2 Cn ) ! (c): For c = 1:96; Pr ( 2 Cn ) ! 0:95: Con…dence intervals are a simple yet e¤ective tool to assess estimation uncertainty. When reading a set of empirical results, look at the estimated coe¢ cient estimates and the standard errors. For a parameter of interest, compute the con…dence interval Cn and consider the meaning of the spread of the suggested values. If the range of values in the con…dence interval are too wide to learn about ; then do not jump to a conclusion about based on the point estimate alone. 6.13 Regression Intervals In the linear regression model the conditional mean of yi given xi = x is m(x) = E (yi j xi = x) = x0 : CHAPTER 6. ASYMPTOTIC THEORY FOR LEAST SQUARES 137 Figure 6.6: Wage on Education Regression Intervals In some cases, we want to estimate m(x) at a particular point x: Notice that this is a (linear) 0 function b = b = x0 b and H = x; so qof : Letting h( ) = x and = h( ); we see that m(x) s(b) = n 1 x0 Vb x: Thus an asymptotic 95% con…dence interval for m(x) is 0b x q 2 n 1 x0 V b x : It is interesting to observe that if this is viewed as a function of x; the width of the con…dence set is dependent on x: To illustrate, we return to the log wage regression (3.11) of Section 3.6. The estimated regression equation is \ log(W age) = x0 b = 0:626 + 0:156x: where x = dducation. The White covariance matrix estimate is Vb b = 7:092 0:445 0:445 0:029 and the sample size is n = 61: Thus the 95% con…dence interval for the regression takes the form r 1 0:626 + 0:156x 2 (7:092 0:89x + 0:029x2 ) : 61 The estimated regression and 95% intervals are shown in Figure 6.6. Notice that the con…dence bands take a hyperbolic shape. This means that the regression line is less precisely estimated for very large and very small values of education. Plots of the estimated regression line and con…dence intervals are especially useful when the regression includes nonlinear terms. To illustrate, consider the log wage regression (3.12) which includes experience and its square. \ log(W age) = 1:06 + 0:116 education + 0:010 experience 0:014 experience2 =100 (6.43) and has n = 2454 observations. We are interested in plotting the regression estimate and regression intervals as a function of experience. Since the regression also includes education, to plot the CHAPTER 6. ASYMPTOTIC THEORY FOR LEAST SQUARES 138 Figure 6.7: Wage on Experience Regression Intervals estimates in a simple graph we need to …x education at a speci…c value. We select education=12. This only a¤ects the level of the estimated regression, since education enters without an interaction. De…ne the points of evaluation 0 1 1 B 12 C C z(x) = B @ A x 2 x =100 where x =experience. The covariance matrix estimate is 0 22:92 1:0601 0:56687 B 1:0601 0:06454 :0080737 Vb b = B @ 0:56687 :0080737 :040736 0:86626 :0066749 :075583 1 0:86626 :0066749 C C: :075583 A 0:14994 Thus the regression interval for education=12, as a function of x =experience is 1:06 + 0:116 12 + 0:010 x 0:014 x2 =100 v 0 u 22:92 1:0601 0:56687 u u B 1:0601 1u 1 0:06454 :0080737 u z(x)0 B @ 0:56687 :0080737 :040736 50 t 2454 0:86626 :0066749 :075583 = 2:452 + 0:010 x :00014 x2 2 p 27:592 3:8304 x + 0:23007 x2 100 1 0:86626 :0066749 C C z(x) :075583 A 0:14994 0:00616 x3 + 0:0000611 x4 The estimated regression and 95% intervals are shown in Figure 6.7. The regression interval widens greatly for small and large values of experience, indicating considerable uncertainty about the e¤ect of experience on mean wages for this population. The con…dence bands take a more complicated shape than in Figure 6.6 due to the nonlinear speci…cation. CHAPTER 6. ASYMPTOTIC THEORY FOR LEAST SQUARES 6.14 139 Forecast Intervals For a given value of xi = x; we may want to forecast (guess) yi out-of-sample. A reasonable rule is the conditional mean m(x) as it is the mean-square-minimizing forecast. A point forecast is the estimated conditional mean m(x) ^ = x0 b . We would also like a measure of uncertainty for the forecast. The forecast error is e^i = yi m(x) ^ = ei x0 b : As the out-of-sample error ei is independent of the in-sample estimate ^ ; this has variance E^ e2i = E e2i j xi = x + x0 E b 2 = (x) + n b 1 0 x V x: 0 x the natural estimate of this variance is ^ 2 + n 1 x0 Vb x; so a standard q error for the forecast is s^(x) = ^ 2 + n 1 x0 Vb x: Notice that this is di¤erent from the standard error for the conditional mean. If we have an estimate of the conditional variance function, e.g. q 0 2 2 ~ (x) = e z from (9.5), then the forecast standard error is s^(x) = ~ (x) + n 1 x0 Vb x Assuming E e2i j xi = 2; It would appear natural to conclude that an asymptotic 95% forecast interval for yi is h i x0 b 2^ s(x) ; but this turns out to be incorrect. In general, the validity of an asymptotic con…dence interval is based on the asymptotic normality of the studentized ratio. In the present case, this would require the asymptotic normality of the ratio x0 b ei : s^(x) But no such asymptotic approximation can be made. The only special exception is the case where ei has the exact distribution N(0; 2 ); which is generally invalid. To get an accurate forecast interval, we need to estimate the conditional distribution of ei given xi = x; which is a much more di¢ culth task. Perhaps i due to this di¢ culty, many applied forecasters 0 b use the simple approximate interval x 2^ s(x) despite the lack of a convincing justi…cation. 6.15 Wald Statistic Let = h( ) : Rk ! Rq be any parameter vector of interest, b its estimate and Vb covariance matrix estimator. Consider the quadratic form 0 Wn ( ) = n b Vb 1 b : its (6.44) When q = 1; then Wn ( ) = tn ( )2 is the square of the t-ratio. When q > 1; Wn ( ) is typically called a Wald statistic. We are interested in its sampling distribution. The asymptotic distribution of Wn ( ) is simple to derive given Theorem 6.9.2 and Theorem 6.9.3, which show that p d n b ! Z N (0; V ) and It follows that Wn ( ) = p n b Vb 0 p !V : Vb 1p n b d ! Z0 V 1 Z (6.45) CHAPTER 6. ASYMPTOTIC THEORY FOR LEAST SQUARES 140 a quadratic in the normal random vector Z: Here we can appeal to a useful result from probability theory. (See Theorem B.9.3 in the Appendix.) Theorem 6.15.1 If Z N (0; A) with A > 0; q q; then Z 0 A a chi-square random variable with q degrees of freedom. 1 Z 2; q The asymptotic distribution in (6.45) takes exactly this form. Note that V > 0 since H is full rank under Assumption 6.9.1 It follows that Wn ( ) converges in distribution to a chi-square random variable. Theorem 6.15.2 Under Assumptions 6.1.2 and 6.9.1, as n ! 1; d Wn ( ) ! 2 q: Theorem 6.15.2 is used to justify multivariate con…dence regions and mutivariate hypothesis tests. 6.16 Con…dence Regions A con…dence region Cn is a set estimator for 2 Rq when q > 1: A con…dence region Cn is a set in Rq intended to cover the true parameter value with a pre-selected probability 1 : Thus an ideal con…dence region has the coverage probability Pr( 2 Cn ) = 1 . In practice it is typically not possible to construct a region with exact coverage, but we can calculate its asymptotic coverage. When the parameter estimate satis…es the conditions of Theorem 6.15.2, a good choice for a con…dence region is the ellipse Cn = f : Wn ( ) c1 g : with c1 the 1 ’th quantile of the 2q distribution. (Thus Fq (c1 can be found from the 2q critical value table. Theorem 6.15.2 implies 2 q Pr ( 2 Cn ) ! Pr c1 )=1 :) These quantiles =1 which shows that Cn has asymptotic coverage (1 )%: To illustrate the construction of a con…dence region, consider the estimated regression (6.43) of the model \ log(W age) = + 1 education + 2 experience + 3 experience2 =100: Suppose that the two parameters of interest are the percentage return to education 1 = 100 1 and the percentage return to experience for individuals with 10 years experience 2 = 100 2 + 20 3 . (We need to condition on the level of experience since the regression is quadratic in experience.) These two parameters are a linear transformation of the regression parameters with point estimates b= 0 100 0 0 0 0 100 20 and have the covariance matrix estimate b= 11:6 0:72 ; CHAPTER 6. ASYMPTOTIC THEORY FOR LEAST SQUARES 141 Figure 6.8: Con…dence Region for Return to Experience and Return to Education Vb = 0 100 0 0 0 0 100 20 = 645:4 67:387 67:387 165 1 0 0 B 100 0 C C Vb b B @ 0 100 A 0 20 0 with inverse Thus the Wald statistic is 1 Vb 0 Wn ( ) = n b Vb 11:6 0:72 = 2454 = 3:97 (11:6 0:0016184 0:00066098 = 1 1 2 2 1) b 0 0:00066098 0:0063306 0:0016184 0:00066098 3:2441 (11:6 : 0:00066098 0:0063306 1 ) (0:72 2) 11:6 0:72 + 15:535 (0:72 1 2 2) 2 The 90% quantile of the 22 distribution is 4.605 (we use the 22 distribution as the dimension of is two), so an asymptotic 90% con…dence region for the two parameters is the interior of the ellipse 3:97 (11:6 1) 2 3:2441 (11:6 1 ) (0:72 2) + 15:535 (0:72 2) 2 = 4:605 which is displayed in Figure 6.8. Since the estimated correlation of the two coe¢ cient estimates is small (about 0.2) the ellipse is close to circular. CHAPTER 6. ASYMPTOTIC THEORY FOR LEAST SQUARES 6.17 142 Semiparametric E¢ ciency in the Projection Model In Section 4.6 we presented the Gauss-Markov theorem, which stated that in the homoskedastic CEF model, in the class of linear unbiased estimators the one with the smallest variance is leastsquares. As we noted in that section, the restriction to linear unbiased estimators is unsatisfactory as it leaves open the possibility that an alternative (non-linear) estimator could have a smaller asymptotic variance. In addition, the restriction to the homoskedastic CEF model is also unsatisfactory as the projection model is more relevant for empirical application. The question remains: what is the most e¢ cient estimator of the projection coe¢ cient (or functions = h( )) in the projection model? It turns out that it is straightforward to show that the projection model falls in the estimator class considered in Proposition 5.13.2. It follows that the least-squares estimator is semiparametrically e¢ cient in the sense that it has the smallest asymptotic variance in the class of semiparametric estimators of . This is a more powerful and interesting result than the Gauss-Markov theorem. To see this, it is worth rephrasing Proposition 5.13.2 with amended notation. Suppose that P a parameter of interest is = g( n) where = Ez i ; for which the moment estimators are b = n1 ni=1 o zi 2 and b = g(b ): Let L2 (g) = F : E kzk < 1; g (u) is continuously di¤erentiable at u = Ez be the set of distributions for which b satis…es the central limit theorem. Proposition 6.17.1 In the class of distributions F 2 L2 (g); b is semiparametrically e¢ cient for in the sense that its asymptotic variance equals the semiparametric e¢ ciency bound. Proposition 6.17.1 says that under the minimal conditions in which b is asymptotically normal, then no semiparametric estimator can have a smaller asymptotic variance than b. To show that an estimator is semiparametrically e¢ cient it is su¢ cient to show that it falls in the class covered by this Proposition. To show that the projection model falls in this class, we write = Qxx1 Qxy = g ( ) where = Ez i and z i = (xi x0i ; xi yi ) : The class L2 (g) equals the class of distributions n o L4 ( ) = F : Ey 4 < 1; E kxk4 < 1; Exi x0i > 0 : Proposition 6.17.2 In the class of distributions F 2 L4 ( ); the leastsquares estimator b is semiparametrically e¢ cient for . The least-squares estimator is an asymptotically e¢ cient estimator of the projection coe¢ cient because the latter is a smooth function of sample moments and the model implies no further restrictions. However, if the class of permissible distributions is restricted to a strict subset of L4 ( ) then least-squares can be ine¢ cient. For example, the linear CEF model with heteroskedastic errors is a strict subset of L4 ( ); and the GLS estimator has a smaller asymptotic variance than OLS. In this case, the knowledge that true conditional mean is linear allows for more e¢ cient estimation of the unknown parameter. From Proposition 6.17.1 we can also deduce that plug-in estimators b = h( b ) are semiparametrically e¢ cient estimators of = h( ) when h is continuously di¤erentiable. We can also deduce CHAPTER 6. ASYMPTOTIC THEORY FOR LEAST SQUARES that other parameters estimators are semiparametrically e¢ cient, such as ^ 2 for note that we can write 2 = E yi 143 2: To see this, 2 x0i = Eyi2 2E yi x0i = Qyy Qyx Qxx1 Qxy + 0 E xi x0i which is a smooth function of the moments Qyy ; Qyx and Qxx : Similarly the estimator ^ 2 equals n ^ 2 = 1X 2 e^i n i=1 b yy = Q b yx Q b 1Q b Q xx xy Since the variables yi2 ; yi x0i and xi x0i all have …nite variances when F 2 L4 ( ); the conditions of Proposition 6.17.1 are satis…ed. We conclude: Proposition 6.17.3 In the class of distributions F 2 L4 ( ); ^ 2 is semiparametrically e¢ cient for 2 . 6.18 Semiparametric E¢ ciency in the Homoskedastic Regression Model* In Section 6.17 we showed that the OLS estimator is semiparametrically e¢ cient in the projection model. What if we restrict attention to the classical homoskedastic regression model? Is OLS still e¢ cient in this class? In this section we derive the asymptotic semiparametric e¢ ciency bound for this model, and show that it is the same as that obtained by the OLS estimator. Therefore it turns out that least-squares is e¢ cient in this class as well. Recall that in the homoskedastic regression model the asymptotic variance of the OLS estimator b for is V 0 = Q 1 2 : Therefore, as described in Section 5.13, it is su¢ cient to …nd a parametric xx submodel whose Cramer-Rao bound for estimation of is V 0 : This would establish that V 0 is the semiparametric variance bound and the OLS estimator b is semiparametrically e¢ cient for : Let the joint density of y and x be written as f (y; x) = f1 (y j x) f2 (x) ; the product of the conditional density of y given x and the marginal density of x. Now consider the parametric submodel f (y; x j ) = f1 (y j x) 1 + y x0 x0 = 2 f2 (x) : (6.46) You can check that in this submodel the marginal density of x is f2 (x) and the conditional density of y given x is f1 (y j x) 1 + (y x0 ) (x0 ) = 2 : To see that the latter is a valid conditional R density, observe that the regression assumption implies that yf1 (y j x) dy = x0 and therefore Z f1 (y j x) 1 + y x0 x0 = 2 dy Z Z = f1 (y j x) dy + f1 (y j x) y x0 dy x0 = 2 = 1: CHAPTER 6. ASYMPTOTIC THEORY FOR LEAST SQUARES In this parametric submodel the conditional mean of y given x is Z E (y j x) = yf1 (y j x) 1 + y x0 x0 = 2 dy Z Z = yf1 (y j x) dy + yf1 (y j x) y x0 x0 Z Z 2 = yf1 (y j x) dy + y x0 f1 (y j x) x0 Z + y x0 f1 (y j x) dy x0 x0 = 2 144 = 2 = 2 dy dy = x0 ( + ) ; R using the homoskedasticity assumption (y x0 )2 f1 (y j x) dy = 2 : This means that in this parametric submodel, the conditional mean is linear in x and the regression coe¢ cient is ( ) = + : We now calculate the score for estimation of : Since @ @ log f (y; x j ) = log 1 + y @ @ x0 x0 = 2 = x (y x0 ) = 2 1 + (y x0 ) (x0 ) = 2 the score is @ log f (y; x j 0 ) = xe= 2 : @ The Cramer-Rao bound for estimation of (and therefore ( ) as well) is s= E ss0 1 = 4 E (xe) (xe)0 1 = 2 Qxx1 = V 0 : We have shown that there is a parametric submodel (6.46) whose Cramer-Rao bound for estimation of is identical to the asymptotic variance of the least-squares estimator, which therefore is the semiparametric variance bound. Theorem 6.18.1 In the homoskedastic regression model, the semiparametric variance bound for estimation of is V 0 = 2 Qxx1 and the OLS estimator is semiparametrically e¢ cient. This result is similar to the Gauss-Markov theorem, in that it asserts the e¢ ciency of the leastsquares estimator in the context of the homoskedastic regression model. The di¤erence is that the Gauss-Markov theorem states that OLS has the smallest variance among the set of unbiased linear estimators, while Theorem 6.18.1 states that OLS has the smallest asymptotic variance among all regular estimators. This is a much more powerful statement. 6.19 Uniformly Consistent Residuals* It seems natural to view the residuals e^i as estimates of the unknown errors ei : Are they consistent estimates? In this section we develop an appropriate convergence result. This is not a widely-used technique, and can safely be skipped by most readers. Notice that we can write the residual as e^i = yi x0i b = ei + x0i = ei x0i b x0i b : (6.47) CHAPTER 6. ASYMPTOTIC THEORY FOR LEAST SQUARES 145 p Since b ! 0 it seems reasonable to guess that e^i will be close to ei if n is large. We can bound the di¤erence in (6.47) using the Schwarz inequality (A.10) to …nd j^ ei ei j = x0i b kxi k b : (6.48) To bound (6.48) we can use b = Op (n 1=2 ) from Theorem 6.3.2, but we also need to bound the random variable kxi k. The key is Theorem 5.12.1 which shows that E kxi kr < 1 implies xi = op n1=r uniformly in i; or p n 1=r max kxi k ! 0: 1 i n Applied to (6.48) we obtain max j^ ei max kxi k b ei j 1 i n 1 i n 1=2+1=r = op (n ): We have shown the following. Theorem 6.19.1 Under Assumption 6.1.2 and E kxi kr < 1, then uniformly in 1 i n e^i = ei + op (n 1=2+1=r ): (6.49) The rate of convergence in (6.49) depends on r: Assumption 6.1.2 requires r 4; so the rate 1=4 of convergence is at least op (n ): As r increases, the rate becomes close to Op (n 1=2 ). If the regressor is bounded, kxi k B < 1, then e^i = ei + op (n 1=2 ): 6.20 Asymptotic Leverage* Recall the de…nition of leverage from (3.21) 1 hii = x0i X 0 X xi : These are the diagonal elements of the projection matrix P and appear in the formula for leaveone-out prediction errors and several covariance matrix estimators. We can show that under iid sampling the leverage values are uniformly asymptotically small. Let min (A) and max (A) denote the smallest and largest eigenvalues of a symmetric square matrix A; and note that max (A 1 ) = ( min (A)) 1 : p p Since n1 X 0 X ! Qxx > 0 then by the CMT, min n1 X 0 X ! min (Qxx ) > 0: (The latter is positive since Qxx is positive de…nite and thus all its eigenvalues are positive.) Then by the Trace Inequality (A.13) hii = x0i X 0 X xi 1 0 XX n = tr max = 1 1 0 XX n 1 0 XX n 1 1 xi x0i n ! ! 1 tr 1 xi x0i n 1 1 kxi k2 n 1 ( min (Qxx ) + op (1)) 1 max kxi k2 : n1 i n min (6.50) CHAPTER 6. ASYMPTOTIC THEORY FOR LEAST SQUARES Theorem 5.12.1 shows that E kxi kr < 1 implies max1 op n2=r 1 . i n kxi k 146 = op n1=r and thus (6.50) is Theorem 6.20.1 If xi is independent and identically distributed and E kxi kr < 1 for some r 2; then uniformly in 1 i n, hii = op n2=r 1 : For any r 2 then hii = op (1) (uniformly in i n): Larger r implies a stronger rate of convergence, for example r = 4 implies hii = op n 1=2 : Theorem (6.20.1) implies that under random sampling with …nite variances and large samples, no individual observation should have a large leverage value. Consequently individual observations should not be in‡uential, unless one of these conditions is violated. CHAPTER 6. ASYMPTOTIC THEORY FOR LEAST SQUARES 147 Exercises Exercise 6.1 Take the model yi = x01i 1 +x02i 2 +ei with Exi ei = 0: Suppose that 1 is estimated by regressing yi on x1i only. Find the probability limit of this estimator. In general, is it consistent for 1 ? If not, under what conditions is this estimator consistent for 1 ? Exercise 6.2 Let y be n regression estimator where 1; X be n b= k (rank k): y = X + e with E(xi ei ) = 0: De…ne the ridge n X xi x0i + I k i=1 ! 1 n X xi yi i=1 ! (6.51) > 0 is a …xed constant. Find the probability limit of b as n ! 1: Is b consistent for Exercise 6.3 For the ridge regression estimator (6.51), set Find the probability limit of b as n ! 1: ? = cn where c > 0 is …xed as n ! 1: Exercise 6.4 Verify some of the calculations reported in Section 6.4. Speci…cally, suppose that x1i and x2i only take the values f 1; +1g; symmetrically, with Pr (x1i = x2i = 1) = Pr (x1i = x2i = Pr (x1i = 1; x2i = 1) = Pr (x1i = 5 E e2i j x1i = x2i = 4 1 2 : E ei j x1i 6= x2i = 4 1; x2i = 1) = 1=8 Verify the following: 1. Ex1i = 0 2. Ex21i = 1 3. Ex1i x2i = 1 2 4. E e2i = 1 5. E x21i e2i = 1 7 6. E x1i x2i e2i = : 8 Exercise 6.5 Show (6.19)-(6.22). Exercise 6.6 The model is yi = x0i + ei E (xi ei ) = 0 = E xi x0i e2i : Find the method of moments estimators ( b ; b ) for ( ; ) : (a) In this model, are ( b ; b ) e¢ cient estimators of ( ; )? (b) If so, in what sense are they e¢ cient? 1) = 3=8 CHAPTER 6. ASYMPTOTIC THEORY FOR LEAST SQUARES 148 Exercise 6.7 Of the variables (yi ; yi ; xi ) only the pair (yi ; xi ) are observed. In this case, we say that yi is a latent variable. Suppose yi = x0i + ei E (xi ei ) = 0 yi = yi + ui where ui is a measurement error satisfying E (xi ui ) = 0 E (yi ui ) = 0 Let b denote the OLS coe¢ cient from the regression of yi on xi : (a) Is the coe¢ cient from the linear projection of yi on xi ? (b) Is b consistent for as n ! 1? (c) Find the asymptotic distribution of p n b as n ! 1: Exercise 6.8 Find the asymptotic distribution of p n ^2 2 as n ! 1: Exercise 6.9 The model is yi = xi + ei E (ei j xi ) = 0 where xi 2 R: Consider the two estimators b = e = Pn x i yi Pi=1 n 2 i=1 xi n 1 X yi : n xi i=1 (a) Under the stated assumptions, are both estimators consistent for ? (b) Are there conditions under which either estimator is e¢ cient? Exercise 6.10 In the homoskedastic regression model y = X + e with E(ei j xi ) = 0 and E(e2i j xi ) = 2 ; suppose ^ is the OLS estimate of with covariance matrix V^ ; based on a sample of size n: Let ^ 2 be the estimate of 2 : You wish to forecast an out-of-sample value of yn+1 given that xn+1 = x: Thus the available information is the sample (y; X); the estimates ( b ; Vb ; ^ 2 ), the residuals e ^; and the out-of-sample value of the regressors, xn+1 : (a) Find a point forecast of yn+1 : (b) Find an estimate of the variance of this forecast. Chapter 7 Restricted Estimation 7.1 Introduction In the linear projection model yi = x0i + ei E (xi ei ) = 0 a common task is to impose a constraint on the coe¢ cient vector . For example, partitioning 0 0 x0i = (x01i ; x02i ) and 0 = 1 ; 2 ; a typical constraint is an exclusion restriction of the form 2 = 0: In this case the constrained model is yi = x01i 1 + ei E (xi ei ) = 0 At …rst glance this appears the same as the linear projection model, but there is one important di¤erence: the error ei is uncorrelated with the entire regressor vector x0i = (x01i ; x02i ) not just the included regressor x1i : In general, a set of q linear constraints on takes the form R0 =c (7.1) where R is k q; rank(R) = q < k and c is q 1: The assumption that R is full rank means that the constraints are linearly independent (there are no redundant or contradictory constraints). The constraint 2 = 0 discussed above is a special case of the constraint (7.1) with R= 0 I ; (7.2) a selector matrix, and c = 0. Another common restriction is that a set of coe¢ cients sum to a known constant, i.e. 1 + 2 = 1: This constraint arises in a constant-return-to-scale production function. Other common restrictions include the equality of coe¢ cients 1 = 2 ; and equal and o¤setting coe¢ cients 1 = 2: A typical reason to impose a constraint is that we believe (or have information) that the constraint is true. By imposing the constraint we hope to improve estimation e¢ ciency. The goal is to obtain consistent estimates with reduced variance relative to the unconstrained estimator. The questions then arise: How should we estimate the coe¢ cient vector imposing the linear restriction (7.1)? If we impose such constraints, what is the sampling distribution of the resulting estimator? How should we calculate standard errors? These are the questions explored in this chapter. 149 CHAPTER 7. RESTRICTED ESTIMATION 7.2 150 Constrained Least Squares An intuitively appealing method to estimate a constrained linear projection is to minimize the least-squares criterion subject to the constraint R0 = c. This estimator is e = argmin SSEn ( ) (7.3) R0 =c where SSEn ( ) = n X x0i yi 2 = y0y 2y 0 X + 0 X 0X : i=1 The estimator e minimizes the sum of squared errors over all such that the restriction (7.1) e holds. We call the constrained least-squares (CLS) estimator. We follow the convention of using a tilde “~” rather than a hat “^” to indicate that e is a restricted estimator in contrast to the unrestricted least-squares estimator b ; and write it as e cls when we want to be clear that the estimation method is CLS. One method to …nd the solution to (7.3) uses the technique of Lagrange multipliers. The problem (7.3) is equivalent to the minimization of the Lagrangian 1 L( ; ) = SSEn ( ) + 2 over ( ; ); where is an s minimization of (7.4) are R0 c (7.4) 1 vector of Lagrange multipliers. The …rst-order conditions for @ L( e ; e ) = @ and 0 Premultiplying (7.5) by R0 (X 0 X) X 0y + X 0X e + R e = 0 @ L( ; ) = R0 e @ 1 c = 0: (7.6) we obtain R0 b + R0 e + R0 X 0 X 1 Re = 0 where b = (X 0 X) X 0 y is the unrestricted least-squares estimator. Imposing R0 e (7.6) and solving for e we …nd h i 1 e = R0 X 0 X 1 R R0 b c : 1 (7.5) (7.7) c = 0 from Substuting this expression into (7.5) and solving for e we …nd the solution to the constrained minimization problem (7.3) h i 1 1 1 e =b R R0 X 0 X R R0 b c : (7.8) X 0X cls This is a general formula for the CLS estimator. It also can be written as h i 1 1 0b 1 e =b Q b R R Q R R0 b c : cls xx xx Given e cls the residuals are The moment estimator of 2 e~i = yi is x0i e cls : n ~ 2cls = 1X 2 e~i : n i=1 CHAPTER 7. RESTRICTED ESTIMATION 7.3 151 Exclusion Restriction While (7.8) is a general formula for the CLS estimator, in most cases the estimator can be found by applying least-squares to a reparameterized equation. To illustrate, let us return to the …rst example presented at the beginning of the chapter –a simple exclusion restriction. Recall the unconstrained model is yi = x01i 1 + x02i 2 + ei (7.9) the exclusion restriction is 2 = 0; and the constrained equation is yi = x01i + ei : 1 (7.10) In this setting the CLS estimator is OLS of yi on x1i : (See Exercise 7.1.) We can write this as e = 1 n X x1i x01i i=1 The CLS estimator of the entire vector 0 = ! 0 1; e e= 1 n X x1i yi i=1 0 2 is 1 : 0 ! : (7.11) (7.12) It is not immediately obvious, but (7.8) and (7.12) are algebraically (and numerically) equivalent. To see this, the …rst component of (7.8) with (7.2) is # " 1 1 1 0 0 b Q e = I 0 b b 0 I b : 0 I Q 1 xx xx I I Using (3.33) this equals e 1 = b1 = = b 12 Q b 22 Q 1 b 2 b +Q b b 1 Q b b 1b 1 11 2 12 Q22 Q22 1 2 1 1 b b 12 Q b b Q b 2y Q Q 11 2 Q1y 22 b 1 Q b b 1b b 1 b +Q 11 2 12 Q22 Q22 1 Q22 1 Q2y b 1 Q b 1y = Q 11 2 b 1 Q b 11 = Q 11 2 b 1Q b 1y = Q 11 b 12 Q b 1Q b 21 Q b 1Q b 1y Q 22 11 b 21 Q b 1Q b 1y Q 11 b 12 Q b 1Q b 21 Q b 1Q b 1y Q 22 11 which is (7.12) as originally claimed. 7.4 Minimum Distance A minimum distance estimator tries to …nd a parameter value which satis…es the constraint which is as close as possible to the unconstrained estimate. Let b be the unconstrained leastsquares estimator, and for some k k positive de…nite weight matricx W n > 0 de…ne Jn ( ) = n b 0 Wn b (7.13) CHAPTER 7. RESTRICTED ESTIMATION 152 This is a (squared) weighted Euclidean distance between b and : Jn ( ) is small if is close to b , and is minimized at zero only if = b . A minimum distance estimator e for e minimizes Jn ( ) subject to the constraint (7.1), that is, e md = argmin Jn ( ) : (7.14) R0 =c 0 Vb The CLS estimator is the special case when W n = 1 : To see this, rewrite the leastsquares criterion as follows. Write the unconstrained least-squares …tted equation as yi = x0i b + e^i and substitute this equation into SSEn ( ) to obtain SSEn ( ) = = = n X yi i=1 n X i=1 n X x0i b + e^i e^2i i=1 = s2 (n x0i + b 2 2 x0i 0 n X xi x0i i=1 ! b k + Jn ( )) (7.15) P where the third equality uses the fact that ni=1 xi e^i = 0; and the last line holds when W n = 1 0 1 Pn = xi x0i s 2 : The expression (7.15) only depends on through Jn ( ) : Thus Vb n i=1 1 0 minimization of SSEn ( ) and Jn ( ) are equivalent, and hence e md = e cls when W n = Vb : We can solve for e explicitly by the method of Lagrange multipliers. The Lagrangian is md 1 L( ; ) = Jn ( ; W n ) + 2 0 R0 c which is minimized over ( ; ): The solution is e md =b W n 1 R R0 W n 1 R 1 R0 b c : (7.16) (See Exercise 7.5.) Examining (7.16) we can see that e md specializes to e cls when we set W n = 1 0 Vb : An obvious question is which weight matrix W n is best. We will address this question after we derive the asymptotic distribution for a general weight matrix. 7.5 Asymptotic Distribution We …rst show that the class of minimum distance estimators are consistent for the population parameters when the constraints are valid. Assumption 7.5.1 R0 = c where R is k q with rank(R) = q: CHAPTER 7. RESTRICTED ESTIMATION 153 p Assumption 7.5.2 W n ! W > 0 Theorem 7.5.1 Consistency p Under Assumptions 6.1.1, 7.5.1, and 7.5.2, e md ! as n ! 1: Theorem 7.5.1 shows that consistency holds for any weight matrix with a positive de…nate limit, so the result includes the CLS estimator. Similarly, the constrained estimators are asymptotically normally distributed. Theorem 7.5.2 Asymptotic Normality Under Assumptions 6.1.2, 7.5.1, and 7.5.2, p as n ! 1; where V (W ) = V W +W and V 1 d n e md 1 ! N (0; V (W )) R R0 W R R0 W 1 R 1 R 1 1 R0 V (7.17) V R R0 W R0 V R R0 W 1 R 1 1 R R0 W 1 R0 W 1 1 (7.18) = Qxx1 Qxx1 : Theorem 7.5.2 shows that the minimum distance estimator is asymptotically normal for all positive de…nite weight matrices. The asymptotic variance depends on W . The theorem includes the CLS estimator as a special case by setting W = Qxx : Theorem 7.5.3 Asymptotic Distribution of CLS Estimator Under Assumptions 6.1.2 and 7.5.1, as n ! 1 p where V cls = V d n e cls ! N (0; V cls ) Qxx1 R R0 Qxx1 R +Qxx1 R R0 Qxx1 R 1 1 R0 V V R R0 Qxx1 R R0 V R R0 Qxx1 R 1 R0 Qxx1 1 R0 Qxx1 CHAPTER 7. RESTRICTED ESTIMATION 7.6 154 E¢ cient Minimum Distance Estimator Theorem 7.5.2 shows that the minimum distance estimators, which include CLS as a special case, are asymptotically normal with an asymptotic covariance matrix which depends on the weight matrix W . The asymptotically optimal weight matrix is the one which minimizes the asymptotic variance V (W ): This turns out to be W = V 1 as is shown in Theorem 7.6.1 below. Since V 1 is unknown this weight matrix cannot be used for a feasible estimator, but we can replace V 1 1 with a consistent estimate Vb and the asymptotic distribution (and e¢ ciency) are unchanged. We call the minimum distance estimator setting W n = Vb estimator and takes the form e emd =b Vb R R0 Vb R 1 1 the e¢ cient minimum distance R0 b (7.19) c The asymptotic distribution of (7.19) can be deduced from Theorem 7.5.2. Theorem 7.6.1 E¢ cient Minimum Distance Estimator Under Assumptions 6.1.2 and 7.5.1, p as n ! 1; where V d n e emd =V ! N 0; V V R R0 V R 1 R0 V : (7.20) Since V V (7.21) the estimator (7.19) has lower asymptotic variance than the unrestricted estimator. Furthermore, for any W ; V V (W ) (7.22) so (7.19) is asymptotically e¢ cient in the class of minimum distance estimators. Theorem 7.6.1 shows that the minimum distance estimator with the smallest asymptotic variance is (7.19). One implication is that the constrained least squares estimator is generally ine¢ cient. The interesting exception is the case of conditional homoskedasticity, in which case the optimal weight matrix is W = V 0 1 so in this case CLS is an e¢ cient minimum distance estimator. Otherwise when the error is conditionally heteroskedastic, there are asymptotic e¢ ciency gains by using minimum distance rather than least squares. The fact that CLS is generally ine¢ cient is counter-intuitive and requires some re‡ection to understand. Standard intuition suggests to apply the same estimation method (least squares) to the unconstrained and constrained models, and this is the most common empirical practice. But Theorem 7.6.1 shows that this is not the e¢ cient estimation method. Instead, the e¢ cient minimum distance estimator has a smaller asymptotic variance. Why? The reason is that the least-squares estimator does not make use of the regressor x2i : It ignores the information E (x2i ei ) = 0. This information is relevant when the error is heteroskedastic and the excluded regressors are correlated with the included regressors. Inequality (7.21) shows that the e¢ cient minimum distance estimator e has a smaller asymptotic variance than the unrestricted least squares estimator b : This means that estimation is more e¢ cient by imposing correct restrictions when we use the minimum distance method. CHAPTER 7. RESTRICTED ESTIMATION 7.7 155 Exclusion Restriction Revisited We return to the example of estimation with a simple exclusion restriction. The model is yi = x01i 1 + x02i 2 + ei with the exclusion restriction 2 = 0: We have introduced three estimators of unconstrained least-squares applied to (7.9), which can be written as 1: The …rst is b =Q b 1 Q b 1 11 2 1y 2 : From Theorem 6.33 and equation (6.20) its asymptotic variance is avar( b 1 ) = Q1112 Q12 Q221 11 The second estimator of 1 1 12 Q22 Q21 21 + Q12 Q221 1 22 Q22 Q21 Q1112 : is the CLS estimator, which can be written as e 1;cls b 1Q b 1y : =Q 11 Its asymptotic variance can be deduced from Theorem 7.5.3, but it is simpler to apply the CLT directly to show that (7.23) avar( e 1;cls ) = Q111 11 Q111 : The third estimator of 1 is the e¢ cient minimum distance estimator. Applying (7.19), it equals e where we have partitioned 1;md Vb 1 Vb 12 Vb 22 b 2 = b1 = " From Theorem 7.6.1 its asymptotic variance is Vb 11 Vb 12 Vb 21 Vb 22 avar( e 1;md ) = V 11 # (7.24) : V 12 V 221 V 21 : (7.25) In general, the three estimators are di¤erent, and they have di¤erent asymptotic variances. It is quite instructive to compare the asymptotic variances of the CLS and unconstrained leastsquares estimators to assess whether or not the constrained estimator is necessarily more e¢ cient than the unconstrained estimator. First, consider the case of conditional homoskedasticity. In this case the two covariance matrices simplify to avar( b 1 ) = 2 Q1112 and avar( e 1;cls ) = 2 Q111 : If Q12 = 0 (so x1i and x2i are orthogonal) then these two variance matrices equal and the two estimators have equal asymptotic e¢ ciency. Otherwise, since Q12 Q221 Q21 0; then Q11 Q11 Q12 Q221 Q21 ; and consequently Q111 2 Q11 Q12 Q221 Q21 1 2 : This means that under conditional homoskedasticity, e 1;cls has a lower asymptotic variance matrix than b 1 : Therefore in this context, constrained least-squares is more e¢ cient than unconstrained least-squares. This is consistent with our intuition that imposing a correct restriction (excluding an irrelevant regressor) improves estimation e¢ ciency. CHAPTER 7. RESTRICTED ESTIMATION 156 However, in the general case of conditional heteroskedasticity this ranking is not guaranteed. In fact what is really amazing is that the variance ranking can be reversed. The CLS estimator can have a larger asymptotic variance than the unconstrained least squares estimator. To see this let’s use the simple heteroskedastic example from Section 6.4. In that example, 1 7 3 Q11 = Q22 = 1; Q12 = ; 11 = 22 = 1; and 12 = : We can calculate that Q11 2 = and 2 8 4 2 3 e avar( 1;cls ) = 1 avar( b 1 ) = (7.26) (7.27) 5 (7.28) avar( e 1;md ) = : 8 Thus the restricted least-squares estimator e 1;cls has a larger variance than the unrestricted leastsquares estimator b 1 ! The minimum distance estimator has the smallest variance of the three, as expected. What we have found is that when the estimation method is least-squares, deleting the irrelevant variable x2i can actually increase estimation variance; or equivalently, adding an irrelevant variable can actually decrease the estimation variance. To repeat this unexpected …nding, we have shown in a very simple example that it is possible for least-squares applied to the short regression (7.10) to be less e¢ cient for estimation of 1 than least-squares applied to the long regression (7.9), even though the constraint 2 = 0 is valid! This result is strongly counter-intuitive. It seems to contradict our initial motivation for pursuing constrained estimation –to improve estimation e¢ ciency. It turns out that a more re…ned answer is appropriate. Constrained estimation is desirable, but not constrained least-squares estimation. While least-squares is asymptotically e¢ cient for estimation of the unconstrained projection model, it is not an e¢ cient estimator of the constrained projection model. 7.8 Variance and Standard Error Estimation The asymptotic covariance matrix (7.20) may be estimated by replacing V estimates such as Vb . This variance estimator is then Vb = Vb Vb R R0 Vb R 1 with a consistent R0 Vb : (7.29) : (7.30) We can calculate standard errors for any linear combination h0 e so long as h does not lie in the range space of R. A standard error for h0 e is 7.9 Misspeci…cation s(h0 e ) = n h Vb h 1 0 1=2 What are the consequences for the constrained estimator e if the constraint (7.1) is incorrect? To be speci…c, suppose that R0 = c where c is not necessarily equal to c: This situation is a generalization of the analysis of “omitted variable bias” from Section 2.22, where we found that the short regression (e.g. (7.11)) is estimating a di¤erent projection coe¢ cient than the long regression (e.g. (7.9)). CHAPTER 7. RESTRICTED ESTIMATION 157 One mechanical answer is that we can use the formula (7.16) for the minimum distance estimator to …nd that p 1 e W 1 R R0 W 1 R (c c) : md ! md = 1 The second term, W 1 R R0 W 1 R (c c), shows that imposing an incorrect constraint leads to inconsistency – an asymptotic bias. We can call the limiting value md the minimum-distance projection coe¢ cient or the pseudo-true value implied by the restriction. However, we can say more. For example, we can describe some characteristics of the approximating projections. The CLS estimator projection coe¢ cient has the representation cls = argmin E yi R 0 2 x0i ; =c the best linear predictor subject to the constraint (7.1). The minimum distance estimator converges to 0 0) W ( 0) md = argmin ( R0 =c where 0 is the true coe¢ cient. That is, md is the coe¢ cient vector satisfying (7.1) closest to the true value ¬n the weighted Euclidean norm. These calculations show that the constrained estimators are still reasonable in the sense that they produce good approximations to the true coe¢ cient, conditional on being required to satisfy the constraint. We can also show that e md has an asymptotic normal distribution. The trick is to de…ne the pseudo-true value 1 W n 1 R R0 W n 1 R (c c) : n = Then p n e md n = p = I 1p n R0 b W n 1 R R0 W n 1 R 1 0 p W n 1 R R0 W n 1 R R n b n b d ! I W 1 R R0 W 1 R 1 = N (0; V (W )) : In particular p n R0 N (0; V ) (7.31) d n e emd c ! N 0; V : This means that even when the constraint (7.1) is misspeci…ed, the conventional covariance matrix estimator (7.29) and standard errors (7.30) are appropriate measures of the sampling variance, though the distributions are centered at the pseudo-true values (or projections) n rather than : The fact that the estimators are biased is an unavoidable consequence of misspeci…cation. There is another way of representing an asymptotic distribution for the estimator under misspeci…cation based on the concept of local alternatives. It is a technical device which might seem a bit arti…cial, but it is a powerful technique which yields useful distributional approximations in a wide variety of contexts. The idea is to index the true coe¢ cient n by n; and suppose that R0 n =c+ n 1=2 : (7.32) The asymptotic theory is then derived as n ! 1 under the sequence of probability distributions with the coe¢ cients n . The expression (7.32) speci…es that n does not satisfy (7.1), but the deviation from the constraint is n 1=2 which depends on and the sample size. The choice to make the deviation of this form is precisely so that the localizing parameter appears in the asymptotic distribution but does not dominate it. CHAPTER 7. RESTRICTED ESTIMATION is the true coe¢ cient value, then yi = x0i ! ! 1 n n X 1X 1 d p ! N (0; V ) xi x0i xi ei n n To see this, …rst observe that since p n b n 158 = n i=1 n + ei and i=1 as conventional. Then under (7.32), e so md = b = b p W n 1 R R0 W n 1 R 1 W n 1 R R0 W n 1 R 1 n e md n = R0 b c R0 b + W n 1 R R0 W n 1 R W n 1 R R0 W n 1 R I ! N (0; V ) + W R0 p 1 +W n 1 R R0 W n 1 R d 1 1 R R0 W 1 n b R 1 n 1=2 n 1 = N ( ; V (W )) (7.33) where =W 1 R R0 W 1 1 R : The asymptotic distribution (7.33) is an alternative way of expressing the sampling distribution of the restricted estimator under misspeci…cation. The distribution (7.33) contains an asymptotic bias component : The approximation is not fundamentally di¤erent from (7.31) –they both have the same asymptotic variances, and both re‡ect the bias due to misspeci…cation. The di¤erence is that (7.31) puts the bias on the left-side of the convergence arrow, while (7.33) has the bias on the right-side. There is no substantive di¤erence between the two, but (7.33) is more convenient for some purposes, such as the analysis of the power of tests, as we will explore in the next chapter. 7.10 Nonlinear Constraints In some cases it is desirable to impose nonlinear constraints on the parameter vector can be written as r( ) = 0 . They (7.34) where r : Rk ! Rq : This includes the linear constraints (7.1) as a special case. An example of (7.34) which cannot be written as (7.1) is 1 2 = 1; which is (7.34) with r( ) = 1 2 1: The minimum distance estimator of subject to (7.34) solves the minimization problem e = argmin Jn ( ) (7.35) r( )=0 where Jn ( ) = n b The solution minimizes the Lagrangian 0 Wn b 1 L( ; ) = Jn ( ) + 2 0 r( ) : (7.36) over ( ; ): Computationally, there is no explicit expression for the solution e so it must be found numerically. Algorithms to numerically solve (7.35) are known as constrained optimization methods, and are available in programming languages including Matlab, Gauss and R. CHAPTER 7. RESTRICTED ESTIMATION 159 Assumption 7.10.1 r( ) = 0 with rank(R) = q; where R = @ r( )0 : @ The asymptotic distribution is a simple generalization of the case of a linear constraint, but the proof is more delicate. Theorem 7.10.1 Under Assumptions 6.1.2, 7.10.1, and 7.5.2, for e de…ned in (7.35) , p d ! N (0; V (W )) n e as n ! 1; where V (W ) is de…ned in (7.18). V (W ) is minimized with W = V 1 ; in which case the asymptotic variance is V V R R0 V R =V 1 R0 V : The asymptotic variance matrix for the e¢ cient minimum distance estimator can be estimated by where Vb b R b 0 Vb R b Vb R = Vb 1 b = @ r( e )0 : R @ b 0 Vb R Standard errors for the elements of e are the square roots of the diagonal elements of n 7.11 Inequality Constraints Inequality constraints on the parameter vector 1V b : take the form r( ) 0 for some function r : Rk ! Rq : The most common example is a non-negative constraint 1 0: The constrained least-squares and minimum distance estimators can be written as and e cls e = argmin SSEn ( ) md (7.37) r( ) 0 = argmin Jn ( ) : (7.38) r( ) 0 Except in special cases the constrained estimators do not have simple algebraic solutions. An important exception is when there is a single non-negativity constraint, e.g. 1 0 with q = 1: In this case the constrained estimator can be found by two-step approach. First compute the uncontrained estimator b . If b 1 0 then e = b : Second, if b 1 < 0 then impose 1 = 0 (eliminate the regressor X1 ) and re-estimate. This yields the constrained least-squares estimator. While this CHAPTER 7. RESTRICTED ESTIMATION 160 method works when there is a single non-negativity constraint, it does not immediately generalize to other contexts. The computational problems (7.37) and (7.38) are examples of quadratic programming problems. Quick and easy computer algorithms are available in programming languages including Matlab, Gauss and R. Inference on inequality-constrained estimators is unfortunately quite challenging. The conventional asymptotic theory gives rise to the following dichotomy. If the true parameter satis…es the strict inequality r( ) > 0, then asymptotically the estimator is not subject to the constraint and the inequality-constrained estimator has an asymptotic distribution equal to the unconstrained case. However if the true parameter is on the boundary, e.g. r( ) = 0, then the estimator has a truncated structure. This is easiest to see in the one-dimensional case. If we have an estimator ^ which p d satis…es n ^ ! Z = N (0; V ) and = 0; then the constrained estimator ~ = max[ ^ ; 0] p d will have the asymptotic distribution n ~ ! max[Z; 0]; a “half-normal” distribution. 7.12 Constrained MLE Recall that the log-likelihood function (3.43) for the normal regression model is log L( ; 2 )= n log 2 2 1 2 2 2 SSEn ( ): The constrained maximum likelihood estimator (CMLE) ( b cmle ; ^ 2cmle ) maximizes log L( ; 2 ) subject to the constraint (7.34) Since log L( ; 2 ) is a function of only through the sum of squared errors SSEn ( ); maximizing the likelihood is identical to minimizing SSEn ( ). Hence b cmle = b cls and ^ 2cmle = ^ 2cls . 7.13 Technical Proofs* Proof of Theorem 7.6.1, Equation (7.22). Let R? be a full rank k (k q) matrix satisfying R0? V R = 0 and then set C = [R; R? ] which is full rank and invertible. Then we can calculate that R0 V R R0 V R? R0? V R R0? V R? C 0V C = 0 0 0 0 R? V R? = and C 0 V (W )C = = R0 V (W )R R0 V (W )R? R0? V (W )R R0? V (W )R? 0 0 1 0 0 0 0 R? V R? + R? W R (R W R) R0 V R (R0 W R) 1 R0 W R? Thus C 0 V (W ) 0 = C V (W )C = 0 V C 0 CV C 0 0 R0? W R (R0 W R) 1 0 R V R (R0 W R) 0 1 R0 W R? : CHAPTER 7. RESTRICTED ESTIMATION Since C is invertible it follows that V (W ) 161 0 which is (7.22). V Proof of Theorem 7.10.1. For simplicity, we assume that the constrained estimator is consistent e p! . This can be shown with more e¤ort, but requires a deeper treatment than appropriate for this textbook. For each element rj ( ) of the q-vector r( ); by the mean value theorem there exists a j on the line segment joining e and such that Let Rn be the k rj ( e ) = rj ( ) + q matrix @ r1 ( @ Rn = p Since e ! 1) @ rj ( @ @ r2 ( @ 0 j) e (7.39) @ rq ( @ 2) p it follows that : q) : p ! , and by the CMT, Rn ! R: Stacking the (7.39), we obtain j r( e ) = r( ) + Rn0 e : Since r( e ) = 0 by construction and r( ) = 0 by Assumption 7.5.1, this implies 0 = Rn0 e : (7.40) The …rst-order condition for (7.36) is Wn b e =R e e: Premultiplying by R 0 W n 1 ; inverting, and using (7.40), we …nd Thus e = R 0W 1R e n n e = 1 Rn0 b e R 0W 1H f W n 1R n n I 1 e = R 0W 1R e n n 1 b Rn0 From Theorem 6.3.2 and Theorem 6.7.1 we …nd p n e = I d e R 0W 1R e W n 1R n n ! I W 1 R R0 W = N (0; V (W )) : 1 1 R Rn0 b Rn0 1 : : p n b R0 N (0; V ) CHAPTER 7. RESTRICTED ESTIMATION 162 Exercises Exercise 7.1 In the model y = X 1 1 + X 2 2 + e; show directly from de…nition (7.3) that the CLS estimate of = ( 1 ; 2 ) subject to the constraint that 2 = 0 is the OLS regression of y on X 1: Exercise 7.2 In the model y = X 1 1 + X 2 2 + e; show directly from de…nition (7.3) that the CLS estimate of = ( 1 ; 2 ); subject to the constraint that 1 = c (where c is some given vector) is the OLS regression of y X 1 c on X 2 : Exercise 7.3 In the model y = X 1 1 + X 2 2 + e; with X 1 and X 2 each n estimate of = ( 1 ; 2 ); subject to the constraint that 1 = 2: k; …nd the CLS Exercise 7.4 Verify that for e de…ned in (7.8) that R0 e = c: Exercise 7.5 Verify (7.16). b xx equals the CLS Exercise 7.6 Verify that the minimum distance estimator e with W n = Q estimator. Exercise 7.7 Prove Theorem 7.5.1. Exercise 7.8 Prove Theorem 7.5.2. Exercise 7.9 Prove Theorem 7.5.3. (Hint: Use that CLS is a special case of Theorem 7.5.2.) Exercise 7.10 Verify that (7.20) is V (W ) with W = V Exercise 7.11 Prove (7.21). Hint: Use (7.20). Exercise 7.12 Verify (7.23), (7.24) and (7.25) Exercise 7.13 Verify (7.26), (7.27), and (7.28). 1 : Chapter 8 Hypothesis Testing 8.1 Hypotheses and Tests It is often the goal of an empirical investigation to determine if one variable a¤ects another. Returning to our example of wage determination, we might be interested if union membership a¤ects wages. Equivalently, we might ask if the hypothesis “Union membership does not a¤ect wages” is true or false. In hypothesis testing, we are interest in testing if a speci…c restriction on the parameters is compatible with the observed data. Letting be the coe¢ cient in a wage regression for union membership, the hypothesis that union membership has no e¤ect on mean wages is the restriction = 0: Hypothesis testing is about making a decision if the restriction = 0 is true or false. In general, a hypothesis is a statement (or assertion) that a restriction is true, where a restriction takes the form 2 0 with 0 is a strict subset of a pararameter space . De…nition 1 A hypothesis is a statement that For every hypothesis 2 is the complement of 0 in 2 0 . the statement 2 c0 is also a hypothesis, where c0 = f 2 . We give a hypothesis and its complement special names. 0 De…nition 2 The null hypothesis is H0 : hypothesis is its complement H1 : 2 c0 : 2 0 : 2 = and the alternative In the example given previously, the alternative hypothesis is that union membership has an e¤ect on mean wages, or 6= 0: A hypothesis can either be true or not true. The goal of hypothesis testing is to provide evidence concerning the truth of the null hypothesis versus its alternative. A hypothesis test makes one of two decisions based on the data: either “Accept H0 ”or “Reject H0 ”. Take again the example about union membership and examine the wage regression reported in Table 5.1. We see that the coe¢ cient for “Male Union Member” is 0.095 (a wage premium of 9.5%) with a standard error of 0.020. Given the magnitude of this estimate, it seems unlikely that the true coe¢ cient could be zero, and so we may be inclined to reject the hypothesis that union membership does not a¤ect wages for males. However, we can also see that the coe¢ cient for “Female Union Member” is 0.022 with a standard error of 0.020. While the point estimate suggests a wage premium of 2.2%, the standard error suggests that it is also plausible that the true 163 0g CHAPTER 8. HYPOTHESIS TESTING 164 coe¢ cient could be zero and the point estimate is merely sampling error. In this case what decision should we make? A hypothesis test consists of a real-valued test statistic Tn = Tn ((y1 ; x1 ) ; :::; (yn ; xn )) ; a critical value c, plus the decision rule 1. Accept H0 if Tn < c; 2. Reject H0 if Tn c: The test statistic Tn should be designed so that small values of Tn are likely when H0 is true and large values of Tn are likely when H1 is true. For example, for a test of H0 : = 0 against H1 : 6= 0 the standard test statistic is the absolute t-statistic Tn = jtn ( 0 )j ; as we expect Tn to have a well-behaved distribution = 0 ; and to be large when 6= 0 : Given the two possible states of the world (H0 or H1 ) and the two possible decisions (Accept H0 or Reject H0 ), there are four possible pairings of states and decisions as is depicted in the following chart. Hypthesis Testing Decisions H0 true H1 true Accept H0 Correct Decision Type II Error Reject H0 Type I Error Correct Decision Hypothesis tests are useful if they avoid making errors. There are two possible errors: (1) Rejecting H0 when H0 is true; and (2) Accepting H0 when H0 is false. These two errors are called Type I and Type II errors, respectively. As the events are random it is constructive to evaluate the tests based on the probability of their making an error. For a given test we de…ne the rejection probability function as the probability of rejecting the null hypothesis n( ) = Pr (Reject H0 j ) = Pr (Tn c j ): The rejection probability in general depends on the unknown parameter and the sample size n: For parameter values in the null hypothesis, 2 0 ; n ( ) is the probability of making a Type I error. We also call n ( ) the power function of the test, as for parameter values in the alternative, 2 c0 , n ( ) equals 1 minus the probability of a Type II error. For the reasons discussed in Chapter 6, in typical econometric models the exact sampling distribution of statistics such as Tn is unknown and hence n ( ) is unknown. Therefore we typically rely on asymptotic approximations which allow a more precise characterization. In particular, we focus on the asymptotic rejection probability ( ) = lim n( n!1 ): Therefore, it is convenient to select a test statistic Tn which has a well speci…ed asymptotic disd tribution under H0 , that is, Tn ! under H0 : Furthermore, if the distribution F of does not depend on 2 0 we say that Tn is asymptotically pivotal. In this case, ( ) = lim Pr (Tn n!1 = Pr ( = 1 c) F (c) cj ) CHAPTER 8. HYPOTHESIS TESTING 165 is only a function of c. For example, if Tn is the absolute t-statistic, then by Theorem 6.11.1, under d N(0; 1) and thus F (c) = (c); the symmetrized normal distribution H0 ; Tn ! jZj where Z function de…ned in (6.40). Given a test statistic Tn , how should we select the critical value c? Larger values of c mean that the test rejects less frequently, which decreases the probability of a Type I error but increases the probability of a Type II error. How can we balance one against the other? The dominant approach is to give special priority to the null hypothesis, and select c so that the Type I error probability is controlled at a speci…ed level, meaning that we select a signi…cance level 2 (0; 1) and then pick c so that ( ) for all 2 0 : When the test statistic Tn is asymptotically pivotal with distribution F this is accomplished by selecting c so that 1 F (c) = : There is no objective scienti…c basis for choice of signi…cance level : However, the common practice is to set = :05 (5%), which implies that the critical value c is selected so that F (c) = 0:95: For example, if Tn is the absolute t-statistic, we …nd from a normal table that (1:96) = 0:95, so that the 5% critical value is thus c = 1:96: The reasonsing behind the choice of a 5% critical value is to ensure that Type I errors should be relatively unlikely –that the decision “Reject H0 ”has scienti…c strength –yet the test retains power against reasonable alternatives. The decision “Reject H0 ” means that the evidence is inconsistent with the null hypothesis, in the sense that it is relatively unlikely (1 in 20) that data generated by the null hypothesis would yield the observed test result. In contrast, the decision “Accept H0 ” is not a strong statement. It does not mean that the evidence supports H0 , only that there is insu¢ cient evidence to reject H0 . Because of this, it is more accurate to use the label “Do not Reject H0 ” instead of “Accept H0 ”. When a test rejects H0 at the 5% signi…cance level it is common to say that the statistic is statistically signi…cant and if the test accepts H0 it is common to say that the statistic is statistically insigni…cant. It is helpful to remember that this is simply a way of saying “Using the statistic Tn , the hypothesis H0 can [cannot] be rejected at the asymptotic 5% level.” When the null hypothesis H0 : = 0 is rejected it is common to say that the coe¢ cient is statistically signi…cant, because the test has shown that the coe¢ cient is not equal to zero. Let us return to the example about the union wage premium. The absolute t-statistic for the coe¢ cient on “Male Union Member”is 0:095=0:020 = 4:75; which is greater than the 5% asymptotic critical value of 1.96. Therefore we reject the hypothesis that union membership does not a¤ect wages for men. However, the absolute t-statistic for the coe¢ cient on “Female Union Member” is 0:022=0:020 = 1:10; which is less than 1.96 and therefore we do not reject the hypothesis that union membership does not a¤ect wages for women. We thus say that the e¤ect of union membership for men is statistically signi…cant, while membership for women is not statistically signi…cant. When a test accepts a null hypothesis, it is commonly interpreted as evidence that the null hypothesis is true. In our wage example, a common interpretation is that “the regression …nds that female union membership has no e¤ect on wages”. This is an incorrect and most unfortunate interpretation. The test has failed to reject the hypothesis that the coe¢ cient is zero, but that does not mean that the coe¢ cient is actually zero. The test could be making a Type II error. Consider another question: Does marriage status a¤ect wages? To test the hypothesis that marriage status has no e¤ect on wages, we examine the t-statistics for the coe¢ cients on “Married Male” and “Married Female” in Table 5.1, which are 0:180=0:008 = 22:5 and 0:016=0:008 = 2:0; respectively. Both exceed the asymptotic 5% critical value of 1.96, so we reject the hypothesis for both men and women. But the statistic for men is exceptionally high, and that for women is only slightly above the critical value. Suppose in contrast that the t-statistic had been 1.9, which is less than the critical value, leading to the decision “Accept H0 ” rather than “Reject H0 ”. Should we really be making a di¤erent decision if the t-statistic is 1.9 rather than 2.0? The di¤erence in values is small, shouldn’t the di¤erence in the decision be also small? Thinking through these examples it seems unsatisfactory to simply report “Accept H0 ” or “Reject H0 ”. These two decisions do not summarize the evidence. Instead, the magnitude of the statistic Tn suggests a “degree of evidence” CHAPTER 8. HYPOTHESIS TESTING 166 against H0 : How can we take this into account? The answer is to report what is known as the asymptotic p-value pn = 1 F (Tn ): Since the distribution function F is monotonically increasing, the p-value is a monotonically decreasing function of Tn and is an equivalent test statistic. Instead of rejecting H0 at the signi…cance level if Tn c; we can reject H0 if pn : Thus it is su¢ cient to report pn ; and let the reader decide. Furthermore, the asymptotic p-value has a very convenient asymptotic null distribution. Since d d Tn ! under H0 ; then pn = 1 F (Tn ) ! 1 F ( ), which has the distribution Pr (1 F( ) u) = Pr (1 = 1 u F ( )) Pr = 1 F F = 1 (1 F 1 (1 1 (1 u) u) u) = u; d which is the uniform distribution on [0; 1]: Thus pn ! U[0; 1]: This means that the “unusualness” of pn is easier to interpret than the “unusualness” of Tn : The implication is that the best empirical practice is to compute and report the asymptotic pvalue pn rather than simply the test statistic Tn or the binary decision Accept/Reject. The p-value is a simple statistic, easy to interpret, and contains more information than the other choices. For example, consider the tests for the e¤ect of marriage status on wages. The p-value for men is 0.000 which we can intrepret as a very strong rejection of the hypothesis of no e¤ect, while the p-value for women is 0.046, which we can describe as “borderline signi…cant”. We now summarize the main features of hypothesis testing. 1. Select a signi…cance level : d 2. Select a test statistic Tn with asymptotic distribution Tn ! 3. Set the critical value c so that 1 under H0 : F (c) = ; where F is the distribution function of : 4. Calculate the asymptotic p-value pn = 1 F (Tn ): 5. Reject H0 if Tn : c; or equivalently pn 6. Accept H0 if Tn < c; or equivalently pn > : 7. Report pn to summarize the evidence concerning H0 versus H1 : 8.2 t tests The most commonly applied statistical tests are hypotheses on individual coe¢ cients and can be written in the form H0 : = 0 = h( ), (8.1) where 0 is some pre-speci…ed value. Quite typically, 0 = 0; as interest focuses on whether or not a coe¢ cient equals zero, but this is not the only possibility. For example, interest may focus on whether an elasticity equals 1, in which case we may wish to test H0 : = 1: As we described in the previous section, tests of H0 typically are based on the t-statistic tn ( ) = b s(b) CHAPTER 8. HYPOTHESIS TESTING 167 where b is the point estimate and s(b) is its standard error. The test will depend on whether the alternative hypothesis is one-sided or two-sided. The two-sided alternative is H1 : 6= (8.2) 0 which is appropriate for general tests of signi…cance. In this case the test statistic is the absolute value of the t-statistic b 0 tn = jtn ( 0 )j = : b s( ) Since Theorem 6.11.1 established that when = 0; d tn ! jZj where Z N(0; 1), then asymptotic critical values can be taken from the normal distribution table. In particular, the asymptotic 5% critical value is c = 1:96: Also, the asymptotic p-value for H0 against H1 is pn = 2 (1 (tn )) : Theorem 8.2.1 Under Assumptions 6.1.2 and 6.9.1, and H0 ; d jtn ( 0 )j ! jZj and for c satisfying = 2 (1 (c)) ; Pr (jtn ( 0 )j so the test “Reject H0 if jtn ( 0 )j c j H0 ) ! c” has asymptotic signi…cance level : We described (8.2) as a “two-sided alternative”. Sometimes we are interested in testing for one-sided alternatives such as H1 : > 0 (8.3) or H1 : < 0: (8.4) Tests of (8.1) against (8.3) or (8.4) are based on the signed t-statistic tn = tn ( 0 ) = b s(b) 0 : The hypothesis (8.1) is rejected in favor of (8.3) if tn c where c satis…es = 1 (c): Negative values of tn are not taken as evidence against H0 ; as point estimates b less than 0 do not point to (8.3). Since the critical values are taken from the single tail of the normal distribution, they are smaller than for two-sided tests. Speci…cally, the asymptotic 5% critical value is = 1:645: Thus we can reject (8.1) is rejected in favor of (8.3) if tn 1:645: Conversely, tests of (8.1) against (8.4) reject H0 for negative t-statistics, e.g. if tn c. For this alternative large positive values of tn are not evidence against H0 : An asymptotic 5% test rejects if tn 1:645: There seems to be an ambiguity. Should we use the two-sided critical value 1.96 or the one-sided critical value 1.645? The answer is that we should use one-sided tests and critical values only when the parameter space is known to satisfy a one-sided restriction such as 0 : This is when the test of (8.1) against (8.3) makes sense. If the restriction is not known a priori, then imposing 0 this restriction to test (8.1) against (8.3) does not makes sense. Since linear regression coe¢ cients do not have a priori sign restrictions, we conclude that two-sided tests are generally appropriate. CHAPTER 8. HYPOTHESIS TESTING 8.3 168 t-ratios and the Abuse of Testing In Section 4.14, we argued that a good applied practice is to report coe¢ cient estimates ^ and standard errors s(^) for all coe¢ cients of interest in estimated models. With ^ and s(^) the reader ^ can easily construct con…dence intervals [^ 2s(^)] and t-statistics ^ 0 =s( ) for hypotheses of interest. Some applied papers (especially older ones) instead report estimates ^ and t-ratios tn = ^=s(^); not standard errors. Reporting t-ratios instead of standard errors is poor econometric practice. While the same information is being reported (you can back out standard errors by division, e.g. s(^) = ^=tn ); standard errors are generally more helpful to readers than t-ratios. Standard errors help the reader focus on the estimation precision and con…dence intervals, while t-ratios focus attention on statistical signi…cance. While statistical signi…cance is important, it is less important that the parameter estimates themselves and their con…dence intervals. The focus should be on the meaning of the parameter estimates, their magnitudes, and their interpretation, not on listing which variables have signi…cant (e.g. non-zero) coe¢ cients. In many modern applications, sample sizes are very large so standard errors can be very small. Consequently t-ratios can be large even if the coe¢ cient estimates are economically small. In such contexts it may not be interesting to announce “The coe¢ cient is non-zero!” Instead, what is interesting to announce is that “The coe¢ cient estimate is economically interesting!” In particular, some applied papers report coe¢ cient estimates and t-ratios, and limit their discussion of the results to describing which variables are “signi…cant”(meaning that their t-ratios exceed 2) and the signs of the coe¢ cient estimates. This is very poor empirical work, and should be studiously avoided. It is also a receipe for banishment of your work to lower tier economics journals. Fundamentally, the common t-ratio is a test for the hypothesis that a coe¢ cient equals zero. This should be reported and discussed when this is an interesting economic hypothesis of interest. But if this is not the case, it is distracting. In general, when a coe¢ cient is of interest, it is constructive to focus on the point estimate, its standard error, and its con…dence interval. The point estimate gives our “best guess” for the value. The standard error is a measure of precision. The con…dence interval gives us the range of values consistent with the data. If the standard error is large then the point estimate is not a good summary about : The endpoints of the con…dence interval describe the bounds on the likely possibilities. If the con…dence interval embraces too broad a set of values for ; then the dataset is not su¢ ciently informative to render inferences about : On the other hand if the con…dence interval is tight, then the data have produced an accurate estimate, and the focus should be on the value and interpretation of this estimate. In contrast, the statement “the t-ratio is highly signi…cant” has little interpretive value. The above discussion requires that the researcher knows what the coe¢ cient means (in terms of the economic problem) and can interpret values and magnitudes, not just signs. This is critical for good applied econometric practice. For example, consider the question about the e¤ect of marriage status on mean log wages. We had found that the e¤ect is “highly signi…cant” for men and “marginally signi…cant” for women. Now, let’s construct asymptotic con…dence intervals for the coe¢ cients. The one for men is [0:16; 0:20] and that for women is [0:00; 0:03]: This shows that average wages for married men are about 16-20% higher than for unmarried men, which is very substantial, while the di¤erence for women about 0-3%, which is small. These magnitudes may be more informative than the results of the hypothesis tests. CHAPTER 8. HYPOTHESIS TESTING 8.4 169 Wald Tests The t-test is appropriate when the null hypothesis is a real-valued restriction. More generally, there may be multiple restrictions on the coe¢ cient vector : For a q 1 vector of functions r, we can write a multiple testing problem as H0 : r( ) = 0 H1 : r( ) 6= 0 It is natural to estimate = r( ) by the plug-in estimate b = r( b ): As this is a q 1 vector, we can assess its magnitude by constructing a quadratic form such as the Wald statistic (6.44) evaluated at the null hypothesis 0 Wn = nb Vb 1 b (8.5) 1 0 b Vb R b = nr( b )0 R where b = @ r( b )0 : R @ r( b ) The Wald statistic Wn is a weighted Euclidean measure of the length of the vector b: When q = 1 then Wn = t2n ; the square of the t-statistic, so hypothesis tests based on Wn and jtn j are equivalent. The Wald statistic (8.5) is a generalization of the t-statistic to the case of multiple restrictions. When r( ) = R0 c is a linear function of ; then the Wald statistic simpli…es to Wn = n R 0 b c 0 1 R0 Vb R R0 b c : As shown in Theorem 6.15.2, when r( ) = 0 then Wn ! 2q ; a chi-square random variable with q degrees of freedom. Let Fq (u) denote the 2q distribution function. For a given signi…cance level ; the asymptotic critical value c satis…es = 1 Fq (c) and can be found from the chi-square distribution table. For example, the 5% critical values for q = 1; q = 2; and q = 3 are 3.84, 5.99, and 7.82, respectively. An asymptotic test rejects H0 in favor of H1 if Wn c: As with t-tests, it is conventional to describe a Wald test as “signi…cant” if Wn exceeds the 5% critical value. Theorem 8.4.1 Under Assumptions 6.1.2 and 6.9.1, rank(H ) = q; and d H0 ; then Wn ! 2, q and for c satisfying Pr (Wn so the test “Reject H0 if Wn =1 Fq (c); c j H0 ) ! c” has asymptotic signi…cance level : Notice that the asymptotic distribution in Theorem 8.4.1 depends solely on q –the number of restrictions being tested. It does not depend on k –the number of parameters estimated. The asymptotic p-value for Wn is pn = 1 Fq (Wn ). For multiple hypothesis tests it is particularly useful to report p-values instead of the Wald statistic. For example, if you write that a Wald test on eight restrictions (q = 8) has the value Wn = 11:2; it is di¢ cult for a reader to assess the magnitude of this statistic without the time-consuming and cumbersome process of looking up the critical values from a table. Instead, if you write that the p-value is pn = 0:19 (as is the case for Wn = 11:2 and q = 8) then it is simple for a reader to intrepret its magnitude. CHAPTER 8. HYPOTHESIS TESTING 170 For example consider the empirical results presented in Table 5.1. The hypothesis “Union membership does not a¤ect wages" is the joint restriction that the coe¢ cients on “Male Union Member” and “Female Union Member” are jointly zero. We calculate the Wald statistic (8.5) for this joint hypothesis and …nd Wn = 23:14 with a p-value of pn = 0:00: Thus we can reject the hypothesis in favor of the alternative that at least one of the coe¢ cients is non-zero. This does not mean that both coe¢ cients are non-zero, just that one of the two is non-zero. Therefore examining the joint Wald statistic and the invididual t-statistics is useful to interpret the coe¢ cients. If the error is known to be homoskedastic, then it is appropriate to replace Vb in (8.5) with 0 b 1 s2 : In this case the Wald statisic equals the homoskedastic covariance matrix estimate Vb = Q xx 1 b 0 Vb 0 R b Wn0 = nr( b )0 R = 1 b 0Q b 1R b nr( b )0 R xx s2 1 0 = r( b ) b X 0X R b r( b )0 R s2 r( b ) r( b ) : (8.6) The Wald statistic is named after the statistician Abraham Wald, who showed that Wn has optimal weighted average power in certain settings. 8.5 Minimum Distance Tests A minimum distance test measures the distance between b and the restricted estimate e . Recall that under the restriction r( ) = 0 the minimum distance estimate solves the minimization problem e = argmin Jn ( ) r( )=0 where 0 Jn ( ) = n b 1 0 and W n is a weight matrix. Setting W n = Vb e Wn b yields the constrained least squares estimator 1 and setting W n = Vb yields the e¢ cient minimum distance estimator e emd : The minimum distance test statistic of H0 against H1 is cls Jn = Jn ( e ) = When r( ) = R0 c is linear then 0 Setting W n = Vb 1 min Jn ( ) r( )=0 = n b Jn = n R0 b c 0 e 0 Wn b R0 W n 1 R we …nd Jn0 := Jn ( e cls ) = n R0 b c 0 0 R0 Vb R 1 e : R0 b 1 R0 b c : c = Wn0 (8.7) CHAPTER 8. HYPOTHESIS TESTING and setting W n = Vb 1 171 we …nd Jn := Jn ( e emd ) = n R0 b c 0 1 R0 Vb R R0 b c = Wn : Thus for linear hypotheses r( ) = R0 c, Jn is identical to the Wald statistic (8.5), and equals the homoskedastic Wald statistic (8.6). When r( ) is non-linear then the Wald and minimum distance statistics are di¤erent. Jn0 Theorem 8.5.1 Under Assumptions 6.1.2 and 6.9.1, rank(H ) = q; and d H0 ; then Jn ! 2. q Testing using the minimum distance statistic Jn is similar to testing using the Wald statistic Wn . Critical values and p-values are computed using the 2q distribution. H0 is rejected in favor of H1 if Jn exceeds the level critical value. The asymptotic p-value is pn = 1 Fq (Jn ): 8.6 F Tests When the hypothesis is the linear restriction H0 : R0 c=0 then a classic statistic for H0 against H1 is the F statistic SSEn ( b ) =q SSEn ( e cls ) Fn = n = k q where SSEn ( b )=(n k) ^2 ~2 (8.8) (8.9) ^2 n ^2 = SSEn ( b ) 1X 2 = e^i n n i=1 is the residual variance estimate under H0 and n ~2 = SSEn ( e cls ) 1X 2 = e~i n n i=1 with e~i = yi x0i e cls is the residual variance estimate under H1 . If we recall equation (7.15) which showed that SSEn ( ) = s2 (n also write Fn as Fn = Jn ( e cls ) = Jn ( e cls )=q = Wn0 =q: k + Jn ( )) ; then we can Jn ( b ) =q The second equality uses the fact that Jn ( b ) = 0 and the third equality is (8.7). Thus the Fn statistic equals the homoskedastic Wald statistic divided by q: It follows that they are equivalent tests for H0 against H1 : CHAPTER 8. HYPOTHESIS TESTING 172 In many statistical packages, linear hypothesis tests are reported as Fn rather than Wn . While they are equivalent, it is important to know which is being reported to know which critical values to use. (If p-values are directly reported this is not an issue.) Most packages will calculate critical values and p-values using the F (q; n k) distribution rather than 2q : This is a prudent small sample adjustment, as the F distribution is exact when the errors are independent of the regressors and normally distributed. However, when the degrees of freedom n k are large then the di¤erence is negligible. More relevantly, if n k is small enough to make a di¤erence, probably we shouldn’t be trusting the asymptotic approximation anyway! An elegant feature about (8.8) is that it is directly computable from the standard output from two simple OLS regressions, as the sum of squared errors (or regression variance) is a typical output from statistical packages, and is often reported in applied tables. Thus Fn can be calculated by hand from standard reported statistics even if you don’t have the original data (or if you are sitting in a seminar and listening to a presentation!). If you are presented with an F statistic (or a Wald statistic, as you can just divide by q) but don’t have access to critical values, a useful rule of thumb is to know that for large n; the 5% asymptotic critical value is decreasing as q increases, and is less than 2 for q 7: In many statistical packages, when an OLS regression is estimated, an “F-statistic”is reported. This is Fn where H0 restricts all coe¢ cients except the intercept to be zero. This was a popular statistic in the early days of econometric reporting, when sample sizes were very small and researchers wanted to know if there was “any explanatory power” to their regression. This is rarely an issue today, as sample sizes are typically su¢ ciently large that this F statistic is nearly always highly signi…cant. While there are special cases where this F statistic is useful, these cases are atypical. As a general rule, there is no reason to report this F statistic. The F statistic is named after the statistician Ronald Fisher, one of the founders of modern statistical theory. 8.7 Likelihood Ratio Test For a model with parameter for H0 : 2 0 versus H1 : 2 2 is and likelihood function Ln ( ) the likelihood ratio statistic c 0 LRn = 2 max log Ln ( ) 2 = 2 log Ln (b) max log Ln ( ) 2 0 log Ln (e) where b and e are the unrestricted and constrained MLE. In the normal linear model the maximized log likelihood (3.44) at the unrestricted and restricted estimates are n n log L b mle ; ^ 2mle = (log (2 ) + 1) log ^ 2 2 2 and n n log L e cmle ; ~ 2cmle = (log (2 ) + 1) log ~ 2 2 2 respectively. Thus the LR statistic is LRn = n log ~ 2 = n log log ^ 2 ~2 ^2 which is a monotonic function of ~ 2 =^ 2 : Recall that the F statistic (8.9) is also a monotonic function of ~ 2 =^ 2 : Thus LRn and Fn are fundamentally the same statistic and have the same information about H0 versus H1 : CHAPTER 8. HYPOTHESIS TESTING 173 Furthermore, by a …rst-order Taylor series approximation LRn =q = n ~2 log 1 + 2 q ^ 1 ~2 ^2 n q ' 1 ' Fn : This shows that the two statistics (LRn and Fn ) will be numerically close. It also shows that the F statistic and the homoskedastic Wald statistic for linear hypotheses can also be interpreted as approximate likelihood ratio statistics under normality. 8.8 Problems with Tests of NonLinear Hypotheses While the t and Wald tests work well when the hypothesis is a linear restriction on ; they can work quite poorly when the restrictions are nonlinear. This can be seen by a simple example introduced by Lafontaine and White (1986). Take the model yi = ei + ei 2 N(0; ) and consider the hypothesis H0 : = 1: Let ^ and ^ 2 be the sample mean and variance of yi : The standard Wald test for H0 is ^ Wn = n 2 1 : ^2 Now notice that H0 is equivalent to the hypothesis H0 (r) : for any positive integer r: Letting h( ) = Wald test for H0 (r) is r r =1 ; and noting H = r ^r Wn (r) = n ^ 2 r2 ^ 1 r 1 ; we …nd that the standard 2 2r 2 : While the hypothesis r = 1 is una¤ected by the choice of r; the statistic Wn (r) varies with r: This is an unfortunate feature of the Wald statistic. To demonstrate this e¤ect, we have plotted in Figure 8.1 the Wald statistic Wn (r) as a function of r; setting n= 2 = 10: The increasing solid line is for the case ^ = 0:8: The decreasing dashed line is for the case ^ = 1:6: It is easy to see that in each case there are values of r for which the test statistic is signi…cant relative to asymptotic critical values, while there are other values of r for which the test statistic is insigni…cant. This is distressing since the choice of r is arbitrary and irrelevant to the actual hypothesis. d Our …rst-order asymptotic theory is not useful to help pick r; as Wn (r) ! 21 under H0 for any r: This is a context where Monte Carlo simulation can be quite useful as a tool to study and compare the exact distributions of statistical procedures in …nite samples. The method uses random simulation to create arti…cial datasets, to which we apply the statistical tools of interest. This produces random draws from the statistic’s sampling distribution. Through repetition, features of this distribution can be calculated. In the present context of the Wald statistic, one feature of importance is the Type I error of the test using the asymptotic 5% critical value 3.84 – the probability of a false rejection, Pr (Wn (r) > 3:84 j = 1) : Given the simplicity of the model, this probability depends only on r; n; and 2 : In Table 8.1 we report the results of a Monte Carlo simulation where we vary these CHAPTER 8. HYPOTHESIS TESTING 174 Figure 8.1: Wald Statistic as a function of r three parameters. The value of r is varied from 1 to 10, n is varied among 20, 100 and 500, and is varied among 1 and 3. The Table reports the simulation estimate of the Type I error probability from 50,000 random samples. Each row of the table corresponds to a di¤erent value of r –and thus corresponds to a particular choice of test statistic. The second through seventh columns contain the Type I error probabilities for di¤erent combinations of n and . These probabilities are calculated as the percentage of the 50,000 simulated Wald statistics Wn (r) which are larger than 3.84. The null hypothesis r = 1 is true, so these probabilities are Type I error. To interpret the table, remember that the ideal Type I error probability is 5% (.05) with deviations indicating distortion. Type I error rates between 3% and 8% are considered reasonable. Error rates above 10% are considered excessive. Rates above 20% are unacceptable. When comparing statistical procedures, we compare the rates row by row, looking for tests for which rejection rates are close to 5% and rarely fall outside of the 3%-8% range. For this particular example the only test which meets this criterion is the conventional Wn = Wn (1) test. Any other choice of r leads to a test with unacceptable Type I error probabilities. In Table 8.1 you can also see the impact of variation in sample size. In each case, the Type I error probability improves towards 5% as the sample size n increases. There is, however, no magic choice of n for which all tests perform uniformly well. Test performance deteriorates as r increases, which is not surprising given the dependence of Wn (r) on r as shown in Figure 8.1. Table 8.1 Type I error Probability of Asymptotic 5% Wn (r) Test CHAPTER 8. HYPOTHESIS TESTING 175 =1 =3 r n = 20 n = 100 n = 500 n = 20 n = 100 n = 500 1 .06 .05 .05 .07 .05 .05 2 .08 .06 .05 .15 .08 .06 3 .10 .06 .05 .21 .12 .07 4 .13 .07 .06 .25 .15 .08 5 .15 .08 .06 .28 .18 .10 6 .17 .09 .06 .30 .20 .11 7 .19 .10 .06 .31 .22 .13 8 .20 .12 .07 .33 .24 .14 9 .22 .13 .07 .34 .25 .15 10 .23 .14 .08 .35 .26 .16 Note: Rejection frequencies from 50,000 simulated random samples In this example it is not surprising that the choice r = 1 yields the best test statistic. Other choices are arbitrary and would not be used in practice. While this is clear in this particular example, in other examples natural choices are not always obvious and the best choices may in fact appear counter-intuitive at …rst. This point can be illustrated through another example which is similar to one developed in Gregory and Veall (1985). Take the model yi = 0 + x1i 1 + x2i 2 + ei (8.10) E (xi ei ) = 0 and the hypothesis 1 H0 : =r 2 where r is a known constant. Equivalently, de…ne = 1 = 2 ; so the hypothesis can be stated as H0 : = r: Let b = ( ^ 0 ; ^ 1 ; ^ 2 ) be the least-squares estimates of (8.10), let Vb be an estimate of the asymptotic covariance matrix for b and set ^ = ^ 1 = ^ 2 . De…ne 1 0 0 C B C B B 1 C C B C c1 = B H B ^2 C B C C B B ^ C @ A 1 ^2 2 so that the standard error for ^ is s(^) = n 1H ^ 01 Vb ^ H 1 ^ 1 ^ : In this case a t-statistic for H0 is r 2 t1n = 1=2 : s(^) An alternative statistic can be constructed through reformulating the null hypothesis as H0 : r 1 2 = 0: A t-statistic based on this formulation of the hypothesis is ^ t2n = n 1 1H 0 V b 2 r ^2 1=2 H2 : CHAPTER 8. HYPOTHESIS TESTING where 176 0 1 0 H2 = @ 1 A : r To compare t1n and t2n we perform another simple Monte Carlo simulation. We let x1i and x2i be mutually independent N(0; 1) variables, ei be an independent N(0; 2 ) draw with = 3, and normalize 0 = 0 and 1 = 1: This leaves 2 as a free parameter, along with sample size n: We vary 2 among :1; .25, .50, .75, and 1.0 and n among 100 and 500: 2 .10 .25 .50 .75 1.00 Table 8.2 Type I error Probability of Asymptotic 5% t-tests n = 100 n = 500 Pr (tn < 1:645) Pr (tn > 1:645) Pr (tn < 1:645) Pr (tn > 1:645) t1n t2n t1n t2n t1n t2n t1n t2n .47 .06 .00 .06 .28 .05 .00 .05 .26 .06 .00 .06 .15 .05 .00 .05 .15 .06 .00 .06 .10 .05 .00 .05 .12 .06 .00 .06 .09 .05 .00 .05 .10 .06 .00 .06 .07 .05 .02 .05 The one-sided Type I error probabilities Pr (tn < 1:645) and Pr (tn > 1:645) are calculated from 50,000 simulated samples. The results are presented in Table 8.2. Ideally, the entries in the table should be 0.05. However, the rejection rates for the t1n statistic diverge greatly from this value, especially for small values of 2 : The left tail probabilities Pr (t1n < 1:645) greatly exceed 5%, while the right tail probabilities Pr (t1n > 1:645) are close to zero in most cases. In contrast, the rejection rates for the linear t2n statistic are invariant to the value of 2 ; and are close to the ideal 5% rate for both sample sizes. The implication of Table 4.2 is that the two t-ratios have dramatically di¤erent sampling behavior. The common message from both examples is that Wald statistics are sensitive to the algebraic formulation of the null hypothesis. A simple solution is to use the minimum distance statistic Jn , which equals Wn with r = 1 in the …rst example, and t2n in the second example. The minimum distance statistic is invariant to the algebraic formulation of the null hypothesis, so is immune to this problem. Whenever possible, the Wald statistic should not be used to test nonlinear hypotheses. 8.9 Monte Carlo Simulation In the Section 8.8 we introduced the method of Monte Carlo simulation to illustrate the small sample problems with tests of nonlinear hypotheses. In this section we describe the method in more detail. Recall, our data consist of observations (yi ; xi ) which are random draws from a population distribution F: Let be a parameter and let Tn = Tn ((y1 ; x1 ) ; :::; (yn ; xn ) ; ) be a statistic of interest, for example an estimator ^ or a t-statistic (^ )=s(^): The exact distribution of Tn is Gn (u; F ) = Pr (Tn u j F): While the asymptotic distribution of Tn might be known, the exact (…nite sample) distribution Gn is generally unknown. Monte Carlo simulation uses numerical simulation to compute Gn (u; F ) for selected choices of F: This is useful to investigate the performance of the statistic Tn in reasonable situations and sample sizes. The basic idea is that for any given F; the distribution function Gn (u; F ) can be calculated CHAPTER 8. HYPOTHESIS TESTING 177 numerically through simulation. The name Monte Carlo derives from the famous Mediterranean gambling resort where games of chance are played. The method of Monte Carlo is quite simple to describe. The researcher chooses F (the distribution of the data) and the sample size n. A “true” value of is implied by this choice, or equivalently the value is selected directly by the researcher which implies restrictions on F . Then the following experiment is conducted by computer simulation: 1. n independent random pairs (yi ; xi ) ; i = 1; :::; n; are drawn from the distribution F using the computer’s random number generator. 2. The statistic Tn = Tn ((y1 ; x1 ) ; :::; (yn ; xn ) ; ) is calculated on this pseudo data. For step 1, most computer packages have built-in procedures for generating U[0; 1] and N(0; 1) random numbers, and from these most random variables can be constructed. (For example, a chi-square can be generated by sums of squares of normals.) For step 2, it is important that the statistic be evaluated at the “true”value of corresponding to the choice of F: The above experiment creates one random draw from the distribution Gn (u; F ): This is one observation from an unknown distribution. Clearly, from one observation very little can be said. So the researcher repeats the experiment B times, where B is a large number. Typically, we set B = 1000 or B = 5000: We will discuss this choice later. Notationally, let the bth experiment result in the draw Tnb ; b = 1; :::; B: These results are stored. After all B experiments have been calculated, these results constitute a random sample of size B from the distribution of Gn (u; F ) = Pr (Tnb u) = Pr (Tn u j F ) : From a random sample, we can estimate any feature of interest using (typically) a method of moments estimator. For example: Suppose we are interested in the bias, mean-squared error (MSE), and/or variance of the distribution of ^ : We then set Tn = ^ ; run the above experiment, and calculate \^) = Bias( M\ SE(^) = B B 1 X 1 X^ Tnb = b B B 1 B b=1 B X b=1 b=1 B 1 X ^ (Tnb ) = b B 2 \ ^) = M\ var( SE(^) 2 b=1 \^) Bias( 2 Suppose we are interested in the Type I error associated with an asymptotic 5% two-sided t-test. We would then set Tn = ^ =s(^) and calculate B 1 X 1 (Tnb P^ = B 1:96) ; (8.11) b=1 the percentage of the simulated t-ratios which exceed the asymptotic 5% critical value. Suppose we are interested in the 5% and 95% quantile of Tn = ^ or Tn = ^ =s(^) We then compute the 5% and 95% sample quantiles of the sample fTnb g: The % sample quantile is a number q such that % of the sample are less than q : A simple way to compute sample quantiles is to sort the sample fTnb g from low to high. Then q is the N ’th number in this ordered sequence, where N = (B + 1) : It is therefore convenient to pick B so that N is an integer. For example, if we set B = 999; then the 5% sample quantile is 50’th sorted value and the 95% sample quantile is the 950’th sorted value. CHAPTER 8. HYPOTHESIS TESTING 178 The typical purpose of a Monte Carlo simulation is to investigate the performance of a statistical procedure (estimator or test) in realistic settings. Generally, the performance will depend on n and F: In many cases, an estimator or test may perform wonderfully for some values, and poorly for others. It is therefore useful to conduct a variety of experiments, for a selection of choices of n and F: As discussed above, the researcher must select the number of experiments, B: Often this is called the number of replications. Quite simply, a larger B results in more precise estimates of the features of interest of Gn ; but requires more computational time. In practice, therefore, the choice of B is often guided by the computational demands of the statistical procedure. Since the results of a Monte Carlo experiment are estimates computed from a random sample of size B; it is straightforward to calculate standard errors for any quantity of interest. If the standard error is too large to make a reliable inference, then B will have to be increased. In particular, it is simple to make inferences about rejection probabilities from statistical tests, such as the percentage estimate reported in (8.11). The random variable 1 (Tnb 1:96) is iid Bernoulli, equalling 1 with probability p = E1 (Tnbp 1:96) : The average (8.11) is therefore an unbiased estimator of p with standard error s (^ p) = p (1 p) =B. As p is unknown, this may be approximated by replacing p with p^ or with an hypothesized value. Forp example, if we are assessing p an asymptotic 5% test, then we can set s (^ p) = (:05) (:95) =B ' :22= B: Hence, standard errors for B = 100; 1000, and 5000, are, respectively, s (^ p) = :022; :007; and .003. Most papers in econometric methods, and some empirical papers, include the results of Monte Carlo simulations to illustrate the performance of their methods. When extending existing results, it is good practice to start by replicating existing (published) results. This is not exactly possible in the case of simulation results, as they are inherently random. For example suppose a paper investigates a statistical test, and reports a simulated rejection probability of 0.07 based on a simulation with B = 100 replications. Suppose you attempt to replicate this result, and …nd a rejection probability of 0.03 (again using B = 100 simulation replications). Should you concluce that you have failed in your attempt? Absolutely not! Under the hypothesis that both simulations are identical, you have two independent estimates, p^1 = 0:07 and p^2 = 0:03, of a common probability p d p: The asymptotic (as B ! 1) distributionpof their di¤erence is B (^ p1 p^2 ) ! N(0; 2p(1 p)); so a standard error for p^1 p^2 = 0:04 is s^ = 2p(1 p)=B ' 0:03; where I estimate p = (^ p1 + p^2 )=2: Since the t-ratio 0:04=0:03 = 1:3 is not statistically signi…cant, it is incorrect to reject the null hypothesis that the two simulations are identical. The di¤erence between the results p^1 = 0:07 and p^2 = 0:03 is consistent with random variation. What should be done? The …rst mistake was to copy the previous paper’s choice of B = 100: Instead, suppose p you set B = 5000 and now obtain p^2 = 0:04: Then p^1 p^2 = 0:03 and a standard error is s^ = p(1 p) (1=100 + 1=5000) ' 0:02: Still we cannot reject the hypothesis that the two simulations are di¤erent. Even though the estimates (0:07 and 0:04) appear to be quite di¤erent, the di¢ culty is that the original simulation used a very small number of replications (B = 100) so the reported estimate is quite imprecise. In this case, it is appropriate to conclude that your results “replicate”the previous study, as there is no statistical evidence to reject the hypothesis that they are equivalent. Most journals have policies requiring authors to make available their data sets and computer programs required for empirical results. They do not have similar politicies regarding simulations. Never-the-less, it is good professional practice to make your simulations available. The best practice is to post your simulation code on your webpage. This invites others to build on and use your results, leading to possible collaboration, citation, and/or advancement. CHAPTER 8. HYPOTHESIS TESTING 8.10 179 Con…dence Intervals by Test Inversion There is a close relationship between hypothesis tests and con…dence intervals. We observed in Section 6.12 that the standard 95% asymptotic con…dence interval for a parameter is h i Cn = b 1:96 s(b); b + 1:96 s(b) (8.12) = f : jtn ( )j 1:96g : That is, we can describe Cn as “The point estimate plus or minus 2 standard errors”or “The set of parameter values not rejected by a two-sided t-test.”The second de…nition, known as “test statistic inversion” is a general method for …nding con…dence intervals, and typically produces con…dence intervals with excellent properties. Given a test statistic Tn ( ) and critical value c, the acceptance region “Accept if Tn ( ) c” is identical to the con…dence interval Cn = f : Tn ( ) cg. Since the regions are identical, the probability of coverage Pr ( 2 Cn ) equals the probability of correct acceptance Pr (Acceptj ) which is exactly 1 minus the Type I error probability. Thus inverting tests with good Type I error probabilities yields a con…dence interval with good coverage probabilities. Now suppose that the parameter of interest = h( ) is a nonlinear function of the coe¢ cient vector . In this case the standard con…denceqinterval for is the set Cn as in (8.12) where ^ = h( b ) is the point estimate and s(^) = n 1=2 H c 0 Vb H c is the delta method standard error. This con…dence interval is inverting the t-test based on the nonlinear hypothesis h( ) = : The trouble is that in Section 8.8 we learned that there is no unique t-statistic for tests of nonlinear hypotheses and that the choice of parameterization matters greatly. For example, if = 1 = 2 then the coverage probability of the standard interval (8.12) is 1 minus the probability of the Type I error, which as shown in Table 8.2 can be far from the nominal 5%. In this example a good solution is the same as discussed in Section 8.8 –to rewrite the hypothesis as a linear restriction. The hypothesis = 1 = 2 is the same as 2 = 1 : The t-statistic for this restriction is ^ ^ 1 2 tn ( ) = 1=2 0 b h Vh where h = 1 and Vb is the covariance matrix for ( ^ 1 ^ 2 ): A 95% con…dence interval for = 1 = 2 is the set of values of such that jtn ( )j 1:96: Since appears in both the numerator and denominator, tn ( ) is a non-linear function of so the easiest method to …nd the con…dence set is by grid search over : For example, in the wage equation log(W age) = 1 Experience + 2 Experience 2 =100 + the highest expected wage occurs at Experience = 50 1 = 2 : From Table 5.1 we have the point estimate ^ = 29:8 and we can calculate the standard error s(^) = 0:022 for a 95% con…dence interval [29:8; 29.9]. However, if we instead invert the linear form of the test we can numerically …nd the interval [29:1; 30.6] which is much larger. From the evidence presented in Section 8.8 we know the …rst interval can be quite inaccurate and the second interval is greatly preferred. CHAPTER 8. HYPOTHESIS TESTING 8.11 180 Asymptotic Power The power of a test is the probability of rejecting H0 when H1 is true. For simplicity suppose that yi is i.i.d. N ( ; 2 ) with 2 known, consider the t-statistic tn ( ) = p n (y ) = ; and tests of H0 : = 0 against H1 : > 1. We reject H0 if tn = tn (0) > c: Note that p tn = tn ( ) + n = and tn ( ) = Z has an exact N(0; 1) distribution. This is because tn ( ) is centered at the true mean ; while the test statistic tn (0) is centered at the (false) hypothesized mean of 0. The power of the test is p p Pr (tn c j ) = Pr Z + n = c = c n = : This function is monotonically increasing in and n; and decreasing in : Notice that for any c and 6= 0; the power increases to 1 as n ! 1: This means that for 2 H1 ; the test will reject H0 with probability approaching 1 as the sample size gets large. We call this property test consistency. test of H0 : 2 0 is consistent 2 1 ; Pr (Reject H0 j ) ! 1 as De…nition 8.11.1 A asymptotic level against …xed alternatives if for all n ! 1: For asymptotic level tests of the form “Reject H0 if Tn c”, a su¢ cient condition for test consistency is that the Tn diverges to positive in…nity with probability one for all 2 1 : p De…nition 8.11.2 Tn ! 1 as n ! 1 if for all M Pr (Tn M ) ! 0 as n ! 1: < 1; In general, t-test and Wald tests are consistent against …xed alternatives. Take a t-statistic for a test of H0 : = 0 b 0 tn = b s( ) p where 0 is a known value and s(b) = n 1 V^ . Note that p b n( 0) p tn = + s(b) V^ where the …rst term converges in distribution to N(0; 1) and the second term converges in probability to +1 if > 0 and converges to 1 if < 0 : Thus the two-sided t-test is consistent against H1 : 6= 0 ; and one-sided t-tests are consistent against the alternatives for which they are designed. The Wald statistic for H0 : = r( ) = 0 against H1 : 6= 0 is 0 Wn = nb Vb 0 1 p p Under H1 ; b ! 6= 0: Thus b Vb b ! 0 V implies that Wald tests are consistent tests. 1 1 b p > 0: Hence under H1 ; Wn ! 1. Again, this CHAPTER 8. HYPOTHESIS TESTING 181 Theorem 8.11.1 Under Assumptions 6.1.2 and 6.9.1, rank(H ) = q; for p = r( ) 6= 0; then Wn ! 1, so the test “Reject H0 if Wn c” is consistent against …xed alternatives. Consistency is a good property for a test, but does not give a useful approximation to the power function. One useful asymptotic method to compute a continuous approximation to the power function is based on local alternatives similar to our analysis of restriction estimation under misspeci…cation (Section 7.9). The technique is to index the parameter by sample size so that the asymptotic distribution of the statistic is continuous in a localizing parameter. Speci…cally, suppose that = r( ) = n 1=2 where is q 1 is the localizing parameter. Then p nb = = p p d n b + n b p n + !Z = N( ; V ) a non-central normal distribution, and 0 1 Wn = nb Vb d 1 ! Z0 V = 2 q( b Z ) where = 0 V 1 . The distribution 2q ( ) is the noncentral chi-square distribution with degrees of freedom q and noncentrality parameter ; and is a function of q and only. This is convenient as we then obtain the following approximation to the power function. As n ! 1; Pr (Wn c) ! Pr 2q ( ) c := q ( ; c): The function q ( ; c) is known as the asymptotic local power function, and for given c and q; depends only on the real-valued parameter 0. Theorem 8.11.2 Under Assumptions 6.1.2 and 6.9.1, rank(H ) = q; and = r( ) = n 1=2 ; then d Wn ! where = 0 V 1 2 q( ) : Furthermore, Pr (Wn c) ! Pr 2 q( ) c = q( ; c) CHAPTER 8. HYPOTHESIS TESTING 182 Exercises Exercise 8.1 Prove that if an additional regressor X k+1 is added to X; Theil’s adjusted R increases if and only if jtk+1 j > 1; where tk+1 = ^ k+1 =s( ^ k+1 ) is the t-ratio for ^ k+1 and s( ^ k+1 ) = s2 [(X 0 X) 1 2 1=2 ]k+1;k+1 is the homoskedasticity-formula standard error. Exercise 8.2 You have two independent samples (y 1 ; X 1 ) and (y 2 ; X 2 ) which satisfy y 1 = X 1 1 + e1 and y 2 = X 2 2 + e2 ; where E (x1i e1i ) = 0 and E (x2i e2i ) = 0; and both X 1 and X 2 have k columns. Let b 1 and b 2 be the OLS estimates of 1 and 2 . For simplicity, you may assume that both samples have the same number of observations n: (a) Find the asymptotic distribution of p n (b) Find an appropriate test statistic for H0 : b 2 2 = b 1 ( 2 1) as n ! 1: 1: (c) Find the asymptotic distribution of this statistic under H0 : Exercise 8.3 The data set invest.dat contains data on 565 U.S. …rms extracted from Compustat for the year 1987. The variables, in order, are Ii Investment to Capital Ratio (multiplied by 100). Qi Total Market Value to Asset Ratio (Tobin’s Q). Ci Cash Flow to Asset Ratio. Di Long Term Debt to Asset Ratio. The ‡ow variables are annual sums for 1987. The stock variables are beginning of year. (a) Estimate a linear regression of Ii on the other variables. Calculate appropriate standard errors. (b) Calculate asymptotic con…dence intervals for the coe¢ cients. (c) This regression is related to Tobin’s q theory of investment, which suggests that investment should be predicted solely by Qi : Thus the coe¢ cient on Qi should be positive and the others should be zero. Test the joint hypothesis that the coe¢ cients on Ci and Di are zero. Test the hypothesis that the coe¢ cient on Qi is zero. Are the results consistent with the predictions of the theory? (d) Now try a non-linear (quadratic) speci…cation. Regress Ii on Qi ; Ci , Di ; Q2i ; Ci2 , Di2 ; Qi Ci ; Qi Di ; Ci Di : Test the joint hypothesis that the six interaction and quadratic coe¢ cients are zero. Exercise 8.4 In a paper in 1963, Marc Nerlove analyzed a cost function for 145 American electric companies. (The problem is discussed in Example 8.3 of Greene, section 1.7 of Hayashi, and the empirical exercise in Chapter 1 of Hayashi). The data …le nerlov.dat contains his data. The variables are described on page 77 of Hayashi. Nerlov was interested in estimating a cost function: T C = f (Q; P L; P F; P K): CHAPTER 8. HYPOTHESIS TESTING 183 (a) First estimate an unrestricted Cobb-Douglass speci…cation log T Ci = 1 + 2 log Qi + 3 log P Li + 4 log P Ki + 5 log P Fi + ei : (8.13) Report parameter estimates and standard errors. You should obtain the same OLS estimates as in Hayashi’s equation (1.7.7), but your standard errors may di¤er. (b) What is the economic meaning of the restriction H0 : (c) Estimate (8.13) by constrained least-squares imposing meter estimates and standard errors. 3 + 3 (d) Estimate (8.13) by e¢ cient minimum distance imposing parameter estimates and standard errors. 4 + + 4 3 + + (e) Test H0 : 3 + 4 + 5 = 1 using a Wald statistic (f) Test H0 : 3 + 4 + 5 = 1 using a minimum distance statistic 5 = 1? 5 4 = 1. Report your para- + 5 = 1. Report your Chapter 9 Regression Extensions 9.1 Generalized Least Squares In the projection model, we know that the least-squares estimator is semi-parametrically e¢ cient for the projection coe¢ cient. However, in the linear regression model yi = x0i + ei E (ei j xi ) = 0; the least-squares estimator is ine¢ cient. The theory of Chamberlain (1987) can be used to show that in this model the semiparametric e¢ ciency bound is obtained by the Generalized Least Squares (GLS) estimator (4.13) introduced in Section 4.6.1. The GLS estimator is sometimes called the Aitken estimator. The GLS estimator (9.1) is infeasible since the matrix D is unknown. ^ = diagf^ 21 ; :::; ^ 2n g: A feasible GLS (FGLS) estimator replaces the unknown D with an estimate D We now discuss this estimation problem. Suppose that we model the conditional variance using the parametric form 2 i = = 0 0 + z 01i 1 zi; where z 1i is some q 1 function of xi : Typically, z 1i are squares (and perhaps levels) of some (or all) elements of xi : Often the functional form is kept simple for parsimony. Let i = e2i : Then E ( i j xi ) = 0 + z 01i 1 and we have the regression equation i E( This regression error i i = 0 + z 01i 1 + (9.1) i j xi ) = 0: is generally heteroskedastic and has the conditional variance var ( i j xi ) = var e2i j xi = E e2i E e2i j xi = E e4i j xi Suppose ei (and thus i) 2 j xi E e2i j xi were observed. Then we could estimate b = Z 0Z 1 Z0 184 p ! 2 : by OLS: CHAPTER 9. REGRESSION EXTENSIONS and p where d n (b ) ! N (0; V ) 1 = E z i z 0i V 185 E z i z 0i 2 i E z i z 0i ^i = = And then n 1 X p zi n n i = i=1 i 2 ei e^2i 2ei x0i b + (b p 2X z i ei x0i n b n )0 xi x0i ( b 1X zi( b n !0 e = Z 0Z be from OLS regression of ^i on z i : Then p p n (e ) = n (b 1 x0i ( b Z0^ )+ n 1 ): Thus ): )0 xi x0i ( b i=1 Let (9.2) n + i=1 p : x0i b = ei While ei is not observed, we have the OLS residual e^i = yi i 1 p ) n (9.3) 1 Z 0Z d ! N (0; V ) n 1=2 Z0 (9.4) Thus the fact that i is replaced with ^i is asymptotically irrelevant. We call (9.3) the skedastic regression, as it is estimating the conditional variance of the regression of yi on xi : We have shown that is consistently estimated by a simple procedure, and hence we can estimate 2i = z 0i by ~ 2i = ~ 0 z i : (9.5) Suppose that ~ 2i > 0 for all i: Then set and e = diagf~ 2 ; :::; ~ 2 g D 1 n e = X 0D e 1 1 X e X 0D 1 y: This is the feasible GLS, or FGLS, estimator of : Since there is not a unique speci…cation for the conditional variance the FGLS estimator is not unique, and will depend on the model (and estimation method) for the skedastic regression. One typical problem with implementation of FGLS estimation is that in the linear speci…cation (9.1), there is no guarantee that ~ 2i > 0 for all i: If ~ 2i < 0 for some i; then the FGLS estimator is not well de…ned. Furthermore, if ~ 2i 0 for some i then the FGLS estimator will force the regression equation through the point (yi ; xi ); which is undesirable. This suggests that there is a need to bound the estimated variances away from zero. A trimming rule takes the form 2 i = max[~ 2i ; c^ 2 ] for some c > 0: For example, setting c = 1=4 means that the conditional variance function is constrained to exceed one-fourth of the unconditional variance. As there is no clear method to select c, this introduces a degree of arbitrariness. In this context it is useful to re-estimate the model with several choices for the trimming parameter. If the estimates turn out to be sensitive to its choice, the estimation method should probably be reconsidered. It is possible to show that if the skedastic regression is correctly speci…ed, then FGLS is asymptotically equivalent to GLS. As the proof is tricky, we just state the result without proof. CHAPTER 9. REGRESSION EXTENSIONS 186 Theorem 9.1.1 If the skedastic regression is correctly speci…ed, p and thus n e GLS e p n e F GLS where ! 0; d ! N (0; V ) ; 2 = E V p F GLS i 1 xi x0i : Examining the asymptotic distribution of Theorem 9.1.1, the natural estimator of the asymptotic variance of e is ! 1 n 1 X 0 1 1 0~ 1 2 0 e V = ~ i xi xi XD X : = n n i=1 0 which is consistent for V as n ! 1: This estimator Ve is appropriate when the skedastic regression (9.1) is correctly speci…ed. It may be the case that 0 z i is only an approximation to the true conditional variance 2i = 2 E(ei j xi ). In this case we interpret 0 z i as a linear projection of e2i on z i : ~ should perhaps be called a quasi-FGLS estimator of : Its asymptotic variance is not that given in Theorem 9.1.1. Instead, V 0 = E zi 1 xi x0i 1 E 0 2 zi 2 0 i xi xi 0 E zi 1 xi x0i 1 : V takes a sandwich form similar to the covariance matrix of the OLS estimator. Unless 2i = 0 z i , 0 Ve is inconsistent for V . 0 An appropriate solution is to use a White-type estimator in place of Ve : This may be written as ! 1 ! ! 1 n n n X X X 1 1 1 ~ i 2 xi x0i ~ i 4 e^2i xi x0i ~ i 2 xi x0i Ve = n n n i=1 = 1 0e XD n i=1 1 1 X 1 0e XD n i=1 1 bD e D 1 X 1 0e XD n 1 1 X b = diagf^ where D e21 ; :::; e^2n g: This is estimator is robust to misspeci…cation of the conditional variance, and was proposed by Cragg (1992). In the linear regression model, FGLS is asymptotically superior to OLS. Why then do we not exclusively estimate regression models by FGLS? This is a good question. There are three reasons. First, FGLS estimation depends on speci…cation and estimation of the skedastic regression. Since the form of the skedastic regression is unknown, and it may be estimated with considerable error, the estimated conditional variances may contain more noise than information about the true conditional variances. In this case, FGLS can do worse than OLS in practice. Second, individual estimated conditional variances may be negative, and this requires trimming to solve. This introduces an element of arbitrariness which is unsettling to empirical researchers. Third, and probably most importantly, OLS is a robust estimator of the parameter vector. It is consistent not only in the regression model, but also under the assumptions of linear projection. The GLS and FGLS estimators, on the other hand, require the assumption of a correct conditional mean. If the equation of interest is a linear projection and not a conditional mean, then the OLS CHAPTER 9. REGRESSION EXTENSIONS 187 and FGLS estimators will converge in probability to di¤erent limits as they will be estimating two di¤erent projections. The FGLS probability limit will depend on the particular function selected for the skedastic regression. The point is that the e¢ ciency gains from FGLS are built on the stronger assumption of a correct conditional mean, and the cost is a loss of robustness to misspeci…cation. 9.2 Testing for Heteroskedasticity The hypothesis of homoskedasticity is that E e2i j xi = H0 : 1 2, or equivalently that =0 in the regression (9.1). We may therefore test this hypothesis by the estimation (9.3) and constructing a Wald statistic. In the classic literature it is typical to impose the stronger assumption that ei is independent of xi ; in which case i is independent of xi and the asymptotic variance (9.2) for ~ simpli…es to 1 V = E z i z 0i E 2i : (9.6) Hence the standard test of H0 is a classic F (or Wald) test for exclusion of all regressors from the skedastic regression (9.3). The asymptotic distribution (9.4) and the asymptotic variance (9.6) under independence show that this test has an asymptotic chi-square distribution. Theorem 9.2.1 Under H0 and ei independent of xi ; the Wald test of H0 is asymptotically 2: q Most tests for heteroskedasticity take this basic form. The main di¤erences between popular tests are which transformations of xi enter z i : Motivated by the form of the asymptotic variance of the OLS estimator b ; White (1980) proposed that the test for heteroskedasticity be based on setting z i to equal all non-redundant elements of xi ; its squares, and all cross-products. BreuschPagan (1979) proposed what might appear to be a distinct test, but the only di¤erence is that they allowed for general choice of z i ; and replaced E 2i with 2 4 which holds when ei is N 0; 2 : If this simpli…cation is replaced by the standard formula (under independence of the error), the two tests coincide. It is important not to misuse tests for heteroskedasticity. It should not be used to determine whether to estimate a regression equation by OLS or FGLS, nor to determine whether classic or White standard errors should be reported. Hypothesis tests are not designed for these purposes. Rather, tests for heteroskedasticity should be used to answer the scienti…c question of whether or not the conditional variance is a function of the regressors. If this question is not of economic interest, then there is no value in conducting a test for heteorskedasticity 9.3 NonLinear Least Squares In some cases we might use a parametric regression function m (x; ) = E (yi j xi = x) which is a non-linear function of the parameters : We describe this setting as non-linear regression. Examples of nonlinear regression functions include x m (x; ) = 1 + 2 1 + 3x m (x; ) = 1 + 2 x 3 m (x; ) = 1 + 2 exp( 3 x) 0 m (x; ) = G(x ); G known m (x; ) = m (x; ) = m (x; ) = 0 1 x1 1 0 2 x1 + + 0 1 x1 2x + 3 (x 1 (x2 < x2 3 4 4 ) 1 (x > 4 ) 0 3) + 2 x1 1 (x2 > 3) CHAPTER 9. REGRESSION EXTENSIONS 188 In the …rst …ve examples, m (x; ) is (generically) di¤erentiable in the parameters : In the …nal two examples, m is not di¤erentiable with respect to 4 and 3 which alters some of the analysis. When it exists, let @ m (x; ) = m (x; ) : @ Nonlinear regression is sometimes adopted because the functional form m (x; ) is suggested by an economic model. In other cases, it is adopted as a ‡exible approximation to an unknown regression function. The least squares estimator b minimizes the normalized sum-of-squared-errors n Sn ( ) = 1X (yi n m (xi ; ))2 : i=1 When the regression function is nonlinear, we call this the nonlinear least squares (NLLS) estimator. The NLLS residuals are e^i = yi m xi ; b : One motivation for the choice of NLLS as the estimation method is that the parameter is the solution to the population problem min E (yi m (xi ; ))2 Since sum-of-squared-errors function Sn ( ) is not quadratic, b must be found by numerical methods. See Appendix E. When m(x; ) is di¤erentiable, then the FOC for minimization are 0= n X xi ; b e^i : m i=1 (9.7) Theorem 9.3.1 Asymptotic Distribution of NLLS Estimator If the model is identi…ed and m (x; ) is di¤ erentiable with respect to , p V where m i n b = E m i m0 i = m (xi ; d ! N (0; V ) 1 E m i m0 i e2i 0 ): Based on Theorem 9.3.1, an estimate of the asymptotic variance V n Vb = 1X m ^ im ^0i n i=1 1 E m i m0 i ! 1 n 1X m ^ im ^ 0 i e^2i n i=1 ! is n 1X m ^ im ^0i n i=1 ! 1 where m ^ i = m (xi ; b) and e^i = yi m(xi ; b): Identi…cation is often tricky in nonlinear regression models. Suppose that m(xi ; ) = 0 1zi + 0 2 xi ( ) where xi ( ) is a function of xi and the unknown parameter : Examples include xi ( ) = xi ; xi ( ) = exp ( xi ) ; and xi ( ) = xi 1 (g (xi ) > ). The model is linear when 2 = 0; and this is often a useful hypothesis (sub-model) to consider. Thus we want to test H0 : 2 = 0: CHAPTER 9. REGRESSION EXTENSIONS However, under H0 , the model is yi = 189 0 1zi + ei and both 2 and have dropped out. This means that under H0 ; is not identi…ed. This renders the distribution theory presented in the previous section invalid. Thus when the truth is that 2 = 0; the parameter estimates are not asymptotically normally distributed. Furthermore, tests of H0 do not have asymptotic normal or chi-square distributions. The asymptotic theory of such tests have been worked out by Andrews and Ploberger (1994) and B. E. Hansen (1996). In particular, Hansen shows how to use simulation (similar to the bootstrap) to construct the asymptotic critical values (or p-values) in a given application. Proof of Theorem 9.3.1 (Sketch). NLLS estimation falls in the class of optimization estimators. For this theory, it is useful to denote the true value of the parameter as 0 : p The …rst step is to show that ^ ! 0 : Proving that nonlinear estimators are consistent is more challenging than for linear estimators. We sketch the main argument. The idea is that ^ minimizes the sample criterion function Sn ( ); which (for any ) converges in probability to the mean-squared error function E (yi m (xi ; ))2 : Thus it seems reasonable that the minimizer ^ will converge in probability to 0 ; the minimizer of E (yi m (xi ; ))2 . It turns out that to show this rigorously, we need to show that Sn ( ) converges uniformly to its expectation E (yi m (xi ; ))2 ; which means that the maximum discrepancy must converge in probability to zero, to exclude the possibility that Sn ( ) is excessively wiggly in . Proving uniform convergence is technically challenging, but it can be shown to hold broadly for relevant nonlinear regression models, especially if the regression function m (xi ; ) is di¤erentiabel in : For a complete treatment of the theory of optimization estimators see Newey and McFadden (1994). p Since ^ ! 0 ; ^ is close to 0 for n large, so the minimization of Sn ( ) only needs to be examined for close to 0 : Let yi0 = ei + m0 i 0 : For close to the true value 0; by a …rst-order Taylor series approximation, m (xi ; ) ' m (xi ; 0) + m0 i ( 0) : Thus yi m (xi ; ) ' (ei + m (xi ; 0 m i( = ei = yi0 m 0 i 0 )) m (xi ; 0) + m0 i ( 0) 0) : Hence the sum of squared errors function is Sn ( ) = n X i=1 (yi 2 m (xi ; )) ' n X yi0 m0 i 2 i=1 and the right-hand-side is the SSE function for a linear regression of yi0 on m i : Thus the NLLS estimator b has the same asymptotic distribution as the (infeasible) OLS regression of yi0 on m i ; which is that stated in the theorem. 9.4 Testing for Omitted NonLinearity If the goal is to estimate the conditional expectation E (yi j xi ) ; it is useful to have a general test of the adequacy of the speci…cation. CHAPTER 9. REGRESSION EXTENSIONS 190 One simple test for neglected nonlinearity is to add nonlinear functions of the regressors to the regression, and test their signi…cance using a Wald test. Thus, if the model yi = x0i b + e^i has been …t by OLS, let z i = h(xi ) denote functions of xi which are not linear functions of xi (perhaps squares of non-binary regressors) and then …t yi = x0i e +z 0i ~ + e~i by OLS, and form a Wald statistic for = 0: Another popular approach is the RESET test proposed by Ramsey (1969). The null model is yi = x0i + ei which is estimated by OLS, yielding predicted values 0 2 y^i B .. zi = @ . y^im be a (m y^i = x0i b : Now let 1 C A 1)-vector of powers of y^i : Then run the auxiliary regression yi = x0i e + z 0i e + e~i by OLS, and form the Wald statistic Wn for d (9.8) = 0: It is easy (although somewhat tedious) to 2 m 1 : Thus the null 2 m 1 distribution. show that under the null hypothesis, Wn ! is rejected at the % level if Wn exceeds the upper % tail critical value of the To implement the test, m must be selected in advance. Typically, small values such as m = 2, 3, or 4 seem to work best. The RESET test appears to work well as a test of functional form against a wide range of smooth alternatives. It is particularly powerful at detecting single-index models of the form yi = G(x0i ) + ei where G( ) is a smooth “link”function. To see why this is the case, note that (9.8) may be written as m 2 3 yi = x0i e + x0i b ~ 1 + x0i b ~ 2 + x0i b ~ m 1 + e~i which has essentially approximated G( ) by a m’th order polynomial. CHAPTER 9. REGRESSION EXTENSIONS 191 Exercises Exercise 9.1 Suppose that yi = g(xi ; )+ei with E (ei j xi ) = 0; ^ is the NLLS estimator, and V^ is the estimate of var ^ : You are interested in the conditional mean function E (yi j xi = x) = g(x) at some x: Find an asymptotic 95% con…dence interval for g(x): Exercise 9.2 In Exercise 8.4, you estimated a cost function on a cross-section of electric companies. The equation you estimated was log T Ci = 1 + 2 log Qi + 3 log P Li + 4 log P Ki + 5 log P Fi + ei : (9.9) (a) Following Nerlove, add the variable (log Qi )2 to the regression. Do so. Assess the merits of this new speci…cation using a hypothesis test. Do you agree with this modi…cation? (b) Now try a non-linear speci…cation. Consider model (9.9) plus the extra term zi = log Qi (1 + exp ( (log Qi 7 ))) 1 6 zi ; where : In addition, impose the restriction 3 + 4 + 5 = 1: This model is called a smooth threshold model. For values of log Qi much below 7 ; the variable log Qi has a regression slope of 2 : For values much above 7 ; the regression slope is 2 + 6 ; and the model imposes a smooth transition between these regimes. The model is non-linear because of the parameter 7 : The model works best when 7 is selected so that several values (in this example, at least 10 to 15) of log Qi are both below and above 7 : Examine the data and pick an appropriate range for 7 : (c) Estimate the model by non-linear least squares. I recommend the concentration method: Pick 10 (or more if you like) values of 7 in this range. For each value of 7 ; calculate zi and estimate the model by OLS. Record the sum of squared errors, and …nd the value of 7 for which the sum of squared errors is minimized. (d) Calculate standard errors for all the parameters ( 1 ; :::; 7 ). Exercise 9.3 The data …le cps78.dat contains 550 observations on 20 variables taken from the May 1978 current population survey. Variables are listed in the …le cps78.pdf. The goal of the exercise is to estimate a model for the log of earnings (variable LNWAGE) as a function of the conditioning variables. (a) Start by an OLS regression of LNWAGE on the other variables. Report coe¢ cient estimates and standard errors. (b) Consider augmenting the model by squares and/or cross-products of the conditioning variables. Estimate your selected model and report the results. (c) Are there any variables which seem to be unimportant as a determinant of wages? You may re-estimate the model without these variables, if desired. (d) Test whether the error variance is di¤erent for men and women. Interpret. (e) Test whether the error variance is di¤erent for whites and nonwhites. Interpret. (f) Construct a model for the conditional variance. Estimate such a model, test for general heteroskedasticity and report the results. (g) Using this model for the conditional variance, re-estimate the model from part (c) using FGLS. Report the results. (h) Do the OLS and FGLS estimates di¤er greatly? Note any interesting di¤erences. (i) Compare the estimated standard errors. Note any interesting di¤erences. Chapter 10 The Bootstrap 10.1 De…nition of the Bootstrap Let F denote a distribution function for the population of observations (yi ; xi ) : Let Tn = Tn ((y1 ; x1 ) ; :::; (yn ; xn ) ; F ) be a statistic of interest, for example an estimator ^ or a t-statistic ^ =s(^): Note that we write Tn as possibly a function of F . For example, the t-statistic is a function of the parameter which itself is a function of F: The exact CDF of Tn when the data are sampled from the distribution F is Gn (u; F ) = Pr(Tn u j F) In general, Gn (u; F ) depends on F; meaning that G changes as F changes. Ideally, inference would be based on Gn (u; F ). This is generally impossible since F is unknown. Asymptotic inference is based on approximating Gn (u; F ) with G(u; F ) = limn!1 Gn (u; F ): When G(u; F ) = G(u) does not depend on F; we say that Tn is asymptotically pivotal and use the distribution function G(u) for inferential purposes. In a seminal contribution, Efron (1979) proposed the bootstrap, which makes a di¤erent approximation. The unknown F is replaced by a consistent estimate Fn (one choice is discussed in the next section). Plugged into Gn (u; F ) we obtain Gn (u) = Gn (u; Fn ): (10.1) We call Gn the bootstrap distribution. Bootstrap inference is based on Gn (u): Let (yi ; xi ) denote random variables with the distribution Fn : A random sample from this distribution is called the bootstrap data. The statistic Tn = Tn ((y1 ; x1 ) ; :::; (yn ; xn ) ; Fn ) constructed on this sample is a random variable with distribution Gn : That is, Pr(Tn u) = Gn (u): We call Tn the bootstrap statistic: The distribution of Tn is identical to that of Tn when the true CDF is Fn rather than F: The bootstrap distribution is itself random, as it depends on the sample through the estimator Fn : In the next sections we describe computation of the bootstrap distribution. 10.2 The Empirical Distribution Function Recall that F (y; x) = Pr (yi y; xi x) = E (1 (yi y) 1 (xi x)) ; where 1( ) is the indicator function. This is a population moment. The method of moments estimator is the corresponding 192 CHAPTER 10. THE BOOTSTRAP 193 sample moment: n Fn (y; x) = 1X 1 (yi n y) 1 (xi x) : (10.2) i=1 Fn (y; x) is called the empirical distribution function (EDF). Fn is a nonparametric estimate of F: Note that while F may be either discrete or continuous, Fn is by construction a step function. The EDF is a consistent estimator of the CDF. To see this, note that for any (y; x); 1 (yi y) 1 (xi p is an iid random variable with expectation F (y; x): Thus by the WLLN (Theorem 5.4.2), Fn (y; x) ! F (y; x) : Furthermore, by the CLT (Theorem 5.7.1), p n (Fn (y; x) d F (y; x)) ! N (0; F (y; x) (1 F (y; x))) : To see the e¤ect of sample size on the EDF, in the Figure below, I have plotted the EDF and true CDF for three random samples of size n = 25; 50, 100, and 500. The random draws are from the N (0; 1) distribution. For n = 25; the EDF is only a crude approximation to the CDF, but the approximation appears to improve for the large n. In general, as the sample size gets larger, the EDF step function gets uniformly close to the true CDF. Figure 10.1: Empirical Distribution Functions The EDF is a valid discrete probability distribution which puts probability mass 1=n at each pair (yi ; xi ), i = 1; :::; n: Notationally, it is helpful to think of a random pair (yi ; xi ) with the distribution Fn : That is, Pr(yi y; xi x) = Fn (y; x): We can easily calculate the moments of functions of (yi ; xi ) : Z Eh (yi ; xi ) = h(y; x)dFn (y; x) = n X h (yi ; xi ) Pr (yi = yi ; xi = xi ) i=1 = n 1X h (yi ; xi ) ; n i=1 the empirical sample average. x) CHAPTER 10. THE BOOTSTRAP 10.3 194 Nonparametric Bootstrap The nonparametric bootstrap is obtained when the bootstrap distribution (10.1) is de…ned using the EDF (10.2) as the estimate Fn of F: Since the EDF Fn is a multinomial (with n support points), in principle the distribution Gn could be calculated by direct methods. However, as there are 2nn 1 possible samples f(y1 ; x1 ) ; :::; (yn ; xn )g; such a calculation is computationally infeasible. The popular alternative is to use simulation to approximate the distribution. The algorithm is identical to our discussion of Monte Carlo simulation, with the following points of clari…cation: The sample size n used for the simulation is the same as the sample size. The random vectors (yi ; xi ) are drawn randomly from the empirical distribution. This is equivalent to sampling a pair (yi ; xi ) randomly from the sample. The bootstrap statistic Tn = Tn ((y1 ; x1 ) ; :::; (yn ; xn ) ; Fn ) is calculated for each bootstrap sample. This is repeated B times. B is known as the number of bootstrap replications. A theory for the determination of the number of bootstrap replications B has been developed by Andrews and Buchinsky (2000). It is desirable for B to be large, so long as the computational costs are reasonable. B = 1000 typically su¢ ces. When the statistic Tn is a function of F; it is typically through dependence on a parameter. For example, the t-ratio ^ =s(^) depends on : As the bootstrap statistic replaces F with Fn ; it similarly replaces with n ; the value of implied by Fn : Typically n = ^; the parameter estimate. (When in doubt use ^:) Sampling from the EDF is particularly easy. Since Fn is a discrete probability distribution putting probability mass 1=n at each sample point, sampling from the EDF is equivalent to random sampling a pair (yi ; xi ) from the observed data with replacement. In consequence, a bootstrap sample f(y1 ; x1 ) ; :::; (yn ; xn )g will necessarily have some ties and multiple values, which is generally not a problem. 10.4 Bootstrap Estimation of Bias and Variance ^ The bias of ^ is n = E(^ : Then n = E(Tn ( 0 )): The bootstrap 0 ): Let Tn ( ) = ^ ^: The bootstrap estimate counterparts are ^ = ^((y1 ; x1 ) ; :::; (yn ; xn )) and Tn = ^ n = of n is n = E(Tn ): If this is calculated by the simulation described in the previous section, the estimate of ^n = = n is B 1 X Tnb B 1 B = ^ b=1 B X ^ b ^ b=1 ^: If ^ is biased, it might be desirable to construct a biased-corrected estimator (one with reduced bias). Ideally, this would be ~=^ n; CHAPTER 10. THE BOOTSTRAP but n 195 is unknown. The (estimated) bootstrap biased-corrected estimator is ~ = ^ ^n = ^ (^ = 2^ ^) ^ : Note, in particular, that the biased-corrected estimator is not ^ : Intuitively, the bootstrap makes the following experiment. Suppose that ^ is the truth. Then what is the average value of ^ calculated from such samples? The answer is ^ : If this is lower than ^; this suggests that the estimator is downward-biased, so a biased-corrected estimator of should be larger than ^; and the best guess is the di¤erence between ^ and ^ : Similarly if ^ is higher than ^; then the estimator is upward-biased and the biased-corrected estimator should be lower than ^. Let Tn = ^: The variance of ^ is Vn = E(Tn ETn )2 : Vn = E(Tn ETn )2 : Let Tn = ^ : It has variance The simulation estimate is B 1 X ^ ^ Vn = b B ^ 2 : b=1 A bootstrap standard error for ^ is the square root of the bootstrap estimate of variance, q s (^) = V^n : While this standard error may be calculated and reported, it is not clear if it is useful. The primary use of asymptotic standard errors is to construct asymptotic con…dence intervals, which are based on the asymptotic normal approximation to the t-ratio. However, the use of the bootstrap presumes that such asymptotic approximations might be poor, in which case the normal approximation is suspected. It appears superior to calculate bootstrap con…dence intervals, and we turn to this next. 10.5 Percentile Intervals For a distribution function Gn (u; F ); let qn ( ; F ) denote its quantile function. This is the function which solves Gn (qn ( ; F ); F ) = : [When Gn (u; F ) is discrete, qn ( ; F ) may be non-unique, but we will ignore such complications.] Let qn ( ) denote the quantile function of the true sampling distribution, and qn ( ) = qn ( ; Fn ) denote the quantile function of the bootstrap distribution. Note that this function will change depending on the underlying statistic Tn whose distribution is Gn : Let Tn = ^; an estimate of a parameter of interest. In (1 )% of samples, ^ lies in the region [qn ( =2); qn (1 =2)]: This motivates a con…dence interval proposed by Efron: C1 = [qn ( =2); qn (1 =2)]: This is often called the percentile con…dence interval. Computationally, the quantile qn ( ) is estimated by q^n ( ); the ’th sample quantile of the simulated statistics fTn1 ; :::; TnB g; as discussed in the section on Monte Carlo simulation. The (1 )% Efron percentile interval is then [^ qn ( =2); q^n (1 =2)]: CHAPTER 10. THE BOOTSTRAP 196 The interval C1 is a popular bootstrap con…dence interval often used in empirical practice. This is because it is easy to compute, simple to motivate, was popularized by Efron early in the history of the bootstrap, and also has the feature that it is translation invariant. That is, if we de…ne = f ( ) as the parameter of interest for a monotonically increasing function f; then percentile method applied to this problem will produce the con…dence interval [f (qn ( =2)); f (qn (1 =2))]; which is a naturally good property. However, as we show now, C1 is in a deep sense very poorly motivated. It will be useful if we introduce an alternative de…nition of C1 . Let Tn ( ) = ^ and let qn ( ) be the quantile function of its distribution. (These are the original quantiles, with subtracted.) Then C1 can alternatively be written as C1 = [^ + qn ( =2); ^ + q (1 n =2)]: This is a bootstrap estimate of the “ideal” con…dence interval C10 = [^ + qn ( =2); ^ + qn (1 =2)]: The latter has coverage probability Pr 0 2 C10 = Pr ^ + qn ( =2) = Pr qn (1 ^ + qn (1 0 ^ =2) = Gn ( qn ( =2); F0 ) qn ( =2) 0 Gn ( qn (1 =2); F0 ) which generally is not 1 ! There is one important exception. If ^ about 0, then Gn ( u; F0 ) = 1 Gn (u; F0 ); so Pr 0 2 C10 = Gn ( qn ( =2); F0 ) = (1 = 1 Gn ( qn (1 Gn (qn ( =2); F0 )) 1 2 1 =2) (1 0 has a symmetric distribution =2); F0 ) Gn (qn (1 =2); F0 )) 2 = 1 and this idealized con…dence interval is accurate. Therefore, C10 and C1 are designed for the case that ^ has a symmetric distribution about 0 : When ^ does not have a symmetric distribution, C1 may perform quite poorly. However, by the translation invariance argument presented above, it also follows that if there exists some monotonically increasing transformation f ( ) such that f (^) is symmetrically distributed about f ( 0 ); then the idealized percentile bootstrap method will be accurate. Based on these arguments, many argue that the percentile interval should not be used unless the sampling distribution is close to unbiased and symmetric. The problems with the percentile method can be circumvented, at least in principle, by an alternative method. Let Tn ( ) = ^ . Then 1 = Pr (qn ( =2) = Pr ^ so an exact (1 qn (1 )% con…dence interval for C20 = [^ Tn ( 0 ) 0 qn (1 =2) 0 ^ =2)) qn ( =2) ; would be qn (1 =2); ^ qn ( =2)]: qn (1 =2); ^ qn ( =2)]: This motivates a bootstrap analog C2 = [^ CHAPTER 10. THE BOOTSTRAP 197 Notice that generally this is very di¤erent from the Efron interval C1 ! They coincide in the special case that Gn (u) is symmetric about ^; but otherwise they di¤er. Computationally, this interval can be estimated from a bootstrap simulation by sorting the ^ ; which are centered at the sample estimate ^: These are sorted bootstrap statistics Tn = ^ to yield the quantile estimates q^n (:025) and q^n (:975): The 95% con…dence interval is then [^ q^ (:975); ^ q^ (:025)]: n n This con…dence interval is discussed in most theoretical treatments of the bootstrap, but is not widely used in practice. 10.6 Percentile-t Equal-Tailed Interval Suppose we want to test H0 : = 0 against H1 : < 0 at size : We would set Tn ( ) = ^ =s(^) and reject H0 in favor of H1 if Tn ( 0 ) < c; where c would be selected so that Pr (Tn ( 0 ) < c) = : Thus c = qn ( ): Since this is unknown, a bootstrap test replaces qn ( ) with the bootstrap estimate qn ( ); and the test rejects if Tn ( 0 ) < qn ( ): ): Similarly, if the alternative is H1 : > 0 ; the bootstrap test rejects if Tn ( 0 ) > qn (1 Computationally, these critical values can be estimated from a bootstrap simulation by sorting ^ =s(^ ): Note, and this is important, that the bootstrap test the bootstrap t-statistics Tn = ^ statistic is centered at the estimate ^; and the standard error s(^ ) is calculated on the bootstrap sample. These t-statistics are sorted to …nd the estimated quantiles q^n ( ) and/or q^n (1 Let Tn ( ) = ^ =s(^). Then taking the intersection of two one-sided intervals, 1 = Pr (qn ( =2) Tn ( 0 ) = Pr qn ( =2) = Pr ^ so an exact (1 s(^)qn (1 )% con…dence interval for C30 = [^ ^ 0 0 qn (1 =2)) =s(^) qn (1 =2) ^ 0 ): =2) s(^)qn ( =2) ; would be s(^)qn (1 =2); ^ s(^)qn ( =2)]: s(^)qn (1 =2); ^ s(^)qn ( =2)]: This motivates a bootstrap analog C3 = [^ This is often called a percentile-t con…dence interval. It is equal-tailed or central since the probability that 0 is below the left endpoint approximately equals the probability that 0 is above the right endpoint, each =2: Computationally, this is based on the critical values from the one-sided hypothesis tests, discussed above. 10.7 ^ Symmetric Percentile-t Intervals Suppose we want to test H0 : = 0 against H1 : 6= 0 at size : We would set Tn ( ) = =s(^) and reject H0 in favor of H1 if jTn ( 0 )j > c; where c would be selected so that Pr (jTn ( 0 )j > c) = : CHAPTER 10. THE BOOTSTRAP 198 Note that Pr (jTn ( 0 )j < c) = Pr ( c < Tn ( 0 ) < c) = Gn (c) Gn ( c) Gn (c); which is a symmetric distribution function. The ideal critical value c = qn ( ) solves the equation Gn (qn ( )) = 1 : Equivalently, qn ( ) is the 1 quantile of the distribution of jTn ( 0 )j : The bootstrap estimate is qn ( ); the 1 quantile of the distribution of jTn j ; or the number which solves the equation Gn (qn ( )) = Gn (qn ( )) Gn ( qn ( )) = 1 : Computationally, qn ( ) is estimated from a bootstrap simulation by sorting the bootstrap t^ =s(^ ); and taking the upper % quantile. The bootstrap test rejects if statistics jTn j = ^ jTn ( 0 )j > qn ( ): Let C4 = [^ s(^)qn ( ); ^ + s(^)qn ( )]; where qn ( ) is the bootstrap critical value for a two-sided hypothesis test. C4 is called the symmetric percentile-t interval. It is designed to work well since Pr ( 0 2 C4 ) = Pr ^ s(^)qn ( ) 0 ^ + s(^)q ( ) n = Pr (jTn ( 0 )j < qn ( )) ' Pr (jTn ( 0 )j < qn ( )) = 1 : If is a vector, then to test H0 : = 0 against H1 : 6= statistic 0 1 ^ Wn ( ) = n ^ V^ 0 at size ; we would use a Wald or some other asymptotically chi-square statistic. Thus here Tn ( ) = Wn ( ): The ideal test rejects if Wn qn ( ); where qn ( ) is the (1 )% quantile of the distribution of Wn : The bootstrap test rejects if Wn qn ( ); where qn ( ) is the (1 )% quantile of the distribution of Wn = n ^ ^ 0 V^ 1 ^ ^ : Computationally, the critical value qn ( ) is found as the quantile from simulated values of Wn : ^ ; not ^ Note in the simulation that the Wald statistic is a quadratic form in ^ 0 : [This is a typical mistake made by practitioners.] 10.8 Asymptotic Expansions Let Tn 2 R be a statistic such that d Tn ! N(0; 2 ): (10.3) CHAPTER 10. THE BOOTSTRAP 199 In some cases, such as when Tn is a t-ratio, then writing Tn Gn (u; F ) then for each u and F 2 = 1: In other cases u lim Gn (u; F ) = n!1 or u Gn (u; F ) = 2 is unknown. Equivalently, ; + o (1) : (10.4) u While (10.4) says that Gn converges to as n ! 1; it says nothing, however, about the rate of convergence; or the size of the divergence for any particular sample size n: A better asymptotic approximation may be obtained through an asymptotic expansion. The following notation will be helpful. Let an be a sequence. De…nition 10.8.1 an = o(1) if an ! 0 as n ! 1 De…nition 10.8.2 an = O(1) if jan j is uniformly bounded. De…nition 10.8.3 an = o(n r) if nr jan j ! 0 as n ! 1. Basically, an = O(n r ) if it declines to zero like n r : We say that a function g(u) is even if g( u) = g(u); and a function h(u) is odd if h( u) = The derivative of an even function is odd, and vice-versa. h(u): Theorem 10.8.1 Under regularity conditions and (10.3), u Gn (u; F ) = + 1 g1 (u; F ) n1=2 + 1 g2 (u; F ) + O(n n 3=2 ) uniformly over u; where g1 is an even function of u; and g2 is an odd function of u: Moreover, g1 and g2 are di¤ erentiable functions of u and continuous in F relative to the supremum norm on the space of distribution functions. The expansion in Theorem 10.8.1 is often called an Edgeworth expansion. We can interpret Theorem 10.8.1 as follows. First, Gn (u; F ) converges to the normal limit at rate n1=2 : To a second order of approximation, Gn (u; F ) u +n 1=2 g1 (u; F ): Since the derivative of g1 is odd, the density function is skewed. To a third order of approximation, Gn (u; F ) u +n 1=2 g1 (u; F ) + n 1 g2 (u; F ) which adds a symmetric non-normal component to the approximate density (for example, adding leptokurtosis). CHAPTER 10. THE BOOTSTRAP [Side Note: When Tn = p 200 n Xn 1 6 g1 (u) = g2 (u) = = ; a standardized sample mean, then 3 u2 1 24 4 1 (u) u3 3u + 1 72 2 3 u5 10u3 + 15u (u) where (u) is the standard normal pdf, and 3 = E (X )3 = 3 4 = E (X )4 = 4 3 the standardized skewness and excess kurtosis of the distribution of X: Note that when 3 = 0 and 4 = 0; then g1 = 0 and g2 = 0; so the second-order Edgeworth expansion corresponds to the normal distribution.] Francis Edgeworth Francis Ysidro Edgeworth (1845-1926) of Ireland, founding editor of the Economic Journal, was a profound economic and statistical theorist, developing the theories of indi¤erence curves and asymptotic expansions. He also could be viewed as the …rst econometrician due to his early use of mathematical statistics in the study of economic data. 10.9 One-Sided Tests Using the expansion of Theorem 10.8.1, we can assess the accuracy of one-sided hypothesis tests and con…dence regions based on an asymptotically normal t-ratio Tn . An asymptotic test is based on (u): To the second order, the exact distribution is Pr (Tn < u) = Gn (u; F0 ) = since 1 (u) + g1 (u; F0 ) n1=2 + O(n 1 ) = 1: The di¤erence is (u) Gn (u; F0 ) = 1 g1 (u; F0 ) n1=2 = O(n 1=2 + O(n 1 ) ); so the order of the error is O(n 1=2 ): A bootstrap test is based on Gn (u); which from Theorem 10.8.1 has the expansion Gn (u) = Gn (u; Fn ) = (u) + 1 g1 (u; Fn ) n1=2 + O(n 1 ): Because (u) appears in both expansions, the di¤erence between the bootstrap distribution and the true distribution is Gn (u) Gn (u; F0 ) = 1 n1=2 (g1 (u; Fn ) g1 (u; F0 )) + O(n 1 ): CHAPTER 10. THE BOOTSTRAP 201 p Since Fn converges to F at rate n; and g1 is continuous with respect to F; the di¤erence p (g1 (u; Fn ) g1 (u; F0 )) converges to 0 at rate n: Heuristically, g1 (u; Fn ) g1 (u; F0 ) @ g1 (u; F0 ) (Fn @F = O(n 1=2 ); F0 ) @ g1 (u; F ) is only heuristic, as F is a function. We conclude that The “derivative” @F Gn (u) Gn (u; F0 ) = O(n 1 ); or Pr (Tn u) = Pr (Tn u) + O(n 1 ); which is an improved rate of convergence over the asymptotic test (which converged at rate O(n 1=2 )). This rate can be used to show that one-tailed bootstrap inference based on the tratio achieves a so-called asymptotic re…nement –the Type I error of the test converges at a faster rate than an analogous asymptotic test. 10.10 Symmetric Two-Sided Tests If a random variable y has distribution function H(u) = Pr(y jyj has distribution function H(u) = H(u) H( u) u); then the random variable since Pr (jyj For example, if Z u) = Pr ( u y u) = Pr (y u) Pr (y = H(u) H( u): u) N(0; 1); then jZj has distribution function (u) = (u) ( u) = 2 (u) 1: Similarly, if Tn has exact distribution Gn (u; F ); then jTn j has the distribution function Gn (u; F ) = Gn (u; F ) Gn ( u; F ): d d A two-sided hypothesis test rejects H0 for large values of jTn j : Since Tn ! Z; then jTn j ! jZj : Thus asymptotic critical values are taken from the distribution, and exact critical values are taken from the Gn (u; F0 ) distribution. From Theorem 10.8.1, we can calculate that Gn (u; F ) = Gn (u; F ) = = Gn ( u; F ) 1 1 (u) + 1=2 g1 (u; F ) + g2 (u; F ) n n 1 1 ( u) + 1=2 g1 ( u; F ) + g2 ( u; F ) + O(n n n 2 (u) + g2 (u; F ) + O(n 3=2 ); n 3=2 ) (10.5) where the simpli…cations are because g1 is even and g2 is odd. Hence the di¤erence between the asymptotic distribution and the exact distribution is (u) Gn (u; F0 ) = 2 g2 (u; F0 ) + O(n n 3=2 ) = O(n 1 ): CHAPTER 10. THE BOOTSTRAP 202 The order of the error is O(n 1 ): Interestingly, the asymptotic two-sided test has a better coverage rate than the asymptotic one-sided test. This is because the …rst term in the asymptotic expansion, g1 ; is an even function, meaning that the errors in the two directions exactly cancel out. Applying (10.5) to the bootstrap distribution, we …nd Gn (u) = Gn (u; Fn ) = (u) + 2 g2 (u; Fn ) + O(n n 3=2 ): Thus the di¤erence between the bootstrap and exact distributions is 2 (g2 (u; Fn ) g2 (u; F0 )) + O(n 3=2 ) n = O(n 3=2 ); p the last equality because Fn converges to F0 at rate n; and g2 is continuous in F: Another way of writing this is Pr (jTn j < u) = Pr (jTn j < u) + O(n 3=2 ) Gn (u) Gn (u; F0 ) = so the error from using the bootstrap distribution (relative to the true unknown distribution) is O(n 3=2 ): This is in contrast to the use of the asymptotic distribution, whose error is O(n 1 ): Thus a two-sided bootstrap test also achieves an asymptotic re…nement, similar to a one-sided test. A reader might get confused between the two simultaneous e¤ects. Two-sided tests have better rates of convergence than the one-sided tests, and bootstrap tests have better rates of convergence than asymptotic tests. The analysis shows that there may be a trade-o¤ between one-sided and two-sided tests. Twosided tests will have more accurate size (Reported Type I error), but one-sided tests might have more power against alternatives of interest. Con…dence intervals based on the bootstrap can be asymmetric if based on one-sided tests (equal-tailed intervals) and can therefore be more informative and have smaller length than symmetric intervals. Therefore, the choice between symmetric and equal-tailed con…dence intervals is unclear, and needs to be determined on a case-by-case basis. 10.11 Percentile Con…dence Intervals To evaluate the coverage rate of the percentile interval, set Tn = p n ^ 0 : We know that d Tn ! N(0; V ); which is not pivotal, as it depends on the unknown V: Theorem 10.8.1 shows that a …rst-order approximation u + O(n 1=2 ); Gn (u; F ) = p where = V ; and for the bootstrap u + O(n ^ Gn (u) = Gn (u; Fn ) = 1=2 ); where ^ = V (Fn ) is the bootstrap estimate of : The di¤erence is Gn (u) Gn (u; F0 ) = u ^ = = O(n u u 1=2 u (^ + O(n 1=2 ) + O(n ) 1=2 ) ) Hence the order of the error is O(n 1=2 ): p The good news is that the percentile-type methods (if appropriately used) can yield nconvergent asymptotic inference. Yet these methods do not require the calculation of standard CHAPTER 10. THE BOOTSTRAP 203 errors! This means that in contexts where standard errors are not available or are di¢ cult to calculate, the percentile bootstrap methods provide an attractive inference method. The bad news is that the rate of convergence is disappointing. It is no better than the rate obtained from an asymptotic one-sided con…dence region. Therefore if standard errors are available, it is unclear if there are any bene…ts from using the percentile bootstrap over simple asymptotic methods. Based on these arguments, the theoretical literature (e.g. Hall, 1992, Horowitz, 2001) tends to advocate the use of the percentile-t bootstrap methods rather than percentile methods. 10.12 Bootstrap Methods for Regression Models The bootstrap methods we have discussed have set Gn (u) = Gn (u; Fn ); where Fn is the EDF. Any other consistent estimate of F may be used to de…ne a feasible bootstrap estimator. The advantage of the EDF is that it is fully nonparametric, it imposes no conditions, and works in nearly any context. But since it is fully nonparametric, it may be ine¢ cient in contexts where more is known about F: We discuss bootstrap methods appropriate for the linear regression model yi = x0i + ei E (ei j xi ) = 0: The non-parametric bootstrap resamples the observations (yi ; xi ) from the EDF, which implies yi = xi 0 ^ + ei E (xi ei ) = 0 but generally E (ei j xi ) 6= 0: The bootstrap distribution does not impose the regression assumption, and is thus an ine¢ cient estimator of the true distribution (when in fact the regression assumption is true.) One approach to this problem is to impose the very strong assumption that the error "i is independent of the regressor xi : The advantage is that in this case it is straightforward to construct bootstrap distributions. The disadvantage is that the bootstrap distribution may be a poor approximation when the error is not independent of the regressors. To impose independence, it is su¢ cient to sample the xi and ei independently, and then create yi = xi 0 ^ + ei : There are di¤erent ways to impose independence. A non-parametric method is to sample the bootstrap errors ei randomly from the OLS residuals f^ e1 ; :::; e^n g: A parametric method is to generate the bootstrap errors ei from a parametric distribution, such as the normal ei N(0; ^ 2 ): For the regressors xi , a nonparametric method is to sample the xi randomly from the EDF or sample values fx1 ; :::; xn g: A parametric method is to sample xi from an estimated parametric distribution. A third approach sets xi = xi : This is equivalent to treating the regressors as …xed in repeated samples. If this is done, then all inferential statements are made conditionally on the observed values of the regressors, which is a valid statistical approach. It does not really matter, however, whether or not the xi are really “…xed” or random. The methods discussed above are unattractive for most applications in econometrics because they impose the stringent assumption that xi and ei are independent. Typically what is desirable is to impose only the regression condition E (ei j xi ) = 0: Unfortunately this is a harder problem. One proposal which imposes the regression condition without independence is the Wild Bootstrap. The idea is to construct a conditional distribution for ei so that E (ei j xi ) = 0 E ei 2 j xi E ei 3 j xi = e^2i = e^3i : CHAPTER 10. THE BOOTSTRAP 204 A conditional distribution with these features will preserve the main important features of the data. This can be achieved using a two-point distribution of the form p p ! ! 1+ 5 5 1 p = Pr ei = e^i 2 2 5 p p ! ! 1 5 5+1 p = Pr ei = e^i 2 2 5 For each xi ; you sample ei using this two-point distribution. CHAPTER 10. THE BOOTSTRAP 205 Exercises Exercise 10.1 Let Fn (x) denote the EDF of a random sample. Show that p d n (Fn (x) F0 (x)) ! N (0; F0 (x) (1 F0 (x))) : Exercise 10.2 Take a random sample fy1 ; :::; yn g with = Eyi and 2 = var (yi ) : Let the statistic of interest be the sample mean Tn = y n : Find the population moments ETn and var (Tn ) : Let fy1 ; :::; yn g be a random sample from the empirical distribution function and let Tn = y n be its sample mean. Find the bootstrap moments ETn and var (Tn ) : Exercise 10.3 Consider the following bootstrap procedure for a regression of yi on xi : Let ^ denote the OLS estimator from the regression of y on X, and e ^ = y X ^ the OLS residuals. (a) Draw a random vector (x ; e ) from the pair f(xi ; e^i ) : i = 1; :::; ng : That is, draw a random integer i0 from [1; 2; :::; n]; and set x = xi0 and e = e^i0 . Set y = x 0 ^ + e : Draw (with replacement) n such vectors, creating a random bootstrap data set (y ; X ): (b) Regress y on X ; yielding OLS estimates ^ and any other statistic of interest. Show that this bootstrap procedure is (numerically) identical to the non-parametric bootstrap. Exercise 10.4 Consider the following bootstrap procedure. Using the non-parametric bootstrap, generate bootstrap samples, calculate the estimate ^ on these samples and then calculate Tn = (^ ^)=s(^); where s(^) is the standard error in the original data. Let qn (:05) and qn (:95) denote the 5% and 95% quantiles of Tn , and de…ne the bootstrap con…dence interval h i C = ^ s(^)qn (:95); ^ s(^)qn (:05) : Show that C exactly equals the Alternative percentile interval (not the percentile-t interval). Exercise 10.5 You want to test H0 : = 0 against H1 : > 0: The test for H0 is to reject if Tn = ^=s(^) > c where c is picked so that Type I error is : You do this as follows. Using the nonparametric bootstrap, you generate bootstrap samples, calculate the estimates ^ on these samples and then calculate Tn = ^ =s(^ ): Let qn (:95) denote the 95% quantile of Tn . You replace c with qn (:95); and thus reject H0 if Tn = ^=s(^) > qn (:95): What is wrong with this procedure? Exercise 10.6 Suppose that in an application, ^ = 1:2 and s(^) = :2: Using the non-parametric bootstrap, 1000 samples are generated from the bootstrap distribution, and ^ is calculated on each sample. The ^ are sorted, and the 2.5% and 97.5% quantiles of the ^ are .75 and 1.3, respectively. (a) Report the 95% Efron Percentile interval for : (b) Report the 95% Alternative Percentile interval for : (c) With the given information, can you report the 95% Percentile-t interval for ? Exercise 10.7 The data…le hprice1.dat contains data on house prices (sales), with variables listed in the …le hprice1.pdf. Estimate a linear regression of price on the number of bedrooms, lot size, size of house, and the colonial dummy. Calculate 95% con…dence intervals for the regression coe¢ cients using both the asymptotic normal approximation and the percentile-t bootstrap. Chapter 11 NonParametric Regression 11.1 Introduction When components of x are continuously distributed then the conditional expectation function E (yi j xi = x) = m(x) can take any nonlinear shape. Unless an economic model restricts the form of m(x) to a parametric function, the CEF is inherently nonparametric, meaning that the function m(x) is an element of an in…nite dimensional class. In this situation, how can we estimate m(x)? What is a suitable method, if we acknowledge that m(x) is nonparametric? There are two main classes of nonparametric regression estimators: kernel estimators, and series estimators. In this chapter we introduce kernel methods. To get started, suppose that there is a single real-valued regressor xi : We consider the case of vector-valued regressors later. 11.2 Binned Estimator For clarity, …x the point x and consider estimation of the single point m(x). This is the mean of yi for random pairs (yi ; xi ) such that xi = x: If the distribution of xi were discrete then we could estimate m(x) by taking the average of the sub-sample of observations yi for which xi = x: But when xi is continuous then the probability is zero that xi exactly equals any speci…c x. So there is no sub-sample of observations with xi = x and we cannot simply take the average of the corresponding yi values. However, if the CEF m(x) is continuous, then it should be possible to get a good approximation by taking the average of the observations for which xi is close to x; perhaps for the observations for which jxi xj h for some small h > 0: Later we will call h a bandwidth. This estimator can be written as Pn 1 (jxi xj h) yi m(x) b = Pi=1 (11.1) n xj h) i=1 1 (jxi where 1( ) is the indicator function. Alternatively, (11.1) can be written as m(x) b = where Notice that Pn i=1 wi (x) n X wi (x)yi i=1 1 (jxi xj h) : wi (x) = Pn xj h) j=1 1 (jxj = 1; so (11.2) is a weighted average of the yi . 206 (11.2) CHAPTER 11. NONPARAMETRIC REGRESSION 207 Figure 11.1: Scatter of (yi ; xi ) and Nadaraya-Watson regression It is possible P that for some values of x there are no values of xi such that jxi xj h; which implies that ni=1 1 (jxi xj h) = 0: In this case the estimator (11.1) is unde…ned for those values of x: To visualize, Figure 11.1 displays a scatter plot of 100 observations on a random pair (yi ; xi ) generated by simulation1 . (The observations are displayed as the open circles.) The estimator (11.1) of the CEF m(x) at x = 2 with h = 1=2 is the average of the yi for the observations such that xi falls in the interval [1:5 xi 2:5]: (Our choice of h = 1=2 is somewhat arbitrary. Selection of h will be discussed later.) The estimate is m(2) b = 5:16 and is shown on Figure 11.1 by the …rst solid square. We repeat the calculation (11.1) for x = 3; 4, 5, and 6, which is equivalent to partitioning the support of xi into the regions [1:5; 2:5]; [2:5; 3:5]; [3:5; 4:5]; [4:5; 5:5]; and [5:5; 6:5]: These partitions are shown in Figure 11.1 by the verticle dotted lines, and the estimates (11.1) by the solid squares. These estimates m(x) b can be viewed as estimates of the CEF m(x): Sometimes called a binned estimator, this is a step-function approximation to m(x) and is displayed in Figure 11.1 by the horizontal lines passing through the solid squares. This estimate roughly tracks the central tendency of the scatter of the observations (yi ; xi ): However, the huge jumps in the estimated step function at the edges of the partitions are disconcerting, counter-intuitive, and clearly an artifact of the discrete binning. If we take another look at the estimation formula (11.1) there is no reason why we need to evaluate (11.1) only on a course grid. We can evaluate m(x) b for any set of values of x: In particular, we can evaluate (11.1) on a …ne grid of values of x and thereby obtain a smoother estimate of the CEF. This estimator with h = 1=2 is displayed in Figure 11.1 with the solid line. This is a generalization of the binned estimator and by construction passes through the solid squares. The bandwidth h determines the degree of smoothing. Larger values of h increase the width of the bins in Figure 11.1, thereby increasing the smoothness of the estimate m(x) b as a function of x. Smaller values of h decrease the width of the bins, resulting in less smooth conditional mean estimates. 1 The distribution is xi N (4; 1) and yi j xi N (m(xi ); 16) with m(x) = 10 log(x): CHAPTER 11. NONPARAMETRIC REGRESSION 11.3 208 Kernel Regression One de…ciency with the estimator (11.1) is that it is a step function in x, as it is discontinuous at each observation x = xi : That is why its plot in Figure 11.1 is jagged. The source of the discontinuity is that the weights wi (x) are constructed from indicator functions, which are themselves discontinuous. If instead the weights are constructed from continuous functions then the CEF estimator will also be continuous in x: To generalize (11.1) it is useful to write the weights 1 (jxi xj h) in terms of the uniform density function on [ 1; 1] 1 k0 (u) = 1 (juj 1) : 2 Then xi x xi x : 1 (jxi xj h) = 1 1 = 2k0 h h and (11.1) can be written as Pn xi i=1 k0 m(x) b = Pn i=1 k0 x h xi x h yi : (11.3) The uniform density k0 (u) is a special case of what is known as a kernel function. De…nition 11.3.1 A second-order kernel function R 1 k(u) satis…es 0 R1 k(u) < 1; k(u) = k( u); 1 k(u)du = 1 and 2k = 1 u2 k(u)du < 1: Essentially, a kernel function is a probability density function which is bounded and symmetric about zero. A generalization of (11.1) is obtained by replacing the uniform kernel with any other kernel function: Pn xi x yi i=1 k h : (11.4) m(x) b = Pn xi x k i=1 h The estimator (11.4) also takes the form (11.2) with x h wi (x) = : Pn xj x j=1 k h k xi The estimator (11.4) is known as the Nadaraya-Watson estimator, the kernel regression estimator, or the local constant estimator. The bandwidth h plays the same role in (11.4) as it does in (11.1). Namely, larger values of h will result in estimates m(x) b which are smoother in x; and smaller values of h will result in estimates which are more erratic. It might be helpful to consider the two extreme cases h ! 0 and h ! 1: As h ! 0 we can see that m(x b i ) ! yi (if the values of xi are unique), so that m(x) b is simply the scatter of yi on xi : In contrast, as h ! 1 then for all x; m(x) b ! y; the sample mean, so that the nonparametric CEF estimate is a constant function. For intermediate values of h; m(x) b will lie between these two extreme cases. CHAPTER 11. NONPARAMETRIC REGRESSION 209 The uniform density is not a good kernel choice as it produces discontinuous CEF estimates: To obtain a continuous CEF estimate m(x) b it is necessary for the kernel k(u) to be continuous. The two most commonly used choices are the Epanechnikov kernel k1 (u) = 3 1 4 u2 1 (juj 1) and the normal or Gaussian kernel u2 2 1 k (u) = p exp 2 : For computation of the CEF estimate (11.4) the scale of the kernel is not important so long as u the bandwidth is selected appropriately. That is, for any b > 0; kb (u) = b 1 k is a valid kernel b function with the identical shape as k(u): Kernel regression with the kernel k(u) and bandwidth h is identical to kernel regression with the kernel kb (u) and bandwidth h=b: The estimate (11.4) using the Epanechnikov kernel and h = 1=2 is also displayed in Figure 11.1 with the dashed line. As you can see, this estimator appears to be much smoother than that using the uniform kernel. Two important constants associated with a kernel function k(u) are its variance 2k and roughness Rk , which are de…ned as Z 1 2 = u2 k(u)du (11.5) k 1 Z 1 Rk = k(u)2 du: (11.6) 1 Some common kernels and their roughness and variance values are reported in Table 9.1. Table 9.1: Common Second-Order Kernels 11.4 Kernel Uniform Epanechnikov Biweight Triweight Equation k0 (u) = 21 1 (juj 1) k1 (u) = 34 1 u2 1 (juj 1) 2 k2 (u) = 15 u2 1 (juj 1) 16 1 3 35 k3 (u) = 32 1 u2 1 (juj 1) Gaussian k (u) = p1 2 u2 2 exp Rk 1=2 3=5 5=7 350=429 p 1= (2 ) 2 k 1=3 1=5 1=7 1=9 1 Local Linear Estimator The Nadaraya-Watson (NW) estimator is often called a local constant estimator as it locally (about x) approximates the CEF m(x) as a constant function. One way to see this is to observe that m(x) b solves the minimization problem m(x) b = argmin n X i=1 k xi x h (yi )2 : This is a weighted regression of yi on an intercept only. Without the weights, this estimation problem reduces to the sample mean. The NW estimator generalizes this to a local mean. This interpretation suggests that we can construct alternative nonparametric estimators of the CEF by alternative local approximations. Many such local approximations are possible. A popular choice is the local linear (LL) approximation. Instead of approximating m(x) locally as a constant, CHAPTER 11. NONPARAMETRIC REGRESSION 210 the local linear approximation approximates the CEF locally by a linear function, and estimates this local approximation by locally weighted least squares. Speci…cally, for each x we solve the following minimization problem n n o X b b (x); (x) = argmin k ; xi x (yi h i=1 (xi x))2 : The local linear estimator of m(x) is the estimated intercept m(x) b = b (x) and the local linear estimator of the regression derivative rm(x) is the estimated slope coe¢ cient d rm(x) = b (x): Computationally, for each x set 1 z i (x) = xi x xi x and ki (x) = k : h Then b (x) b (x) = = n X i=1 0 0 ki (x)z i (x)z i (x) Z KZ 1 0 Z Ky ! 1 n X ki (x)z i (x)yi i=1 where K = diagfk1 (x); :::; kn (x)g: To visualize, Figure 11.2 displays the scatter plot of the same 100 observations from Figure 11.1, divided into three regions depending on the regressor xi : [1; 3]; [3; 5]; [5; 7]: A linear regression is …t to the observations in each region, with the observations weighted by the Epanechnikov kernel with h = 1: The three …tted regression lines are displayed by the three straight solid lines. The values of these regression lines at x = 2; x = 4 and x = 6; respectively, are the local linear estimates m(x) b at x = 2; 4, and 6. This estimation is repeated for all x in the support of the regressors, and plotted as the continuous solid line in Figure 11.2. One interesting feature is that as h ! 1; the LL estimator approaches the full-sample linear least-squares estimator m(x) b ! ^ + ^ x. That is because as h ! 1 all observations receive equal weight regardless of x: In this sense we can see that the LL estimator is a ‡exible generalization of the linear OLS estimator. Which nonparametric estimator should you use in practice: NW or LL? The theoretical literature shows that neither strictly dominates the other, but we can describe contexts where one or the other does better. Roughly speaking, the NW estimator performs better than the LL estimator when m(x) is close to a ‡at line, but the LL estimator performs better when m(x) is meaningfully non-constant. The LL estimator also performs better for values of x near the boundary of the support of xi : 11.5 Nonparametric Residuals and Regression Fit The …tted regression at x = xi is m(x b i ) and the …tted residual is e^i = yi m(x b i ): CHAPTER 11. NONPARAMETRIC REGRESSION 211 Figure 11.2: Scatter of (yi ; xi ) and Local Linear …tted regression As a general rule, but especially when the bandwidth h is small, it is hard to view e^i as a good measure of the …t of the regression. As h ! 0 then m(x b i ) ! yi and therefore e^i ! 0: This clearly indicates over…tting as the true error is not zero. In general, since m(x b i ) is a local average which includes yi ; the …tted value will be necessarily close to yi and the residual e^i small, and the degree of this over…tting increases as h decreases. A standard solution is to measure the …t of the regression at x = xi by re-estimating the model excluding the i’th observation. For Nadaraya-Watson regression, the leave-one-out estimator of m(x) excluding observation i is P j6=i k m e i (x) = xj x h xj x h P j6=i k yj : Notationally, the “ i” subscript is used to indicate that the i’th observation is omitted. The leave-one-out predicted value for yi at x = xi equals P j6=i k y~i = m e i (xi ) = P j6=i k xj xi h xj xi yj : h The leave-one-out residuals (or prediction errors) are the di¤erence between the leave-one-out predicted values and the actual observation e~i = yi y~i : Since y~i is not a function of yi ; there is no tendency for y~i to over…t for small h: Consequently, e~i is a good measure of the …t of the estimated nonparametric regression. Similarly, the leave-one-out local-linear residual is e~i = yi e i with 0 1 1 X X ei 0 A @ = k z z kij z ij yj ; ij ij ij e i j6=i j6=i CHAPTER 11. NONPARAMETRIC REGRESSION z ij = and kij = k 11.6 212 1 xj xi xj xi h : Cross-Validation Bandwidth Selection As we mentioned before, the choice of bandwidth h is crucial. As h increases, the kernel regression estimators (both NW and LL) become more smooth, ironing out the bumps and wiggles. This reduces estimation variance but at the cost of increased bias and oversmoothing. As h decreases the estimators become more wiggly, erratic, and noisy. It is desirable to select h to trade-o¤ these features. How can this be done systematically? To be explicit about the dependence of the estimator on the bandwidth, let us write the estimator of m(x) with a given bandwidth h as m(x; b h); and our discussion will apply equally to the NW and LL estimators. Ideally, we would like to select h to minimize the mean-squared error (MSE) of m(x; b h) as a estimate of m(x): For a given value of x the MSE is M SEn (x; h) = E (m(x; b h) m(x))2 : We are typically interested in estimating m(x) for all values in the support of x: A common measure for the average …t is the integrated MSE Z IM SEn (h) = M SEn (x; h)fx (x)dx Z = E (m(x; b h) m(x))2 fx (x)dx where fx (x) is the marginal density of xi : Notice that we have de…ned the IMSE as an integral with respect to the density fx (x): Other weight functions could be used, but it turns out that this is a convenient choice: The IMSE is closely related with the MSFE of Section 4.9. Let (yn+1 ; xn+1 ) be out-of-sample observations (and thus independent of the sample) and consider predicting yn+1 given xn+1 and the nonparametric estimate m(x; b h): The natural point estimate for yn+1 is m(x b n+1 ; h) which has mean-squared forecast error M SF En (h) = E (yn+1 m(x b n+1 ; h))2 = E (en+1 + m(xn+1 ) = = m(x b n+1 ; h))2 + E (m(xn+1 ) m(x b n+1 ; h))2 Z 2 + E (m(x; b h) m(x))2 fx (x)dx 2 where the …nal equality uses the fact that xn+1 is independent of m(x; b h): We thus see that M SF En (h) = 2 + IM SEn (h): Since 2 is a constant independent of the bandwidth h; M SF En (h) and IM SEn (h) are equivalent measures of the …t of the nonparameric regression. The optimal bandwidth h is the value which minimizes IM SEn (h) (or equivalently M SF En (h)): While these functions are unknown, we learned in Theorem 4.9.1 that (at least in the case of linear regression) M SF En can be estimated by the sample mean-squared prediction errors. It turns out that this fact extends to nonparametric regression. The nonparametric leave-one-out residuals are e~i (h) = yi m e i (xi ; h) CHAPTER 11. NONPARAMETRIC REGRESSION 213 where we are being explicit about the dependence on the bandwidth h: The mean squared leaveone-out residuals is n 1X CV (h) = e~i (h)2 : n i=1 This function of h is known as the cross-validation criterion. The cross-validation bandwidth b h is the value which minimizes CV (h) b h = argmin CV (h) (11.7) h h` for some h` > 0: The restriction h h` is imposed so that CV (h) is not evaluated over unreasonably small bandwidths. There is not an explicit solution to the minimization problem (11.7), so it must be solved numerically. A typical practical method is to create a grid of values for h; e.g. [h1 ; h2 ; :::; hJ ], evaluate CV (hj ) for j = 1; :::; J; and set b h= argmin CV (h): h2[h1 ;h2 ;:::;hJ ] Evaluation using a coarse grid is typically su¢ cient for practical application. Plots of CV (h) against h are a useful diagnostic tool to verify that the minimum of CV (h) has been obtained. We said above that the cross-validation criterion is an estimator of the MSFE. This claim is based on the following result. Theorem 11.6.1 E (CV (h)) = M SF En 1 (h) = IM SEn 1 (h) + 2 (11.8) Theorem 11.6.1 shows that CV (h) is an unbiased estimator of IM SEn 1 (h) + 2 : The …rst term, IM SEn 1 (h); is the integrated MSE of the nonparametric estimator using a sample of size n 1: If n is large, IM SEn 1 (h) and IM SEn (h) will be nearly identical, so CV (h) is essentially unbiased as an estimator of IM SEn (h) + 2 . Since the second term ( 2 ) is una¤ected by the bandwidth h; it is irrelevant for the problem of selection of h. In this sense we can view CV (h) as an estimator of the IMSE, and more importantly we can view the minimizer of CV (h) as an estimate of the minimizer of IM SEn (h): To illustrate, Figure 11.3 displays the cross-validation criteria CV (h) for the Nadaraya-Watson and Local Linear estimators using the data from Figure 11.1, both using the Epanechnikov kernel. The CV functions are computed on a grid with intervals 0.01. The CV-minimizing bandwidths are h = 1:09 for the Nadaraya-Watson estimator and h = 1:59 for the local linear estimator. Figure 11.3 shows the minimizing bandwidths by the arrows. It is not surprising that the CV criteria recommend a larger bandwidth for the LL estimator than for the NW estimator, as the LL employs more smoothing for a given bandwidth. The CV criterion can also be used to select between di¤erent nonparametric estimators. The CV-selected estimator is the one with the lowest minimized CV criterion. For example, in Figure 11.3, the NW estimator has a minimized CV criterion of 16.88, while the LL estimator has a minimized CV criterion of 16.81. Since the LL estimator achieves a lower value of the CV criterion, LL is the CV-selected estimator. The di¤erence (0.07) is small, suggesting that the two estimators are near equivalent in IMSE. Figure 11.4 displays the …tted CEF estimates (NW and LL) using the bandwidths selected by cross-validation. Also displayed is the true CEF m(x) = 10 ln(x). Notice that the nonparametric CHAPTER 11. NONPARAMETRIC REGRESSION 214 Figure 11.3: Cross-Validation Criteria, Nadaraya-Watson Regression and Local Linear Regression Figure 11.4: Nonparametric Estimates using data-dependent (CV) bandwidths CHAPTER 11. NONPARAMETRIC REGRESSION 215 estimators with the CV-selected bandwidths (and especially the LL estimator) track the true CEF quite well. Proof of Theorem 11.6.1. Observe that m(xi ) m e i (xi ; h) is a function only of (x1 ; :::; xn ) and (e1 ; :::; en ) excluding ei ; and is thus uncorrelated with ei : Since e~i (h) = m(xi ) m e i (xi ; h) + ei ; then E (CV (h)) = E e~i (h)2 = E e2i + E (m e i (xi ; h) = +2E ((m e i (xi ; h) 2 m(xi ))2 m(xi )) ei ) + E (m e i (xi ; h) m(xi ))2 : (11.9) The second term is an expectation over the random variables xi and m e i (x; h); which are independent as the second is not a function of the i’th observation. Thus taking the conditional expectation given the sample excluding the i’th observation, this is the expectation over xi only, which is the integral with respect to its density Z 2 e i (x; h) m(x))2 fx (x)dx: E i (m e i (xi ; h) m(xi )) = (m Taking the unconditional expecation yields E (m e i (xi ; h) m(xi ))2 = E Z (m e i (x; h) = IM SEn m(x))2 fx (x)dx 1 (h) where this is the IMSE of a sample of size n 1 as the estimator m e Combined with (11.9) we obtain (11.8), as desired. 11.7 i uses n 1 observations. Asymptotic Distribution There is no …nite sample distribution theory for kernel estimators, but there is a well developed asymptotic distribution theory. The theory is based on the approximation that the bandwidth h decreases to zero as the sample size n increases. This means that the smoothing is increasingly localized as the sample size increases. So long as the bandwidth does not decrease to zero too quickly, the estimator can be shown to be asymptotically normal, but with a non-trivial bias. Let fx (x) denote the marginal density of xi and 2 (x) = E e2i j xi = x denote the conditional variance of ei = yi m(xi ): CHAPTER 11. NONPARAMETRIC REGRESSION 216 Theorem 11.7.1 Let m(x) b denote either the Nadarya-Watson or Local Linear estimator of m(x): If x is interior to the support of xi and fx (x) > 0; then as n ! 1 and h ! 0 such that nh ! 1; p nh m(x) b m(x) h2 2 k B(x) d ! N 0; Rk 2 (x) fx (x) (11.10) where 2k and Rk are de…ned in (11.5) and (11.6). For the NadarayaWatson estimator 1 B(x) = m00 (x) + fx (x) 2 1 0 fx (x)m0 (x) and for the local linear estimator 1 B(x) = fx (x)m00 (x) 2 There are several interesting features about the asymptotic distribution which are p noticeably p di¤erent than for the estimator converges at the rate nh; not n: p parametric estimators. First, p Since h ! 0; nh diverges slower than n; thus the nonparametric estimator converges more slowly than a parametric estimator. Second, the asymptotic distribution contains a non-neglible bias term h2 2k B(x): This term asymptotically disappears since h ! 0: Third, the assumptions that nh ! 1 and h ! 0 mean that the estimator is p consistent for the CEF m(x). The fact that the estimator converges at the rate nh has led to the interpretation of nh as the “e¤ective sample size”. This is because the number of observations being used to construct m(x) b is proportional to nh; not n as for a parametric estimator. It is helpful to understand that the nonparametric estimator has a reduced convergence rate because the object being estimated – m(x) – is nonparametric. This is harder than estimating a …nite dimensional parameter, and thus comes at a cost. Unlike parametric estimation, the asymptotic distribution of the nonparametric estimator includes a term representing the bias of the estimator. The asymptotic distribution (11.10) shows the form of this bias. Not only is it proportional to the squared bandwidth h2 (the degree of smoothing), it is proportional to the function B(x) which depends on the slope and curvature of the CEF m(x): Interestingly, when m(x) is constant then B(x) = 0 and the kernel estimator has no asymptotic bias. The bias is essentially increasing in the curvature of the CEF function m(x): This is because the local averaging smooths m(x); and the smoothing induces more bias when m(x) is curved. Theorem 11.7.1 shows that the asymptotic distributions of the NW and LL estimators are similar, with the only di¤erence arising in the bias function B(x): The bias term for the NW estimator has an extra component which depends on the …rst derivative of the CEF m(x) while the bias term of the LL estimator is invariant to the …rst derivative. The fact that the bias formula for the LL estimator is simpler and is free of dependence on the …rst derivative of m(x) suggests that the LL estimator will generally have smaller bias than the NW estimator (but this is not a precise ranking). Since the asymptotic variances in the two distributions are the same, this means that the LL estimator achieves a reduced bias without an e¤ect on asymptotic variance. This analysis has led to the general preference for the LL estimator over the NW estimator in the nonparametrics literature. One implication of Theorem 11.7.1 is that we can de…ne the asymptotic MSE (AMSE) of m(x) b as the squared bias plus the asymptotic variance AM SE (m(x)) b = h2 2 2 k B(x) + Rk 2 (x) : nhfx (x) (11.11) CHAPTER 11. NONPARAMETRIC REGRESSION 217 Focusing on rates, this says AM SE (m(x)) b h4 + 1 nh (11.12) which means that the AMSE is dominated by the larger of h4 and (nh) 1 : Notice that the bias is increasing in h and the variance is decreasing in h: (More smoothing means more observations are used for local estimation: this increases the bias but decreases estimation variance.) To select h to minimize the AMSE, these two components should balance each other. Setting h4 / (nh) 1 means setting h / n 1=5 : Another way to see this is to pick h to minimize the right-hand-side of (11.12). The …rst-order condition for h is @ @h h4 + 1 nh = 4h3 1 =0 nh2 which when solved for h yields h = n 1=5 : What this means is that for AMSE-e¢ cient estimation of m(x); the optimal rate for the bandwidth is h / n 1=5 : Theorem 11.7.2 The bandwidth which minimizes the AMSE (11.12) is of order h / n 1=5 . With h / n 1=5 then AM SE (m(x)) b = O n 4=5 and 2=5 m(x) b = m(x) + Op n : This result means that the bandwidth should take the form h = cn 1=5 : The optimal constant c depends on the kernel k; the bias function B(x) and the marginal density fx (x): A common misinterpretation is to set h = n 1=5 ; which is equivalent to setting c = 1 and is completely arbitrary. Instead, an empirical bandwidth selection rule such as cross-validation should be used in practice. When h = cn 1=5 we can rewrite the asymptotic distribution (11.10) as n2=5 (m(x) b d m(x)) ! N c2 Rk 2 (x) 2 k B(x); 1=2 c fx (x) In this representation, we see that m(x) b is asymptotically normal, but with a n2=5 rate of convergence and non-zero mean. The asymptotic distribution depends on the constant c through the bias (positively) and the variance (inversely). The asymptotic distribution in Theorem 11.7.1 allows for the optimal rate h = cn 1=5 but this rate is not required. In particular, consider an undersmoothing (smaller than optimal) bandwith with rate h = o n 1=5 . For example, we could specify that h = cn for some c > 0 and p 2 (1 5 )=2 1=5 < < 1: Then nhh = O(n ) = o(1) so the bias term in (11.10) is asymptotically negligible so Theorem 11.7.1 implies p nh (m(x) b d m(x)) ! N 0; Rk 2 (x) fx (x) : That is, the estimator is asymptotically normal without a bias component. Not having an asymptotic bias component is convenient for some theoretical manipuations, so many authors impose the undersmoothing condition h = o n 1=5 to ensure this situation. This convenience comes at a cost. First, the resulting estimator is ine¢ cient as its convergence rate is is Op n (1 )=2 > Op n 2=5 since > 1=5: Second, the distribution theory is an inherently misleading approximation as it misses a critically key ingredient of nonparametric estimation – the trade-o¤ between bias and variance. The approximation (11.10) is superior precisely because it contains the asymptotic bias component which is a realistic implication of nonparametric estimation. Undersmoothing assumptions should be avoided when possible. CHAPTER 11. NONPARAMETRIC REGRESSION 11.8 218 Conditional Variance Estimation Let’s consider the problem of estimation of the conditional variance 2 (x) = var (yi j xi = x) = E e2i j xi = x : Even if the conditional mean m(x) is parametrically speci…ed, it is natural to view 2 (x) as inherently nonparametric as economic models rarely specify the form of the conditional variance. Thus it is quite appropriate to estimate 2 (x) nonparametrically. We know that 2 (x) is the CEF of e2i given xi : Therefore if e2i were observed, 2 (x) could be nonparametrically estimated using NW or LL regression. For example, the NW estimator is Pn ki (x)e2i 2 : ~ (x) = Pi=1 n i=1 ki (x) Since the errors ei are not observed, we need to replace them with an empirical residual, such as e^i = yi m(x b i ) where m(x) b is the estimated CEF. (The latter could be a nonparametric estimator such as NW or LL, or even a parametric estimator.) Even better, use the leave-one-out predication errors e~i = yi m b i (xi ); as these are not subject to over…tting. With this substitution the NW estimator of the conditional variance is Pn ki (x)~ e2i 2 ^ (x) = Pi=1 : (11.13) n i=1 ki (x) This estimator depends on a set of bandwidths h1 ; :::; hq , but there is no reason for the bandwidths to be the same as those used to estimate the conditional mean. Cross-validation can be used to select the bandwidths for estimation of ^ 2 (x) separately from cross-validation for estimation of m(x): b There is one subtle di¤erence between CEF and conditional variance estimation. The conditional variance is inherently non-negative 2 (x) 0 and it is desirable for our estimator to satisfy this property. Interestingly, the NW estimator (11.13) is necessarily non-negative, since it is a smoothed average of the non-negative squared residuals, but the LL estimator is not guarenteed to be nonnegative for all x. For this reason, the NW estimator is preferred for conditional variance estimation. Fan and Yao (1998, Biometrika) derive the asymptotic distribution of the estimator (11.13). They obtain the surprising result that the asymptotic distribution of this two-step estimator is identical to that of the one-step idealized estimator ~ 2 (x). 11.9 Standard Errors Theorem 11.7.1 shows the asymptotic variances of both the NW and LL nonparametric regression estimators equal Rk 2 (x) V (x) = : fx (x) For standard errors we need an estimate of V (x) : A plug-in estimate replaces the unknowns by estimates. The roughness Rk can be found from Table 9.1. The conditional variance can be estimated using (11.13). The density of xi can be estimated using the methods from Section 21.1. Replacing these estimates into the formula for V (x) we obtain the asymptotic variance estimate Rk ^ 2 (x) V^ (x) = : f^x (x) CHAPTER 11. NONPARAMETRIC REGRESSION 219 Then an asymptotic standard error for the kernel estimate m(x) b is r 1 ^ s^(x) = V (x): nh Plots of the estimated CEF m(x) b can be accompanied by con…dence intervals m(x) b 2^ s(x): These are known as pointwise con…dence intervals, as they are designed to have correct coverage at each x; not uniformly in x: One important caveat about the interpretation of nonparametric con…dence intervals is that they are not centered at the true CEF m(x); but rather are centered at the biased or pseudo-true value m (x) = m(x) + h2 2k B(x): Consequently, a correct statement about the con…dence interval m(x) b 2^ s(x) is that it asymptotically contains m (x) with probability 95%, not that it asymptotically contains m(x) with probability 95%. The discrepancy is that the con…dence interval does not take into account the bias h2 2k B(x): Unfortunately, nothing constructive can be done about this. The bias is di¢ cult and noisy to estimate, so making a bias-correction only in‡ates estimation variance and decreases overall precision. A technical “trick” is to assume undersmoothing h = o n 1=5 but this does not really eliminate the bias, it only assumes it away. The plain fact is that once we honestly acknowledge that the true CEF is nonparametric, it then follows that any …nite sample estimate will have …nite sample bias, and this bias will be inherently unknown and thus impossible to incorporate into con…dence intervals. 11.10 Multiple Regressors Our analysis has focus on the case of real-valued xi for simplicity of exposition, but the methods of kernel regression extend easily to the multiple regressor case, at the cost of a reduced rate of convergence. In this section we consider the case of estimation of the conditional expectation function E (yi j xi = x) = m(x) when 0 1 x1i B C xi = @ ... A xdi is a d-vector. For any evaluation point x and observation i; de…ne the kernel weights ki (x) = k x1i x1 h1 k x2i x2 h2 k xdi xd hd ; a d-fold product kernel. The kernel weights ki (x) assess if the regressor vector xi is close to the evaluation point x in the Euclidean space Rd . These weights depend on a set of d bandwidths, hj ; one for each regressor. We can group them together into a single vector for notational convenience: 0 1 h1 B C h = @ ... A : hd Given these weights, the Nadaraya-Watson estimator takes the form Pn ki (x)yi : m(x) b = Pi=1 n i=1 ki (x) CHAPTER 11. NONPARAMETRIC REGRESSION 220 For the local-linear estimator, de…ne 1 z i (x) = xi x and then the local-linear estimator can be written as m(x) b = b (x) where b (x) b (x) = = n X i=1 0 ki (x)z i (x)z i (x) Z KZ 1 0 ! 0 Z Ky 1 n X ki (x)z i (x)yi i=1 where K = diagfk1 (x); :::; kn (x)g: In multiple regressor kernel regression, cross-validation remains a recommended method for bandwidth selection. The leave-one-out residuals e~i and cross-validation criterion CV (h) are de…ned identically as in the single regressor case. The only di¤erence is that now the CV criterion is a function over the d-dimensional bandwidth h. This is a critical practical di¤erence since …nding b which minimizes CV (h) can be computationally di¢ cult when h is high the bandwidth vector h dimensional. Grid search is cumbersome and costly, since G gridpoints per dimension imply evaulation of CV (h) at Gd distinct points, which can be a large number. Furthermore, plots of CV (h) against h are challenging when d > 2: The asymptotic distribution of the estimators in the multiple regressor case is an extension of the single regressor case. Let fx (x) denote the marginal density of xi and 2 (x) = E e2i j xi = x the conditional variance of ei = yi m(xi ): Let jhj = h1 h2 hd : Theorem 11.10.1 Let m(x) b denote either the Nadarya-Watson or Local Linear estimator of m(x): If x is interior to the support of xi and fx (x) > 0; then as n ! 1 and hj ! 0 such that n jhj ! 1; 0 1 d X p Rd 2 (x) d 2 n jhj @m(x) b m(x) h2j B;j (x)A ! N 0; k k fx (x) j=1 where for the Nadaraya-Watson estimator Bj (x) = 1 @2 m(x) + fx (x) 2 @x2j 1 @ @ fx (x) m(x) @xj @xj and for the Local Linear estimator Bj (x) = 1 @2 m(x) 2 @x2j For notational simplicity consider the case that there is a single common bandwidth h: In this case the AMSE takes the form 1 AM SE(m(x)) b h4 + nhd That is, the squared bias is of order h4 ; the same as in the single regressor case, but the variance is of larger order (nhd ) 1 : Setting h to balance these two components requires setting h n 1=(4+d) : CHAPTER 11. NONPARAMETRIC REGRESSION 221 Theorem 11.10.2 The bandwidth which minimizes the AMSE is of order h / n 1=(4+d) . With h / n 1=(4+d) then AM SE (m(x)) b = O n 4=(4+d) and m(x) b = m(x) + Op n 2=(4+d) In all estimation problems an increase in the dimension decreases estimation precision. For example, in parametric estimation an increase in dimension typically increases the asymptotic variance. In nonparametric estimation an increase in the dimension typically decreases the convergence rate, which is a more fundamental decrease in precision. For example, in kernel regression the convergence rate Op n 2=(4+d) decreases as d increases. The reason is the estimator m(x) b is a local average of the yi for observations such that xi is close to x, and when there are multiple regressors the number of such observations is inherently smaller. This phenomenon – that the rate of convergence of nonparametric estimation decreases as the dimension increases –is called the curse of dimensionality. Chapter 12 Series Estimation 12.1 Approximation by Series As we mentioned at the beginning of Chapter 11, there are two main methods of nonparametric regression: kernel estimation and series estimation. In this chapter we study series methods. Series methods approximate an unknown function (e.g. the CEF m(x)) with a ‡exible parametric function, with the number of parameters treated similarly to the bandwidth in kernel regression. A series approximation to m(x) takes the form mK (x) = mK (x; K ) where mK (x; K ) is a known parametric family and K is an unknown coe¢ cient. The integer K is the dimension of K and indexes the complexity of the approximation. A linear series approximation takes the form mK (x) = K X zjK (x) jK j=1 = z K (x)0 (12.1) K where zjK (x) are (nonlinear) functions of x; and are known as basis functions or basis function transformations of x: For real-valued x; a well-known linear series approximation is the p’th-order polynomial mK (x) = p X xj jK j=0 where K = p + 1: When x 2 Rd is vector-valued, a p’th-order polynomial is mK (x) = p X j1 =0 p X xj11 xjdd j1 ;:::;jd K : jd =0 This includes all powers and cross-products, and the coe¢ cient vector has dimension K = (p + 1)d : In general, a common method to create a series approximation for vector-valued x is to include all non-redundant cross-products of the basis function transformations of the components of x: 12.2 Splines Another common series approximation is a continuous piecewise polynomial function known as a spline. While splines can be of any polynomial order (e.g. linear, quadratic, cubic, etc.), a common choice is cubic. To impose smoothness it is common to constrain the spline function to have continuous derivatives up to the order of the spline. Thus a quadratic spline is typically 222 CHAPTER 12. SERIES ESTIMATION 223 constrained to have a continuous …rst derivative, and a cubic spline is typically constrained to have a continuous …rst and second derivative. There is more than one way to de…ne a spline series expansion. All are based on the number of knots –the join points between the polynomial segments. To illustrate, a piecewise linear function with two segments and a knot at t is 8 x<t < m1 (x) = 00 + 01 (x t) mK (x) = : m2 (x) = 10 + 11 (x t) x t (For convenience we have written the segments functions as polyomials in x t.) The function mK (x) equals the linear function m1 (x) for x < t and equals m2 (t) for x > t. Its left limit at x = t is 00 and its right limit is 10 ; so is continuous if (and only if) 00 = 10 : Enforcing this constraint is equivalent to writing the function as mK (x) = 0 + 1 (x mK (x) = 0 t) + 2 (x t) 1 (x t) or after transforming coe¢ cients, as + 1x + 2 (x t) 1 (x t) Notice that this function has K = 3 coe¢ cients, the same as a quadratic polynomial. A piecewise quadratic function with one knot at t is 8 2 x<t < m1 (x) = 00 + 01 (x t) + 02 (x t) mK (x) = : x t m2 (x) = 10 + 11 (x t) + 12 (x t)2 This function is continuous at x = t if 00 = 10 ; and has a continuous …rst derivative if Imposing these contraints and rewriting, we obtain the function mK (x) = 0 + 1x 2 2x + + 3 (x t)2 1 (x 01 = 11 : t) : Here, K = 4: Furthermore, a piecewise cubic function with one knot and a continuous second derivative is mK (x) = 0 + 1x + 2x 2 + 3x 3 + 4 (x t)3 1 (x t) which has K = 5: The polynomial order p is selected to control the smoothness of the spline, as mK (x) has continuous derivatives up to p 1. In general, a p’th-order spline with N knots at t1 , t2 ; :::; tN with t1 < t2 < < tN is mK (x) = p X j=0 j jx + N X k (x tk )p 1 (x tk ) k=1 which has K = N + p + 1 coe¢ cients. In spline approximation, the typical approach is to treat the polynomial order p as …xed, and select the number of knots N to determine the complexity of the approximation. The knots tk are typically treated as …xed. A common choice is to set the knots to evenly partition the support X of xi : CHAPTER 12. SERIES ESTIMATION 12.3 224 Partially Linear Model A common use of a series expansion is to allow the CEF to be nonparametric with respect to one variable, yet linear in the other variables. This allows ‡exibility in a particular variable of interest. A partially linear CEF with vector-valued regressor x1 and real-valued continuous x2 takes the form m (x1 ; x2 ) = x01 1 + m2 (x2 ): This model is commonly used when x1 are discrete (e.g. binary variables) and x2 is continuously distributed. Series methods are particularly convenient for estimation of partially linear models, as we can replace the unknown function m2 (x2 ) with a series expansion to obtain m (x) ' mK (x) = x01 = x0K 1 + z 0K 2K K where z K = z K (x2 ) are the basis transformations of x2 (typically polynomials or splines) and 2K are coe¢ cients. After transformation the regressors are xK = (x01 ; z 0K ): and the coe¢ cients are 0 0 0 K = ( 1 ; 2K ) : 12.4 Additively Separable Models When x is multivariate a common simpli…cation is to treat the regression function m (x) as additively separable in the individual regressors, which means that m (x) = m1 (x1 ) + m2 (x2 ) + + md (xd ) : Series methods are quite convenient for estimation of additively separable models, as we simply apply series expansions (polynomials or splines) separately for each component mj (xj ) : The advantage of additive separability is the reduction in dimensionality. While an unconstrained p’th order polynomial has (p + 1)d coe¢ cients, an additively separable polynomial model has only (p + 1)d coe¢ cients. This can be a major reduction in the number of coe¢ cients. The disadvatage of this simpli…cation is that the interaction e¤ects have been eliminated. The decision to impose additive separability can be based on an economic model which suggests the absence of interaction e¤ects, or can be a model selection decision similar to the selection of the number of series terms. We will discuss model selection methods below. 12.5 Uniform Approximations A good series approximation mK (x) will have the property that it gets close to the true CEF m(x) as the complexity K increases. Formal statements can be derived from the theory of functional analysis. An elegant and famous theorem is the Stone-Weierstrass theorem, (Weierstrass, 1885, Stone 1937, 1948) which states that any continuous function can be arbitrarily uniformly well approximated by a polynomial of su¢ ciently high order. Speci…cally, the theorem states that for x 2 Rd ; if m(x) is continuous on a compact set X , then for any " > 0 there exists a polynomial mK (x) of some order K which is uniformly within " of m(x): sup jmK (x) m(x)j ": (12.2) x2X Thus the true unknown m(x) can be aribitrarily well approximately by selecting a suitable polynomial. CHAPTER 12. SERIES ESTIMATION 225 Figure 12.1: True CEF and Best Approximations The result (12.2) can be stengthened. In particular, if the sth derivative of m(x) is continuous then the uniform approximation error satis…es sup jmK (x) m(x)j = O K (12.3) x2X as K ! 1 where = s=d. This result is more useful than (12.2) because it gives a rate at which the approximation mK (x) approaches m(x) as K increases. Both (12.2) and (12.3) hold for spline approximations as well. Intuitively, the number of derivatives s indexes the smoothness of the function m(x): (12.3) says that the best rate at which a polynomial or spline approximates the CEF m(x) depends on the underlying smoothness of m(x): The more smooth is m(x); the fewer series terms (polynomial order or spline knots) are needed to obtain a good approximation. To illustrate polynomial approximation, Figure 12.1 displays the CEF m(x) = x1=4 (1 x)1=2 on x 2 [0; 1]: In addition, the best approximations using polynomials of order K = 3; K = 4; and K = 6 are displayed. You can see how the approximation with K = 3 is fairly crude, but improves with K = 4 and especially K = 6: Approximations obtained with cubic splines are quite similar so not displayed. As a series approximation can be written as mK (x) = z K (x)0 K as in (12.1), then the coe¢ cient of the best uniform approximation (12.3) is then K = argmin sup z K (x)0 K x2X m(x) : K (12.4) The approximation error is rK (x) = m(x) z K (x)0 m(x) = z K (x)0 K K: We can write this as + rK (x) (12.5) to emphasize that the true conditional mean can be written as the linear approximation plus error. A useful consequence of equation (12.3) is sup jrK (x)j x2X O K : (12.6) CHAPTER 12. SERIES ESTIMATION 226 Figure 12.2: True CEF, polynomial interpolation, and spline interpolation 12.6 Runge’s Phenomenon Despite the excellent approximation implied by the Stone-Weierstrass theorem, polynomials have the troubling disadvantage that they are very poor at simple interpolation. The problem is known as Runge’s phenomenon, and is illustrated in Figure 12.2. The solid line is the CEF m(x) = (1 + x2 ) 1 displayed on [ 5; 5]: The circles display the function at the K = 11 integers in this interval. The long dashes display the 10’th order polynomial …t through these points. Notice that the polynomial approximation is erratic and far from the smooth CEF. This discrepancy gets worse as the number of evaluation points increases, as Runge (1901) showed that the discrepancy increases to in…nity with K: In contrast, splines do not exhibit Runge’s phenomenon. In Figure 12.2 the short dashes display a cubic spline with seven knots …t through the same points as the polynomial. While the …tted spline displays some oscillation relative to the true CEF, they are relatively moderate. Because of Runge’s phenomenon, high-order polynomials are not used for interpolation, and are not popular choices for high-order series approximations. Instead, splines are widely used. 12.7 Approximating Regression For each observation i we observe (yi ; xi ) and then construct the regressor vector z Ki = z K (xi ) using the series transformations. Stacking the observations in the matices y and Z K ; the least squares estimate of the coe¢ cient K in the series approximation z K (x)0 K is b K = Z 0K Z K 1 Z 0K y; and the least squares estimate of the regression function is m b K (x) = z K (x)0 b K : (12.7) As we learned in Chapter 2, the least-squares coe¢ cient is estimating the best linear predictor of yi given z Ki : This is 1 0 E (z Ki yi ) : K = E z Ki z Ki CHAPTER 12. SERIES ESTIMATION 227 Given this coe¢ cient, the series approximation is z K (x)0 K z K (x)0 rK (x) = m(x) with approximation error K: (12.8) The true CEF equation for yi is yi = m(xi ) + ei (12.9) with ei the CEF error. De…ning rKi = rK (xi ); we …nd yi = z 0Ki K + eKi where the equation error is eKi = rKi + ei : Observe that the error eKi includes the approximation error and thus does not have the properties of a CEF error. In matrix notation we can write these equations as y = ZK K + rK + e = ZK K + eK : (12.10) We now impose some regularity conditions on the regression model to facilitate the theory. De…ne the K K expected design matrix QK = E z Ki z 0Ki ; let X denote the support of xi ; and de…ne the largest normalized length of the regressor vector in the support of xi 1=2 1 0 : (12.11) K = sup z K (x) QK z K (x) x2X K will increase with K. For example, p if the support of the variables z K (xi ) is the unit cube [0; 1] , then you can compute that K = K. As discussed in Newey (1997) and Li and Racine (2007, Corollary 15.1) if the support of xi is compact then K = O(K) for polynomials and K = O(K 1=2 ) for splines. K Assumption 12.7.1 1. For some > 0 the series approximation satis…es (12.3): 2. E e2i j xi 2 3. > 0: min (QK ) < 1: 4. K = K(n) is a function of n which satis…es K=n ! 0 and 0 as n ! 1: 2 K K=n ! Assumptions 12.7.1.1 through 12.7.1.3 concern properties of the regression model. Assumption 12.7.1.1 holds with = s=d if X is compact and the s’th derivative of m(x) is continuous. Assumption 12.7.1.2 allows for conditional heteroskedasticity, but requires the conditional variance to be bounded. Assumption 12.7.1.3 excludes near-singular designs. Since estimates of the conditional mean are unchanged if we replace z Ki with z Ki = B K z Ki for any non-singular B K ; Assumption 12.7.1.3 can be viewed as holding after transformation by an appropriate non-singular B K . CHAPTER 12. SERIES ESTIMATION 228 Assumption 12.7.1.4 concerns the choice of the number of series terms, which is under the control of the user. It speci…es that K can increase with sample size, but at a controlled rate of growth. Since K = O(K) for polynomials and K = O(K 1=2 ) for splines, Assumption 12.7.1.4 is satis…ed if K 3 =n ! 0 for polynomials and K 2 =n ! 0 for splines. This means that while the number of series terms K can increase with the sample size, K must increase at a much slower rate. In Section 12.5 we introduced the best uniform approximation, and in this section we introduced the best linear predictor. What is the relationship? They may be similar in practice, but they are not the same and we should be careful to maintain the distinction. Note that from (12.5) we can write m(xi ) = z 0Ki K + rKi where rKi = rK (xi ) satis…es supi jrKi j = O (K ) from (12.6). Then the best linear predictor equals K = E z Ki z 0Ki 1 = E z Ki z 0Ki 1 = E z Ki z 0Ki 1 = +E K E (z Ki yi ) E (z Ki m(xi )) E z Ki (z 0Ki z Ki z 0Ki 1 K + rKi ) E (z Ki rKi ) : Thus the di¤erence between the two approximations is rK (x) rK (x) = z K (x)0 ( 0 = z K (x) E K) 1 0 z Ki z Ki E (z Ki rKi ) : K (12.12) Observe that by the properties of projection E (r Ki z Ki )0 E z Ki z 0Ki 2 E r Ki and by (12.6) E 2 rKi = Z 1 rK (x)2 fx (x)dx E (z Ki rKi ) 2 O K 0 (12.13) : (12.14) Then applying the Schwarz inequality to (12.12), De…nition (12.11), (12.13) and (12.14), we …nd jrK (x) z K (x)0 E z Ki z 0Ki rK (x)j 1 z K (x) 1 E (rKi z Ki )0 E z Ki z 0Ki O KK 1=2 E (z Ki rKi ) : 1=2 (12.15) It follows that the best linear predictor approximation error satis…es sup jrK (x)j O x2X KK : (12.16) The bound (12.16) is probably not the best possible, but it shows that the best linear predictor satis…es a uniform approximation bound. Relative to (12.6), the rate is slower by the factor K : The bound (12.16) term is o(1) as K ! 1 if K K ! 0. A su¢ cient condition is that > 1 (s > d) for polynomials and > 1=2 (s > d=2) for splines; where d = dim(x) and s is the number of continuous derivatives of m(x): It is also useful to observe that since K is the best linear approximation to m(xi ) in meansquare (see Section 2.23), then z 0Ki K E m(xi ) z 0Ki K O K the …nal inequality by (12.14). 2 2 ErKi = E m(xi ) 2 2 (12.17) CHAPTER 12. SERIES ESTIMATION 12.8 229 Residuals and Regression Fit The …tted regression at x = xi is m b K (xi ) = z 0Ki b K and the …tted residual is m b K (xi ): e^iK = yi The leave-one-out prediction errors are e~iK m b K; i (xi ) z0 b = yi = yi where b K; also write i Ki K; i is the least-squares coe¢ cient with the i’th observation omitted. Using (3.38) we can e~iK = e^iK (1 1 hKii ) 1 where hKii = z 0Ki (Z 0K Z K ) z Ki : As for kernel regression, the prediction errors e~iK are better estimates of the errors than the …tted residuals e^iK ; as they do not have the tendency to “over-…t”when the number of series terms is large. To assess the …t of the nonparametric regression, the estimate of the mean-square prediction error is n n 1X 2 1X 2 ~ 2K = e~iK = e^iK (1 hKii ) 2 n n i=1 and the prediction 12.9 R2 is 2 eK R =1 i=1 Pn Pn ~2iK i=1 e i=1 (yi y)2 : Cross-Validation Model Selection The cross-validation criterion for selection of the number of series terms is the MSPE n CV (K) = ~ 2K = 1X 2 e^iK (1 n hKii ) 2 : i=1 e2 ; we have a dataBy selecting the series terms to minimize CV (K); or equivalently maximize R K dependent rule which is designed to produce estimates with low integrated mean-squared error (IMSE) and mean-squared forecast error (MSFE). As shown in Theorem 11.6.1, CV (K) is an approximately unbiased estimated of the MSFE and IMSE, so …nding the model which produces the smallest value of CV (K) is a good indicator that the estimated model has small MSFE and IMSE. The proof of the result is the same for all nonparametric estimators (series as well as kernels) so does not need to be repeated here. As a practical matter, an estimator corresponds to a set of regressors z Ki , that is, a set of transformations of the original variables xi : For each set of regressions, the regression is estimated and CV (K) calculated, and the estimator is selected which has the smallest value of CV (K): If there are p ordered regressors, then there are p possible estimators. Typically, this calculation is simple even if p is large. However, if the p regressors are unordered (and this is typical) then there are 2p possible subsets of conceivable models. If p is even moderately large, 2p can be immensely large so brute-force computation of all models may be computationally demanding. CHAPTER 12. SERIES ESTIMATION 12.10 230 Convergence in Mean-Square The series estimate b K are indexed by K. The point of nonparametric estimation is to let K be ‡exible so as to incorporate greater complexity when the data are su¢ ciently informative. This means that K will typically be increasing with sample size n: This invalidates conventional asymptotic distribution theory. However, we can develop extensions which use appropriate matrix norms, and by focusing on real-valued functions of the parameters including the estimated regression function itself. The asymptotic theory we present in this and the next several sections is largely taken from Newey (1997). Our …rst main result shows that the least-squares estimate converges to K in mean-square distance. Theorem 12.10.1 Under Assumption 12.7.1, as n ! 1, b K K 0 QK b K K = Op K n + op K 2 (12.18) The proof of Theorem 12.10.1 is rather technical and deferred to Section 12.16. The rate of convergence in (12.18) has two terms. The Op (K=n) term is due to estimation variance. Note in contrast that the corresponding rate would be Op (1=n) in the parametric case. The di¤erence is that in the parametric case we assume that the number of regressors K is …xed as n increases, while in the nonparametric case we allow the number of regressors K to be ‡exible. As K increases, the estimation variance increases. The op K 2 term in (12.18) is due to the series approximation error. Using Theorem 12.10.1 we can establish the following convergence rate for the estimated regression function. Theorem 12.10.2 Under Assumption 12.7.1, as n ! 1, Z K + Op K 2 (m b K (x) m(x))2 fx (x)dx = Op n (12.19) Theorem 12.10.2 shows that the integrated squared di¤erence between the …tted regression and the true CEF converges in probability to zero if K ! 1 as n ! 1: The convergence results of Theorem 12.10.2 show that the number of series terms K involves a trade-o¤ similar to the role of the bandwidth h in kernel regression. Larger K implies smaller approximation error but increased estimation variance. The optimal rate which minimizes the average squared error in (12.19) is K = O n1=(1+2 ) ; yielding an optimal rate of convergence in (12.19) of Op n 2 =(1+2 ) : This rate depends on the unknown smoothness of the true CEF (the number of derivatives s) and so does not directly syggest a practical rule for determining K: Still, the implication is that when the function being estimated is less smooth ( is small) then it is necessary to use a larger number of series terms K to reduce the bias. In contrast, when the function is more smooth then it is better to use a smaller number of series terms K to reduce the variance. To establish (12.19), using (12.7) and (12.8) we can write m b K (x) m(x) = z K (x)0 b K K rK (x): (12.20) CHAPTER 12. SERIES ESTIMATION 231 Since eRKi are projection errors, they satisfy E (z Ki eKi ) = R0 and thus E (z Ki rKi ) = 0: This 0 2 Rmeans 2 z K (x)rK (x)fx (x)dx = 0: Also observe that QK = z K (x)z K (x) fx (x)dx and ErKi = rK (x) fx (x)dx. Then Z (m b K (x) m(x))2 fx (x)dx = b K Op K K n 0 QK b K 2 + ErKi K 2 + Op K by (12.18) and (12.17), establishing (12.19). 12.11 Uniform Convergence Theorem 12.10.2 established conditions under which m b K (x) is consistent in a squared error norm. It is also of interest to know the rate at which the largest deviation converges to zero. We have the following rate. Theorem 12.11.1 Under Assumption 12.7.1, then as n ! 1; 1 0s 2 K K A + Op K K sup jm b K (x) m(x)j = Op @ : n x2X (12.21) Relative to Theorem 12.10.2, the error has been increased multiplicatively by K : This slower convergence rate is a penalty for the stronger uniform convergence, though it is probably not the best possible rate. Examining the bound in (12.21) notice that the …rst term is op (1) under ! 0; which requires that K ! 1 and Assumption 12.7.1.4. The second term is op (1) if K K that be su¢ ciently large. A su¢ cient condition is that s > d for polynomials and s > d=2 for splines; where d = dim(x) and s is the number of continuous derivatives of m(x): Thus higher dimensional x require a smoother CEF m(x) to ensure that the series estimate m b K (x) is uniformly consistent. The convergence (12.21) is straightforward to show using (12.18). Using (12.20), the Triangle Inequality, the Schwarz inequality (A.10), De…nition (12.11), (12.18) and (12.16), sup jm b K (x) m(x)j x2X sup z K (x)0 b K x2X + sup jrK (x)j K sup z K (x)0 QK1 z K (x) x2X 1=2 x2X +O KK K Op 0s This is (12.21). = Op @ K n K K 0 QK b K 1=2 K 1=2 + Op K 1 2 KK A n b + Op 2 KK +O : KK ; (12.22) CHAPTER 12. SERIES ESTIMATION 12.12 232 Asymptotic Normality One advantage of series methods is that the estimators are (in …nite samples) equivalent to parametric estimators, so it is easy to calculate covariance matrix estimates. We now show that we can also justify normal asymptotic approximations. The theory we present in this section will apply to any linear function of the regression function. That is, we allow the parameter of interest to be aany non-trivial real-valued linear function of the entire regression function m( ) = a (m) : This includes the regression function m(x) at a given point x; derivatives of m(x), and integrals over m(x). Given m b K (x) = z K (x)0 b K as an estimator for m(x); the estimator for is ^K = a (m b K ) = a0K b K for some K 1 vector of constants aK 6= 0: (The relationship a (m b K ) = a0K b K follows since a is linear in m and m b K is linear in b K .) If K were …xed as n ! 1; then by standard asymptotic theory we would expect ^K to be asymptotically normal with variance vK = a0K QK1 1 K QK aK where K = E z Ki z 0Ki e2Ki : The standard justi…cation, however, is not valid in the nonparametric case, in part because vK may diverge as K ! 1; and in part due to the …nite sample bias due to the approximation error. Therefore a new theory is required. Interestingly, it turns out that in the nonparametric case ^K is still asymptotically normal, and vK is still the appropriate variance for ^K . The proof is di¤erent than the parametric case as the dimensions of the matrices are increasing with K; and we need to be attentive to the estimator’s bias due to the series approximation. Theorem 12.12.1 Under Assumption 12.7.1, if in addition E e4i jxi 2 2 > 0; and = O(1); then as n ! 1; 4 < 1, E ei jxi KK p n ^K + a (rK ) 1=2 vK d ! N (0; 1) (12.23) The proof of Theorem 12.12.1 can be found in Section 12.16. Theorem 12.12.1 shows that the estimator ^K is approximately normal with bias a (rK ) and variance vK =n: The variance is the same as in the parametric case, but the asymptotic distribution contains an asymptotic bias, similar as is found in kernel regression. We discuss the bias in more detail below. Notice that Theorem 12.12.1 requires K K = O(1); which is similar to that found in Theorem 12.11.1 to establish uniform convergence. The the bound K K = O(1) allows K to be constant with n or to increase with n. However, when K is increasing the bound requires that be su¢ cient large so that K grows faster than K : A su¢ cient condition is that s = d for polynomials and s = d=2 for splines. The fact that the condition allows for K to be constant means that Theorem 12.12.1 includes parametric least-squares as a special case with explicit attention to estimation bias. CHAPTER 12. SERIES ESTIMATION 233 One useful message from Theorem 12.12.1 is that the classic variance formula vK for ^K still applies for series regression. Indeed, we can estimate the asymptotic variance using the standard White formula v^K bK Hence a standard error for ^K is bK Q s^( K) b 1 b KQ b 1 aK = a0K Q K K n X 1 = z Ki z 0Ki e^2iK n i=1 n 1X z Ki z 0Ki : n = i=1 = r 1 0 b 1b b 1 a Q K QK aK : n K K p It can be shown (Newey, 1997) that v^K =vK ! 1 as n ! 1 and thus the distribution in (12.23) is unchanged if vK is replaced with v^K : Theorem 12.12.1 shows that the estimator ^K has a bias term a (rK ) : What is this? It is the same transformation of the function rK (x) as = a (m) is of the regression function m(x). For example, if = m(x) is the regression at a …xed point x , then a (rK ) = rK (x); the approximation d d error at the same point. If = m(x) is the regression derivative, then a (rK ) = rK (x) is the dx dx derivative of the approximation error. This means that the bias in the estimator ^K for shown in Theorem 12.12.1 is simply the approximation error, transformed by the functional of interest. If we are estimating the regression function then the bias is the error in approximating the regression function; if we are estimating the regression derivative then the bias is the error in the derivative in the approximation error for the regression function. 12.13 Asymptotic Normality with Undersmoothing An unpleasant aspect about Theorem 12.12.1 is the bias term. An interesting trick is that this bias term can be made asymptotically negligible if we assume that K increases with n at a su¢ ciently fast rate. Theorem 12.13.1 Under Assumption 12.7.1, if in addition E e4i jxi 2 2 > 0, a (r ) O (K ) ; nK 2 ! 0; and 4 < 1, E ei jxi K 1 0 aK QK aK is bounded away from zero, then p ^ n K 1=2 vK d ! N (0; 1) : (12.24) The condition a (rK ) O (K ) states that the function of interest (for example, the regression function, its derivative, or its integral) applied to the uniform approximation error converges to zero as the number of terms K in the series approximation increases. If a (m) = m(x) then this condition holds by (12.6). The condition that a0K QK1 aK is bounded away from zero is simply a technical requirement to exclude degeneracy. CHAPTER 12. SERIES ESTIMATION 234 The critical condition is the assumption that nK 2 ! 0: This requires that K ! 1 at a rate faster than n1=2 : This is a troubling condition. The optimal rate for estimation of m(x) is K = O n1=(1+2 ) : If we set K = n1=(1+2 ) by this rule then nK 2 = n1=(1+2 ) ! 1; not zero. Thus this assumption is equivalent to assuming that K is much larger than optimal. The reason why this trick works (that is, why the bias is negligible) is that by increasing K; the asymptotic bias decreases and the asymptotic variance increases and thus the variance dominates. Because K is larger than optimal, we typically say that m b K (x) is undersmoothed relative to the optimal series estimator. Many authors like to focus their asymptotic theory on the assumptions in Theorem 12.13.1, as the distribution (12.24) appears cleaner. However, it is a poor use of asymptotic theory. There are three problems with the assumption nK 2 ! 0 and the approximation (12.24). First, it says that if we intentionally pick K to be larger than optimal, we can increase the estimation variance relative to the bias so the variance will dominate the bias. But why would we want to intentionally use an estimator which is sub-optimal? Second, the assumption nK 2 ! 0 does not eliminate the asymptotic bias, it only makes it of lower order than the variance. So the approximation (12.24) is technically valid, but the missing asymptotic bias term is just slightly smaller in asymptotic order, and thus still relevant in …nite samples. Third, the condition nK 2 ! 0 is just an assumption, it has nothing to do with actual empirical practice. Thus the di¤erence between (12.23) and (12.24) is in the assumptions, not in the actual reality or in the actual empirical practice. Eliminating a nuisance (the asymptotic bias) through an assumption is a trick, not a substantive use of theory. My strong view is that the result (12.23) is more informative than (12.24). It shows that the asymptotic distribution is normal but has a non-trivial …nite sample bias. 12.14 Regression Estimation A special yet important example of a linear estimator of the regression function is the regression function at a …xed point x. In the notation of the previous section, a (m) = m(x) and aK = z K (x): The series estimator of m(x) is ^K = m b K (x) = z K (x)0 b K : As this is a key problem of interest, we restate the asymptotic results of Theorems 12.12.1 and 12.13.1 for this estimator. Theorem 12.14.1 Under Assumption 12.7.1, if in addition E e4i jxi 2 2 > 0; and = O(1); then as n ! 1; 4 < 1; E ei jxi KK p n (m b K (x) m(x) + r K (x)) 1=2 vK (x) d ! N (0; 1) (12.25) where vK (x) = z K (x)0 QK1 1 K QK z K (x): If K K = O(1) is replaced by nK 2 ! 0; and z K (x)0 QK1 z K (x) is bounded away from zero, then p n (m b K (x) m(x)) d ! N (0; 1) (12.26) 1=2 vK (x) There are two important features about the asymptotic distribution (12.25). First, as mentioned in the previous section, it shows how to construct asymptotic standard errors for the CEF m(x): These are r 1 b 1 b KQ b 1 z K (x): s^(x) = z K (x)0 Q K K n CHAPTER 12. SERIES ESTIMATION 235 Second, (12.25) shows that the estimator has the asymptotic bias component r K (x): This is due to the fact that the …nite order series is an approximation to the unknown CEF m(x); and this results in …nite sample bias. The asymptotic distribution (12.26) shows that the bias term is negligable if K diverges fast enough so that nK 2 ! 0: As discussed in the previous section, this means that K is larger than optimal. The assumption that z K (x)0 QK1 z K (x) is bounded away from zero is a technical condition to exclude degenerate cases, and is automatically satis…ed if z K (x) includes an intercept. Plots of the CEF estimate m b K (x) can be accompanied by 95% con…dence intervals m b K (x) 2^ s(x): As we discussed in the chapter on kernel regression, this can be viewed as a con…dence interval for the pseudo-true CEF mK (x) = m(x) r K (x); not for the true m(x). As for kernel regression, the di¤erence is the unavoidable consequence of nonparametric estimation. 12.15 Kernel Versus Series Regression In this and the previous chapter we have presented two distinct methods of nonparametric regression based on kernel methods and series methods. Which should be used in practice? Both methods have advantages and disadvantages and there is no clear overall winner. First, while the asymptotic theory of the two estimators appear quite di¤erent, they are actually rather closely related. When the regression function m(x) is twice di¤erentiable (s = 2) then the rate of convergence of both the MSE of the kernel regression estimator with optimal bandwidth h and the series estimator with optimal K is n 2=(d+4) : There is no di¤erence. If the regression function is smoother than twice di¤erentiable (s > 2) then the rate of the convergence of the series estimator improves. This may appear to be an advantage for series methods, but kernel regression can also take advantage of the higher smoothness by using so-called higher-order kernels or local polynomial regression, so perhaps this advantage is not too large. Both estimators are asymptotically normal and have straightforward asymptotic standard error formulae. The series estimators are a bit more convenient for this purpose, as classic parametric standard error formula work without ammendment. An advantage of kernel methods is that their distributional theory is easier to derive. The theory is all based on local averages which is relatively straightforward. In contrast, series theory is more challenging, dealing with increasing parameter spaces. An important di¤erence in the theory is that for kernel estimators we have explicit representations for the bias while we only have rates for series methods. This means that plug-in methods can be used for bandwidth selection in kernel regression. However, typically we rely on cross-validation, which is equally applicable in both kernel and series regression. Kernel methods are also relatively easy to implement when the dimension d is large. There is not a major change in the methodology as d increases. In contrast, series methods become quite cumbersome as d increases as the number of cross-terms increases exponentially. A major advantage of series methods is that it has inherently a high degree of ‡exibility, and the user is able to implement shape restrictions quite easily. For example, in series estimation it is relatively simple to implement a partial linear CEF, an additively separable CEF, monotonicity, concavity or convexity. These restrictions are harder to implement in kernel regression. 12.16 Technical Proofs 1=2 De…ne z Ki = z K (xi ) and let QK denote the positive de…nite square root of QK : As mentioned before Theorem 12.10.1, the regression problem is unchanged if we replace z Ki with a rotated 1=2 0 ) = I : For regressor such as z Ki = QK z Ki . This is a convenient choice for then E (z Ki z Ki K notational convenience we will simply write the transformed regressors as z Ki and set QK = I K : CHAPTER 12. SERIES ESTIMATION 236 We start with some convergence results for the sample design matrix n X b K = 1 Z0 ZK = 1 Q z Ki z 0Ki : K n n i=1 Theorem 12.16.1 Under Assumption 12.7.1 and QK = I K , as n ! 1, bK Q and I K = op (1) (12.27) p b min (QK ) ! 1: (12.28) Proof. Since bK Q then 2 = IK K X K X j=1 `=1 2 bK E Q IK Ez jKi z `Ki ) i=1 K X K X = !2 n 1X (z jKi z `Ki n n var i=1 j=1 `=1 = n 1 1X z jKi z `Ki n K X K X ! var (z jKi z `Ki ) j=1 `=1 n 1 = n 1 E K X z 2jKi j=1 Since z 0Ki z Ki 2 K K X z 2`Ki `=1 2 0 z Ki z Ki : E (12.29) by de…nition (12.11) and using (A.1) we …nd E z 0Ki z Ki = tr Ez Ki z 0Ki = tr I K = K; so that E z 0Ki z Ki 2 (12.30) 2 KK (12.31) and hence (12.29) is o(1) under Assumption 12.7.1.4. Theorem 5.11.1 shows that this implies (12.27). b K I K which are real as Q b K I K is symmetric. Then Let 1 ; 2 ; :::; K be the eigenvalues of Q b min (QK ) 1 = b min (QK IK) K X `=1 2 ` !1=2 bK = Q IK where the second equality is (A.8). This is op (1) by (12.27), establishing (12.28): Proof of Theorem 12.10.1. As above, assume that the regressors have been transformed so that QK = I K : From expression (12.10) we can substitute to …nd b K K 1 Z 0K Z K Z 0K eK : b 1 1 Z 0 eK = Q K n K = (12.32) CHAPTER 12. SERIES ESTIMATION 237 Using (12.32) and the Quadratic Inequality (A.14), = n Observe that (12.28) implies max b K K 0 b K K 2 b 1Q b 1 Z 0 eK e0K Z K Q K K K 2 1 b n 2 e0K Z K Z 0K eK max QK b 1 = Q K 1 bK Q max : (12.33) = Op (1): (12.34) Since eKi = ei + rKi ; and using Assumption 12.7.1.2 and (12.16), then sup E e2Ki jxi = i 2 2 + sup rKi 2 2 2 KK +O i : (12.35) As eKi are projection errors, they satisfy E (z Ki eKi ) = 0: Since the observations are independent, using (12.30) and (12.35), then 0 1 n n X X n 2 E e0K Z K Z 0K eK = n 2 E @ eKi z 0Ki z Kj eKj A i=1 = n 2 n 1 n X ij=1 E z 0Ki z Ki e2Ki i=1 E z 0Ki z Ki sup E e2Ki jxi 2K +O n 2K +o K n = since 2 K K=n i 2 1 2 KK n 2 (12.36) = o(1) by Assumption 12.7.1.4. Theorem 5.11.1 shows that this implies n 2 0 eK Z K Z 0K eK = Op n 2 + op K 2 : (12.37) Together, (12.33), (12.34) and (12.37) imply (12.18). Proof of Theorem 12.12.1. As above, assume that the regressors have been transformed so that QK = I K : Using m(x) = z K (x)0 K + rK (x) and linearity = a (m) = a z K (x)0 = a0K K K + a (rK ) + a (rK ) Combined with (12.32) we …nd ^K + a (rK ) = a0K b K = K 1 0 b 1 0 a Q Z eK n K K K CHAPTER 12. SERIES ESTIMATION 238 and thus r n ^ K vk K + a (rK ) = r r n 0 b a K vk K K 1 0 b 1 0 a Q Z eK nvk K K K 1 = p a0 Z 0 eK nvK K K 1 b 1 IK Z0 e +p a0K Q K K nvK 1 b 1 I K Z 0 rK : a0 Q +p K K nvK K = (12.38) (12.39) (12.40) where we have used eK = e + r K : We now take the terms in (12.38)-(12.40) separately. First, take (12.38). We can write p n 1 1 X 0 aK z Ki eKi : a0K Z 0K eK = p nvK nvK (12.41) i=1 Observe that a0K z Ki eKi are independent across i, mean zero, and have variance E a0K z Ki eKi 2 = a0K E z Ki z 0Ki e2Ki aK = vK : We will apply the Lindeberg CLT 5.7.2, for which it is su¢ cient to verify Lyapunov’s condition (5.6): n 1 X 1 4 4 E a0K z Ki eKi = a0K z Ki e4Ki ! 0: (12.42) 2 2 E 2 n vK i=1 nvK The assumption that inequality and E e4i jxi KK = O(1) means sup E e4Ki jxi i KK 1 for some 4 8 sup E e4i jxi + rKi 8( + 1 < 1: Then by the cr 1) : (12.43) i Using (12.43), the Schwarz Inequality, and (12.31) E a0K z Ki 4 4 eKi 2 Since E e2Ki jxi = E e2i jxi + rKi vK = E 4 a0K z Ki E e4Ki jxi a0K z Ki 4 8( + 1) E 8( + 1) a0K aK 2 = 8( + 1) a0K aK 2 2 K K: E z 0Ki z Ki (12.44) 2; = a0K E z Ki z 0Ki e2Ki aK = 2 0 aK E z Ki z 0Ki 2 0 aK aK : aK (12.45) Equation (12.44) and (12.45) combine to show that 1 2 E nvK 2 a0K z Ki 4 4 eKi 8( + 4 2 1) K K n = o(1) CHAPTER 12. SERIES ESTIMATION 239 under Assumption 12.7.1.4. This establishes Lyapunov’s condition (12.42). Hence the Lindeberg CLT applies to (12.41) and we conclude p 1 d a0 Z 0 eK ! N (0; 1) : nvK K K Second, take (12.39). Since E (e j X) = 0, then applying E e2i jxi Inequalities, (12.45), (12.34) and (12.27), ! 2 1 1 b E a0 Q I K Z 0K e j X p K nvK K = 1 0 b 1 a QK nvK K 2 vK 2 = b 1 a0K Q K b 1 I K Z 0K E ee0 j X Z K Q K bK Q b 1 IK Q K (12.46) 2; the Schwarz and Norm I K aK I K aK b K IK Q b 1 Q b K I K aK a0 Q K vK K 2 a0 a 2 K K b K IK b 1 Q max QK vK 2 o (1): 2 p This establishes p 1 b 1 a0K Q K nvK p I K Z 0K e ! 0: (12.47) Third, take (12.40). By the Cauchy-Schwarz inequality, (12.45), and the Quadratic Inequality, 2 1 1 0 0 b a QK I K Z K rK p nvK K a0K aK 0 b 1 IK Q b 1 I K Z 0 rK rK Z K Q K K K nvK 2 1 b 1 I K 1 r0 Z K Z 0 rK : Q K K 2 max n K Observe that since the observations are independent and Ez Ki rKi = 0; z 0Ki z Ki 0 1 n n X X 1 0 1 r Z K Z 0K r K = E@ E rKi z 0Ki z Kj rKj A n K n i=1 ij=1 ! n 1X 0 2 = E z Ki z Ki rKi n = (12.48) 2 K, and (12.17) i=1 2 2 K E rKi O 2K K 2 = O(1) since K K implies 2 = O(1): Thus 1 0 r Z K Z 0K r K = Op (1): This means that (12.48) is op (1) since (12.28) n K max Equivalently, b 1 Q K p IK = 1 b 1 a0 Q K nvK K max b 1 Q K 1 = op (1): p I K Z 0K r K ! 0: (12.49) (12.50) CHAPTER 12. SERIES ESTIMATION 240 Equations (12.46), (12.47) and (12.50) applied to (12.38)-(12.40) show that r n ^ d ! N (0; 1) K K + a (rK ) vk completing the proof. 2 Proof of Theorem 12.13.1. The assumption that nK KK 1=2 2 K o n ! = o(1) implies K 1=2 2 KK o n ! =o n 1=2 . Thus = o(1) so the conditions of Theorem 12.12.1 are satis…ed. It is thus su¢ cient to show that r n a (rK ) = o(1): vk From (12.12) rK (x) = rK (x) + z K (x)0 K = E z Ki z 0Ki 1 K E (z Ki rKi ) : Thus by linearity, applying (12.45), and the Schwarz inequality r r n n a (rK ) = a (rK ) + a0K vk vk n1=2 2 + By assumption, n1=2 a (rK ) = O n1=2 K n 0 K K a0K aK (n 1=2 1=2 0 K K) a (rK ) (12.51) : (12.52) = o(1): By (12.14) and nK = nE rKi z 0Ki E z Ki z 0Ki nO K K 2 = o(1): Together, both (12.51) and (12.52) are o(1); as required. 1 2 E (z Ki rKi ) = o(1) Chapter 13 Quantile Regression 13.1 Least Absolute Deviations We stated that a conventional goal in econometrics is estimation of impact of variation in xi on the central tendency of yi : We have discussed projections and conditional means, but these are not the only measures of central tendency. An alternative good measure is the conditional median. To recall the de…nition and properties of the median, let y be a continuous random variable. The median = med(y) is the value such that Pr(y ) = Pr (y ) = :5: Two useful facts about the median are that = argmin E jy j (13.1) and E sgn (y )=0 where sgn (u) = 1 1 if u 0 if u < 0 is the sign function. These facts and de…nitions motivate three estimators ofP : The …rst de…nition is the 50th empirical quantile. The second is the value which minimizes n1 ni=1 jyi j ; and the third de…nition P n 1 ) : These distinctions are illusory, however, is the solution to the moment equation n i=1 sgn (yi as these estimators are indeed identical. Now let’s consider the conditional median of y given a random vector x: Let m(x) = med (y j x) denote the conditional median of y given x: The linear median regression model takes the form yi = x0i + ei med (ei j xi ) = 0 In this model, the linear function med (yi j xi = x) = x0 is the conditional median function, and the substantive assumption is that the median function is linear in x: Conditional analogs of the facts about the median are Pr(yi x0 j xi = x) = Pr(yi > x0 j xi = x) = :5 E (sgn (ei ) j xi ) = 0 E (xi sgn (ei )) = 0 = min E jyi x0i j 241 CHAPTER 13. QUANTILE REGRESSION 242 These facts motivate the following estimator. Let n LADn ( ) = 1X yi n x0i i=1 be the average of absolute deviations. The least absolute deviations (LAD) estimator of minimizes this function b = argmin LADn ( ) Equivalently, it is a solution to the moment condition n 1X xi sgn yi n i=1 x0i b = 0: (13.2) The LAD estimator has an asymptotic normal distribution. Theorem 13.1.1 Asymptotic Distribution of LAD Estimator When the conditional median is linear in x p d n b ! N (0; V ) where V = 1 E xi x0i f (0 j xi ) 4 1 Exi x0i E xi x0i f (0 j xi ) 1 and f (e j x) is the conditional density of ei given xi = x: The variance of the asymptotic distribution inversely depends on f (0 j x) ; the conditional density of the error at its median. When f (0 j x) is large, then there are many innovations near to the median, and this improves estimation of the median. In the special case where the error is independent of xi ; then f (0 j x) = f (0) and the asymptotic variance simpli…es V = (Exi x0i ) 4f (0)2 1 (13.3) This simpli…cation is similar to the simpli…cation of the asymptotic covariance of the OLS estimator under homoskedasticity. Computation of standard error for LAD estimates typically is based on equation (13.3). The main di¢ culty is the estimation of f (0); the height of the error density at its median. This can be done with kernel estimation techniques. See Chapter 21. While a complete proof of Theorem 13.1.1 is advanced, we provide a sketch here for completeness. Proof of Theorem 13.1.1: Similar to NLLS, LAD is an optimization estimator. Let 0 denote the true value of 0 : p The …rst step is to show that ^ ! 0 : The general nature of the proof is similar to that for the p NLLS estimator, and is sketched here. For any …xed ; by the WLLN, LADn ( ) ! E jyi x0i j : Furthermore, it can be shown that this convergence is uniform in : (Proving uniform convergence is more challenging than for the NLLS criterion since the LAD criterion is not di¤erentiable in .) It follows that ^ ; the minimizer of LADn ( ); converges in probability to 0 ; the minimizer of E jyi x0i j. CHAPTER 13. QUANTILE REGRESSION 243 P Since sgn (a) = 1 2 1 (a 0) ; (13.2) is equivalent to g n ( b ) = 0; where g n ( ) = n 1 ni=1 g i ( ) and g i ( ) = xi (1 2 1 (yi x0i )) : Let g( ) = Eg i ( ). We need three preliminary results. First, by the central limit theorem (Theorem 5.7.1) p n (g n ( 0) g( 0 )) = n n X 1=2 gi( d ! N 0; Exi x0i 0) i=1 since Eg i ( 0 )g i ( di¤erentiation, 0 0) = Exi x0i : Second using the law of iterated expectations and the chain rule of @ g( ) = @ 0 = = = so @ Exi 1 2 1 yi x0i @ 0 @ 2 0 E xi E 1 ei x0i x0i 0 j xi @ # " Z 0 xi x0i 0 @ f (e j xi ) de 2 0 E xi @ 1 2E xi x0i f x0i @ g( ) = @ 0 x0i 0 j xi 2E xi x0i f (0 j xi ) : Third, by a Taylor series expansion and the fact g( ) = 0 Together p n b 0 ' = g( b ) ' @ g( ) b @ 0 @ g( @ 0 1 0) p : ng( ^ ) 2E xi x0i f (0 j xi ) 1 E xi x0i f (0 j xi ) 2 d 1 ! E xi x0i f (0 j xi ) 2 = N (0; V ) : ' 1p n g( ^ ) 1p 1 n (g n ( 0) gn( ^ ) g( 0 )) 0 N 0; Exi xi p The third line follows from an asymptotic empirical process argument and the fact that b ! 13.2 0. Quantile Regression Quantile regression has become quite popular in recent econometric practice. For ’th quantile Q of a random variable with distribution function F (u) is de…ned as Q = inf fu : F (u) 2 [0; 1] the g When F (u) is continuous and strictly monotonic, then F (Q ) = ; so you can think of the quantile as the inverse of the distribution function. The quantile Q is the value such that (percent) of the mass of the distribution is less than Q : The median is the special case = :5: CHAPTER 13. QUANTILE REGRESSION 244 The following alternative representation is useful. If the random variable U has ’th quantile Q ; then Q = argmin E (U ): (13.4) where (q) is the piecewise linear function q (1 ) q<0 q q 0 = q( 1 (q < 0)) : (q) = (13.5) This generalizes representation (13.1) for the median to all quantiles. For the random variables (yi ; xi ) with conditional distribution function F (y j x) the conditional quantile function q (x) is Q (x) = inf fy : F (y j x) g: Again, when F (y j x) is continuous and strictly monotonic in y, then F (Q (x) j x) = : For …xed ; the quantile regression function q (x) describes how the ’th quantile of the conditional distribution varies with the regressors. As functions of x; the quantile regression functions can take any shape. However for computational convenience it is typical to assume that they are (approximately) linear in x (after suitable transformations). This linear speci…cation assumes that Q (x) = 0 x where the coe¢ cients vary across the quantiles : We then have the linear quantile regression model yi = x0i + ei where ei is the error de…ned to be the di¤erence between yi and its ’th conditional quantile x0i : By construction, the ’th conditional quantile of ei is zero, otherwise its properties are unspeci…ed without further restrictions. Given the representation (13.4), the quantile regression estimator b for solves the minimization problem b = argmin S ( ) n where n 1X Sn ( ) = n yi x0i i=1 and (q) is de…ned in (13.5). Since the quanitle regression criterion function Sn ( ) does not have an algebraic solution, numerical methods are necessary for its minimization. Furthermore, since it has discontinuous derivatives, conventional Newton-type optimization methods are inappropriate. Fortunately, fast linear programming methods have been developed for this problem, and are widely available. An asymptotic distribution theory for the quantile regression estimator can be derived using similar arguments as those for the LAD estimator in Theorem 13.1.1. Theorem 13.2.1 Asymptotic Distribution of the Quantile Regression Estimator When the ’th conditional quantile is linear in x p where V = (1 n b ) E xi x0i f (0 j xi ) d ! N (0; V ) ; 1 Exi x0i E xi x0i f (0 j xi ) and f (e j x) is the conditional density of ei given xi = x: 1 CHAPTER 13. QUANTILE REGRESSION 245 In general, the asymptotic variance depends on the conditional density of the quantile regression error. When the error ei is independent of xi ; then f (0 j xi ) = f (0) ; the unconditional density of ei at 0, and we have the simpli…cation V = (1 ) E xi x0i 2 f (0) 1 : A recent monograph on the details of quantile regression is Koenker (2005). CHAPTER 13. QUANTILE REGRESSION 246 Exercises Exercise 13.1 For any predictor g(xi ) for yi ; the mean absolute error (MAE) is E jyi g(xi )j : Show that the function g(x) which minimizes the MAE is the conditional median m (x) = med(yi j xi ): Exercise 13.2 De…ne g(u) = 1 (u < 0) where 1 ( ) is the indicator function (takes the value 1 if the argument is true, else equals zero). Let satisfy Eg(yi ) = 0: Is a quantile of the distribution of yi ? Exercise 13.3 Verify equation (13.4). Chapter 14 Generalized Method of Moments 14.1 Overidenti…ed Linear Model Consider the linear model yi = x0i + ei = x01i 1 + x02i 2 + ei E (xi ei ) = 0 where x1i is k 1 and x2i is r 1 with ` = k + r: We know that without further restrictions, an asymptotically e¢ cient estimator of is the OLS estimator. Now suppose that we are given the information that 2 = 0: Now we can write the model as yi = x01i 1 + ei E (xi ei ) = 0: In this case, how should 1 be estimated? One method is OLS regression of yi on x1i alone. This method, however, is not necessarily e¢ cient, as there are ` restrictions in E (xi ei ) = 0; while 1 is of dimension k < `. This situation is called overidenti…ed. There are ` k = r more moment restrictions than free parameters. We call r the number of overidentifying restrictions. This is a special case of a more general class of moment condition models. Let g(y; x; z; ) be an ` 1 function of a k 1 parameter with ` k such that Eg(yi ; xi ; z i ; 0) =0 (14.1) where 0 is the true value of : In our previous example, g(y; z; ) = z (y x01 1 ): In econometrics, this class of models are called moment condition models. In the statistics literature, these are known as estimating equations. As an important special case we will devote special attention to linear moment condition models, which can be written as yi = x0i + ei E (z i ei ) = 0: where the dimensions of xi and z i are k 1 and ` 1 , with ` k: If k = ` the model is just identi…ed, otherwise it is overidenti…ed. The variables xi may be components and functions of z i ; but this is not required. This model falls in the class (14.1) by setting g(y; x; z; 0) = z (y 247 x0 ) (14.2) CHAPTER 14. GENERALIZED METHOD OF MOMENTS 14.2 248 GMM Estimator De…ne the sample analog of (14.2) n gn( ) = n 1X 1X gi( ) = z i yi n n i=1 x0i = i=1 1 Z 0y n Z 0X : (14.3) The method of moments estimator for is de…ned as the parameter value which sets g n ( ) = 0. This is generally not possible when ` > k; as there are more equations than free parameters. The idea of the generalized method of moments (GMM) is to de…ne an estimator which sets g n ( ) “close” to zero. For some ` ` weight matrix W n > 0; let Jn ( ) = n g n ( )0 W n g n ( ): This is a non-negative measure of the “length”of the vector g n ( ): For example, if W n = I; then, Jn ( ) = n g n ( )0 g n ( ) = n kg n ( )k2 ; the square of the Euclidean length. The GMM estimator minimizes Jn ( ). De…nition 14.2.1 b GM M = argmin Jn ( ) : Note that if k = `; then g n ( b ) = 0; and the GMM estimator is the method of moments estimator: The …rst order conditions for the GMM estimator are @ Jn ( b ) @ @ = 2 g n ( b )0 W n g n ( b ) @ 1 0 1 0 = 2 X Z Wn Z y n n 0 = so Xb 2 X 0Z W n Z 0X b = 2 X 0Z W n Z 0y which establishes the following. Proposition 14.2.1 b GM M = X 0Z W n Z 0X 1 X 0Z W n Z 0y : While the estimator depends on W n ; the dependence is only up to scale, for if W n is replaced by cW n for some c > 0; b GM M does not change. CHAPTER 14. GENERALIZED METHOD OF MOMENTS 14.3 249 Distribution of GMM Estimator p Assume that W n ! W > 0: Let Q = E z i x0i and = E z i z 0i e2i = E g i g 0i ; where g i = z i ei : Then 1 0 X Z Wn n and 1 0 X Z Wn n 1 0 ZX n 1 p Z 0e n p ! Q0 W Q d ! Q0 W N (0; ) : We conclude: Theorem 14.3.1 Asymptotic Distribution of GMM Estimator p where V n b = Q0 W Q d ! N (0; V ) ; 1 Q0 W W Q Q0 W Q 1 : In general, GMM estimators are asymptotically normal with “sandwich form”asymptotic variances. 1 The optimal weight matrix W 0 is one which minimizes V : This turns out to be W 0 = : The proof is left as an exercise. This yields the e¢ cient GMM estimator: Thus we have b = X 0Z 1 Z 0X 1 X 0Z 1 Z 0 y: Theorem 14.3.2 Asymptotic Distribution of E¢ cient GMM Estimator p 1 d n b ! N 0; Q0 1 Q : p 1 W0 = is not known in practice, but it can be estimated consistently. For any W n ! W 0 ; b we still call the e¢ cient GMM estimator, as it has the same asymptotic distribution. By “e¢ cient”, we mean that this estimator has the smallest asymptotic variance in the class of GMM estimators with this set of moment conditions. This is a weak concept of optimality, as we are only considering alternative weight matrices W n : However, it turns out that the GMM estimator is semiparametrically e¢ cient, as shown by Gary Chamberlain (1987). If it is known that E (g i ( )) = 0; and this is all that is known, this is a semi-parametric problem, as the distribution of the data is unknown. Chamberlain showed that in this context, no semiparametric estimator (one which is consistent globally for the class of models considered) 1 can have a smaller asymptotic variance than G0 1 G where G = E @@ 0 g i ( ): Since the GMM estimator has this asymptotic variance, it is semiparametrically e¢ cient. This result shows that in the linear model, no estimator has greater asymptotic e¢ ciency than the e¢ cient linear GMM estimator. No estimator can do better (in this …rst-order asymptotic sense), without imposing additional assumptions. CHAPTER 14. GENERALIZED METHOD OF MOMENTS 14.4 250 Estimation of the E¢ cient Weight Matrix Given any weight matrix W n > 0; the GMM estimator b is consistent yet ine¢ cient. For 1 example, we can set W n = I ` : In the linear model, a better choice is W n = (Z 0 Z) : Given any such …rst-step estimator, we can de…ne the residuals e^i = yi x0i b and moment equations g ^i = z i e^i = g(yi ; xi ; z i ; b ): Construct n gn = gn( b ) = g ^i = g ^i and de…ne n Wn = 1X g ^i g ^i 0 n i=1 p ! 1 1X g ^i ; n i=1 gn; n = 1X g ^i g ^0i n g n g 0n i=1 ! 1 : (14.4) 1 Then W n ! = W 0 ; and GMM using W n as the weight matrix is asymptotically e¢ cient. A common alternative choice is to set ! 1 n 1X 0 Wn = g ^i g ^i n i=1 which uses the uncentered moment conditions. Since Eg i = 0; these two estimators are asymptotically equivalent under the hypothesis of correct speci…cation. However, Alastair Hall (2000) has shown that the uncentered estimator is a poor choice. When constructing hypothesis tests, under the alternative hypothesis the moment conditions are violated, i.e. Eg i 6= 0; so the uncentered estimator will contain an undesirable bias term and the power of the test will be adversely a¤ected. A simple solution is to use the centered moment conditions to construct the weight matrix, as in (14.4) above. Here is a simple way to compute the e¢ cient GMM estimator for the linear model. First, set W n = (Z 0 Z) 1 , estimate b using this weight matrix, and construct the residual e^i = yi x0i b : ^ be the associated n ` matrix: Then the e¢ cient GMM estimator is Then set g ^i = z i e^i ; and let g b = X 0Z g ^0 g ^ ng n g 0n 1 Z 0X 1 X 0Z g ^0 g ^ ng n g 0n 1 Z 0 y: In most cases, when we say “GMM”, we actually mean “e¢ cient GMM”. There is little point in using an ine¢ cient GMM estimator when the e¢ cient estimator is easy to compute. An estimator of the asymptotic variance of ^ can be seen from the above formula. Set Vb = n X 0 Z g ^0 g ^ ng n g 0n 1 Z 0X 1 : 1 Asymptotic standard errors are given by the square roots of the diagonal elements of Vb : n There is an important alternative to the two-step GMM estimator just described. Instead, we can let the weight matrix be considered as a function of : The criterion function is then ! 1 n 1X 0 0 g i ( )g i ( ) g n ( ): J( ) = n g n ( ) n i=1 where gi ( ) = gi( ) gn( ) The b which minimizes this function is called the continuously-updated GMM estimator, and was introduced by L. Hansen, Heaton and Yaron (1996). The estimator appears to have some better properties than traditional GMM, but can be numerically tricky to obtain in some cases. This is a current area of research in econometrics. CHAPTER 14. GENERALIZED METHOD OF MOMENTS 14.5 ` 251 GMM: The General Case In its most general form, GMM applies whenever an economic or statistical model implies the 1 moment condition E (g i ( )) = 0: Often, this is all that is known. Identi…cation requires l k = dim( ): The GMM estimator minimizes J( ) = n g n ( )0 W n g n ( ) where n gn( ) = 1X gi( ) n i=1 and n 1X g ^i g ^0i n Wn = g n g 0n i=1 ! 1 ; with g ^i = g i ( e ) constructed using a preliminary consistent estimator e , perhaps obtained by …rst setting W n = I: Since the GMM estimator depends upon the …rst-stage estimator, often the weight matrix W n is updated, and then b recomputed. This estimator can be iterated if needed. Theorem 14.5.1 Distribution of Nonlinear GMM Estimator Under general regularity conditions, p where d n b ! N 0; G0 1 G 1 ; = E g i g 0i and G=E @ g ( ): @ 0 i The variance of b may be estimated by where Vb ^0 ^ = G ^ =n 1 X 1 ^ G 1 g ^i g ^i 0 i and ^=n G 1 X @ ^ 0 g i ( ): @ i The general theory of GMM estimation and testing was exposited by L. Hansen (1982). 14.6 Over-Identi…cation Test Overidenti…ed models (` > k) are special in the sense that there may not be a parameter value such that the moment condition CHAPTER 14. GENERALIZED METHOD OF MOMENTS 252 Eg(yi ; xi ; z i ; ) = 0 holds. Thus the model –the overidentifying restrictions –are testable. For example, take the linear model yi = 01 x1i + 02 x2i +ei with E (x1i ei ) = 0 and E (x2i ei ) = 0: It is possible that 2 = 0; so that the linear equation may be written as yi = 01 x1i + ei : However, it is possible that 2 6= 0; and in this case it would be impossible to …nd a value of 1 so that both E (x1i (yi x01i 1 )) = 0 and E (x2i (yi x01i 1 )) = 0 hold simultaneously. In this sense an exclusion restriction can be seen as an overidentifying restriction. p Note that g n ! Eg i ; and thus g n can be used to assess whether or not the hypothesis that Eg i = 0 is true or not. The criterion function at the parameter estimates is Jn = n g 0n W n g n ^0 g ^ = n2 g 0n g ng n g 0n 1 gn: is a quadratic form in g n ; and is thus a natural test statistic for H0 : Eg i = 0: Theorem 14.6.1 (Sargan-Hansen). Under the hypothesis of correct speci…cation, and if the weight matrix is asymptotically e¢ cient, d Jn = Jn ( b ) ! 2 ` k: The proof of the theorem is left as an exercise. This result was established by Sargan (1958) for a specialized case, and by L. Hansen (1982) for the general case. The degrees of freedom of the asymptotic distribution are the number of overidentifying restrictions. If the statistic J exceeds the chi-square critical value, we can reject the model. Based on this information alone, it is unclear what is wrong, but it is typically cause for concern. The GMM overidenti…cation test is a very useful by-product of the GMM methodology, and it is advisable to report the statistic J whenever GMM is the estimation method. When over-identi…ed models are estimated by GMM, it is customary to report the J statistic as a general test of model adequacy. 14.7 Hypothesis Testing: The Distance Statistic We described before how to construct estimates of the asymptotic covariance matrix of the GMM estimates. These may be used to construct Wald tests of statistical hypotheses. If the hypothesis is non-linear, a better approach is to directly use the GMM criterion function. This is sometimes called the GMM Distance statistic, and sometimes called a LR-like statistic (the LR is for likelihood-ratio). The idea was …rst put forward by Newey and West (1987). For a given weight matrix W n ; the GMM criterion function is Jn ( ) = n g n ( )0 W n g n ( ) For h : Rk ! Rr ; the hypothesis is H0 : h( ) = 0: The estimates under H1 are b = argmin Jn ( ) CHAPTER 14. GENERALIZED METHOD OF MOMENTS and those under H0 are 253 e = argmin J( ): h( )=0 The two minimizing criterion functions are Jn ( b ) and Jn ( e ): The GMM distance statistic is the di¤erence Dn = Jn ( e ) Jn ( b ): Proposition 14.7.1 If the same weight matrix W n is used for both null and alternative, 1. D 0 d 2. D ! 2 r 3. If h is linear in ; then D equals the Wald statistic. If h is non-linear, the Wald statistic can work quite poorly. In contrast, current evidence suggests that the Dn statistic appears to have quite good sampling properties, and is the preferred test statistic. Newey and West (1987) suggested to use the same weight matrix W n for both null and alternative, as this ensures that Dn 0. This reasoning is not compelling, however, and some current research suggests that this restriction is not necessary for good performance of the test. This test shares the useful feature of LR tests in that it is a natural by-product of the computation of alternative models. 14.8 Conditional Moment Restrictions In many contexts, the model implies more than an unconditional moment restriction of the form Eg i ( ) = 0: It implies a conditional moment restriction of the form E (ei ( ) j z i ) = 0 where ei ( ) is some s 1 function of the observation and the parameters. In many cases, s = 1. It turns out that this conditional moment restriction is much more powerful, and restrictive, than the unconditional moment restriction discussed above. Our linear model yi = x0i + ei with instruments z i falls into this class under the stronger assumption E (ei j z i ) = 0: Then ei ( ) = yi x0i : It is also helpful to realize that conventional regression models also fall into this class, except that in this case xi = z i : For example, in linear regression, ei ( ) = yi x0i , while in a nonlinear regression model ei ( ) = yi g(xi ; ): In a joint model of the conditional mean and variance 8 yi x0i < ei ( ; ) = : : 2 0 0 (yi xi ) f (xi ) Here s = 2: Given a conditional moment restriction, an unconditional moment restriction can always be constructed. That is for any ` 1 function (xi ; ) ; we can set g i ( ) = (xi ; ) ei ( ) which satis…es Eg i ( ) = 0 and hence de…nes a GMM estimator. The obvious problem is that the class of functions is in…nite. Which should be selected? CHAPTER 14. GENERALIZED METHOD OF MOMENTS 254 This is equivalent to the problem of selection of the best instruments. If xi 2 R is a valid instrument satisfying E (ei j xi ) = 0; then xi ; x2i ; x3i ; :::; etc., are all valid instruments. Which should be used? One solution is to construct an in…nite list of potent instruments, and then use the …rst k instruments. How is k to be determined? This is an area of theory still under development. A recent study of this problem is Donald and Newey (2001). Another approach is to construct the optimal instrument. The form was uncovered by Chamberlain (1987). Take the case s = 1: Let @ ei ( ) j z i @ Ri = E and 2 i = E ei ( )2 j z i : Then the “optimal instrument” is 2 Ai = i Ri so the optimal moment is g i ( ) = Ai ei ( ): Setting g i ( ) to be this choice (which is k 1; so is just-identi…ed) yields the best GMM estimator possible. In practice, Ai is unknown, but its form does help us think about construction of optimal instruments. In the linear model ei ( ) = yi x0i ; note that Ri = E (xi j z i ) and 2 i = E e2i j z i ; so Ai = 2 i E (xi j z i ) : In the case of linear regression, xi = z i ; so Ai = i 2 z i : Hence e¢ cient GMM is GLS, as we discussed earlier in the course. In the case of endogenous variables, note that the e¢ cient instrument Ai involves the estimation of the conditional mean of xi given z i : In other words, to get the best instrument for xi ; we need the best conditional mean model for xi given z i , not just an arbitrary linear projection. The e¢ cient instrument is also inversely proportional to the conditional variance of ei : This is the same as the GLS estimator; namely that improved e¢ ciency can be obtained if the observations are weighted inversely to the conditional variance of the errors. 14.9 Bootstrap GMM Inference Let b be the 2SLS or GMM estimator of . Using the EDF of (yi ; z i ; xi ), we can apply the bootstrap methods discussed in Chapter 10 to compute estimates of the bias and variance of b ; and construct con…dence intervals for ; identically as in the regression model. However, caution should be applied when interpreting such results. A straightforward application of the nonparametric bootstrap works in the sense of consistently achieving the …rst-order asymptotic distribution. This has been shown by Hahn (1996). However, it fails to achieve an asymptotic re…nement when the model is over-identi…ed, jeopardizing the theoretical justi…cation for percentile-t methods. Furthermore, the bootstrap applied J test will yield the wrong answer. CHAPTER 14. GENERALIZED METHOD OF MOMENTS 255 The problem is that in the sample, b is the “true”value and yet g n ( ^ ) 6= 0: Thus according to random variables (yi ; z i ; xi ) drawn from the EDF Fn ; E gi b = g n ( ^ ) 6= 0: This means that (yi ; z i ; xi ) do not satisfy the same moment conditions as the population distribution. A correction suggested by Hall and Horowitz (1996) can solve the problem. Given the bootstrap sample (y ; Z ; X ); de…ne the bootstrap GMM criterion Jn ( ) = n gn( ) gn( ^ ) 0 W n gn( ) gn( ^ ) where g n ( b ) is from the in-sample data, not from the bootstrap data. Let b minimize Jn ( ); and de…ne all statistics and tests accordingly. In the linear model, this implies that the bootstrap estimator is b = X 0Z W Z 0X n n 1 X 0Z W n Z 0y Z 0e ^ : where e ^ = y X b are the in-sample residuals. The bootstrap J statistic is Jn ( b ): Brown and Newey (2002) have an alternative solution. They note that we can sample from the observations with the empirical likelihood probabilities p^i described in Chapter 15. Since Pn ^i g i b = 0; this sampling scheme preserves the moment conditions of the model, so no i=1 p recentering or adjustments is needed. Brown and Newey argue that this bootstrap procedure will be more e¢ cient than the Hall-Horowitz GMM bootstrap. CHAPTER 14. GENERALIZED METHOD OF MOMENTS 256 Exercises Exercise 14.1 Take the model yi = x0i + ei E (xi ei ) = 0 e2i = z 0i + i E (z i i ) = 0: Find the method of moments estimators ^ ; ^ for ( ; ) : Exercise 14.2 Take the single equation y = X +e E (e j Z) = 0 Assume E e2i j z i = then 2: Show that if ^ is estimated by GMM with weight matrix W n = (Z 0 Z) p n b d ! N 0; 2 Q0 M 1 Q 1 ; 1 where Q = E (z i x0i ) and M = E (z i z 0i ) : Exercise 14.3 Take the model yi = x0i + ei with E (z i ei ) = 0: Let e^i = yi x0i b where b is consistent for (e.g. a GMM estimator with arbitrary weight matrix). De…ne the estimate of the optimal GMM weight matrix ! 1 n 1X z i z 0i e^2i Wn = : n i=1 p Show that W n ! 1 where = E z i z 0i e2i : Exercise 14.4 In the linear model estimated by GMM with general weight matrix W ; the asymptotic variance of ^ GM M is V = Q0 W Q 1 1 (a) Let V 0 be this matrix when W = Q0 W W Q Q0 W Q : Show that V 0 = Q0 1 1 Q 1 : (b) We want to show that for any W ; V V 0 is positive semi-de…nite (for then V 0 is the smaller 1 possible covariance matrix and W = is the e¢ cient weight matrix). To do this, start by …nding matrices A and B such that V = A0 A and V 0 = B 0 B: (c) Show that B 0 A = B 0 B and therefore that B 0 (d) Use the expressions V = A0 A; A = B + (A V V 0: (A B) = 0: B) ; and B 0 (A B) = 0 to show that Exercise 14.5 The equation of interest is yi = m(xi ; ) + ei E (z i ei ) = 0: The observed data is (yi ; z i ; xi ). z i is ` GMM estimator for . 1 and is k 1; ` k: Show how to construct an e¢ cient CHAPTER 14. GENERALIZED METHOD OF MOMENTS 257 Exercise 14.6 In the linear model y = X + e with E(xi ei ) = 0; a Generalized Method of Moments (GMM) criterion function for is de…ned as 1 (y n Jn ( ) = 1 X )0 X b X 0 (y X ) (14.5) P where b = n1 ni=1 xi x0i e^2i ; e^i = yi x0i ^ are the OLS residuals, and b = (X 0 X) GMM estimator of ; subject to the restriction h( ) = 0; is de…ned as 1 X 0 y is LS: The e = argmin Jn ( ): h( )=0 The GMM test statistic (the distance statistic) of the hypothesis h( ) = 0 is D = Jn ( e ) = min Jn ( ): (14.6) h( )=0 (a) Show that you can rewrite Jn ( ) in (14.5) as 0 b Jn ( ) = n V^ 1 b thus e is the same as the minimum distance estimator. (b) Show that in this setting, the distance statistic D in (14.6) equals the Wald statistic. Exercise 14.7 Take the linear model yi = x0i + ei E (z i ei ) = 0: and consider the GMM estimator ^ of : Let 1 Jn = ng n ( b )0 b gn( b ) d denote the test of overidentifying restrictions. Show that Jn ! each of the following: (a) Since > 0; we can write (b) Jn = n C 0 g n ( b ) 0 C0 ^ C (c) C 0 g n ( b ) = D n C 0 g n ( Dn = I ` 0) 1 = CC 0 and 1 C 0gn( b ) where 1 0 ZX n C0 1 0 XZ n gn( p (d) D n ! I ` (e) n1=2 C 0 g n ( d R (R0 R) 0) (f) Jn ! u0 I ` (g) u0 I ` Hint: I ` d !u 1 1 R (R0 R) b = 1 N (0; I ` ) R0 u 1 1 1 C R0 u 2 : ` k R0 is a projection matrix. as n ! 1 by demonstrating 1 1 0 ZX n 1 0 Z e: n R0 where R = C 0 E (z i x0i ) R (R0 R) R (R0 R) 0) = C0 2 ` k 1 1 0 XZ n b 1 C0 1 Chapter 15 Empirical Likelihood 15.1 Non-Parametric Likelihood An alternative to GMM is empirical likelihood. The idea is due to Art Owen (1988, 2001) and has been extended to moment condition models by Qin and Lawless (1994). It is a non-parametric analog of likelihood estimation. The idea is to construct a multinomial distribution F (p1 ; :::; pn ) which places probability pi at each observation. To be a valid multinomial distribution, these probabilities must satisfy the requirements that pi 0 and n X pi = 1: (15.1) i=1 Since each observation is observed once in the sample, the log-likelihood function for this multinomial distribution is n X log L (p1 ; :::; pn ) = log(pi ): (15.2) i=1 First let us consider a just-identi…ed model. In this case the moment condition places no additional restrictions on the multinomial distribution. The maximum likelihood estimators of the probabilities (p1 ; :::; pn ) are those which maximize the log-likelihood subject to the constraint (15.1). This is equivalent to maximizing ! n n X X log(pi ) pi 1 i=1 i=1 where is a Lagrange multiplier. The n …rst order conditions are 0 = pi 1 : Combined with the 1 constraint (15.1) we …nd that the MLE is pi = n yielding the log-likelihood n log(n): Now consider the case of an overidenti…ed model with moment condition Eg i ( 0) =0 where g is ` 1 and is k 1 and for simplicity we write g i ( ) = g(yi ; z i ; xi ; ): The multinomial distribution which places probability pi at each observation (yi ; xi ; z i ) will satisfy this condition if and only if n X pi g i ( ) = 0 (15.3) i=1 The empirical likelihood estimator is the value of which maximizes the multinomial loglikelihood (15.2) subject to the restrictions (15.1) and (15.3). 258 CHAPTER 15. EMPIRICAL LIKELIHOOD 259 The Lagrangian for this maximization problem is L ( ; p1 ; :::; pn ; ; ) = where are and n X n X log(pi ) i=1 pi 1 i=1 ! n 0 n X pi g i ( ) i=1 are Lagrange multipliers. The …rst-order-conditions of L with respect to pi , ; and 1 pi n X + n 0gi ( ) = pi = 1 i=1 n X pi g i ( ) = 0: i=1 Multiplying the …rst equation by pi , summing over i; and using the second and third equations, we …nd = n and 1 : pi = n 1 + 0gi ( ) Substituting into L we …nd R( ; ) = n log (n) n X log 1 + 0 gi ( ) : (15.4) i=1 For given ; the Lagrange multiplier ( ) minimizes R ( ; ) : ( ) = argmin R( ; ): (15.5) This minimization problem is the dual of the constrained maximization problem. The solution (when it exists) is well de…ned since R( ; ) is a convex function of : The solution cannot be obtained explicitly, but must be obtained numerically (see section 6.5). This yields the (pro…le) empirical log-likelihood function for . R( ) = R( ; ( )) = n log (n) n X log 1 + ( )0 g i ( ) i=1 The EL estimate ^ is the value which maximizes R( ); or equivalently minimizes its negative ^ = argmin [ R( )] (15.6) Numerical methods are required for calculation of ^ (see Section 15.5). As a by-product of estimation, we also obtain the Lagrange multiplier ^ = ( ^ ); probabilities p^i = 1 0 n 1 + ^ gi ^ : and maximized empirical likelihood R( ^ ) = n X i=1 log (^ pi ) : (15.7) CHAPTER 15. EMPIRICAL LIKELIHOOD 15.2 Let 260 Asymptotic Distribution of EL Estimator 0 denote the true value of and de…ne Gi ( ) = @ g ( ) @ 0 i G = EGi ( = E gi ( 0) 0 0) gi ( 0) and V = G0 V 1 For example, in the linear model, Gi ( ) = 1 G G G0 = (15.8) 1 (15.9) 1 G z i x0i ; G = G0 (15.10) E (z i x0i ), and = E z i z 0i e2i : Theorem 15.2.1 Under regularity conditions, p d n ^ 0 p d n^ ! ! N (0; V ) 1 N (0; V ) where V and V are de…ned in (15.9) and (15.10), and p ^ n are asymptotically independent. p n ^ 0 and The theorem shows that the asymptotic variance V for ^ is the same as for e¢ cient GMM. Thus the EL estimator is asymptotically e¢ cient. Chamberlain (1987) showed that V is the semiparametric e¢ ciency bound for in the overidenti…ed moment condition model. This means that no consistent estimator for this class of models can have a lower asymptotic variance than V . Since the EL estimator achieves this bound, it is an asymptotically e¢ cient estimator for . Proof of Theorem 15.2.1. ( ^ ; ^ ) jointly solve @ R( ; ) = @ 0 = @ R( ; ) = @ 0 = gi ^ n X 0 n Gi ^ X i=1 (15.11) 0 1 + ^ gi ^ i=1 0 1 + ^ gi ^ : P P P Let Gn = n1 ni=1 Gi ( 0 ) ; g n = n1 ni=1 g i ( 0 ) and n = n1 ni=1 g i ( Expanding (15.12) around = 0 and = 0 = 0 yields 0 ' G0n ^ Expanding (15.11) around = 0 and 0' = gn 0 0 : (15.12) 0 0) gi ( 0) : (15.13) = 0 yields Gn ^ 0 + n ^ (15.14) CHAPTER 15. EMPIRICAL LIKELIHOOD Premultiplying by G0n n 1 261 and using (15.13) yields 0 ' G0n n = G0n n 1 gn G0n n 1 gn G0n n 1 Gn ^ 0 1 Gn ^ 0 Solving for ^ and using the WLLN and CLT yields p G0n n 1 Gn n ^ 0 ' d ! G0 1 1 1 G + G0n G0n n G0 1 n 1 n ^ 1p ng n (15.15) N (0; ) = N (0; V ) Solving (15.14) for ^ and using (15.15) yields p n ^ ' n 1 I Gn G0n d 1 ! 1 = n G G0 I 1 1 Gn 1 1 G G0n G0 n p 1 1 ng n (15.16) N (0; ) N (0; V ) Furthermore, since 1 G0 I p n ^ 15.3 0 and G G0 1 1 G G0 = 0 p ^ n are asymptotically uncorrelated and hence independent. Overidentifying Restrictions In a parametric likelihood context, tests are based on the di¤erence in the log likelihood functions. The same statistic can be constructed for empirical likelihood. Twice the di¤erence between the unrestricted empirical log-likelihood n log (n) and the maximized empirical log-likelihood for the model (15.7) is n X 0 LRn = 2 log 1 + ^ g i ^ : (15.17) i=1 Theorem 15.3.1 If Eg i ( d 0) 2 : ` k = 0 then LRn ! The EL overidenti…cation test is similar to the GMM overidenti…cation test. They are asymptotically …rst-order equivalent, and have the same interpretation. The overidenti…cation test is a very useful by-product of EL estimation, and it is advisable to report the statistic LRn whenever EL is the estimation method. Proof of Theorem 15.3.1. First, by a Taylor expansion, (15.15), and (15.16), n 1 X p gi ^ n i=1 ' p n g n + Gn ^ ' I ' n Gn G0n p n^: n 1 0 Gn 1 G0n n 1 p ng n CHAPTER 15. EMPIRICAL LIKELIHOOD Second, since log(1 + u) ' u 262 u2 =2 for u small, LRn = n X 0 2 log 1 + ^ g i ^ i=1 ' 2^ n 0X i=1 0 ' n^ n X gi ^ gi ^ 0 ^ i=1 ^ n d ! N (0; V )0 = ^0 gi ^ 1 N (0; V ) 2 ` k where the proof of the …nal equality is left as an exercise. 15.4 Testing Let the maintained model be Eg i ( ) = 0 (15.18) where g is ` 1 and is k 1: By “maintained” we mean that the overidentfying restrictions contained in (15.18) are assumed to hold and are not being challenged (at least for the test discussed in this section). The hypothesis of interest is h( ) = 0: where h : Rk ! Ra : The restricted EL estimator and likelihood are the values which solve ~ = argmax R( ) h( )=0 R( ~ ) = max R( ): h( )=0 Fundamentally, the restricted EL estimator ~ is simply an EL estimator with ` k+a overidentifying restrictions, so there is no fundamental change in the distribution theory for ~ relative to ^ : To test the hypothesis h( ) while maintaining (15.18), the simple overidentifying restrictions test (15.17) is not appropriate. Instead we use the di¤erence in log-likelihoods: LRn = 2 R( ^ ) R( ~ ) : This test statistic is a natural analog of the GMM distance statistic. d Theorem 15.4.1 Under (15.18) and H0 : h( ) = 0; LRn ! The proof of this result is more challenging and is omitted. 2: a CHAPTER 15. EMPIRICAL LIKELIHOOD 15.5 263 Numerical Computation Gauss code which implements the methods discussed below can be found at http://www.ssc.wisc.edu/~bhansen/progs/elike.prc Derivatives The numerical calculations depend on derivatives of the dual likelihood function (15.4). De…ne gi ( ) 1 + 0gi ( ) gi ( ; ) = Gi ( ; ) = Gi ( )0 1 + 0gi ( ) The …rst derivatives of (15.4) are @ R( ; ) = @ = R @ R( ; ) = @ = R n X gi ( ; ) i=1 n X Gi ( ; ) : i=1 The second derivatives are R R R = = = @2 @ @ @2 @ @ @2 @ @ 0R( ; )= n X g i ( ; ) g i ( ; )0 i=1 0R( ; )= n X i=1 0R( ; )= n X i=1 g i ( ; ) G i ( ; )0 0 @Gi ( ; ) Gi ( ; )0 Gi ( ) 1 + 0gi ( ) @2 @ @ 0 1+ g i ( )0 0 gi ( ) 1 A Inner Loop The so-called “inner loop” solves (15.5) for given : The modi…ed Newton method takes a quadratic approximation to Rn ( ; ) yielding the iteration rule j+1 = j (R ( ; j )) 1 R ( ; j) : (15.19) where > 0 is a scalar steplength (to be discussed next). The starting value 1 can be set to the zero vector. The iteration (15.19) is continued until the gradient R ( ; j ) is smaller than some prespeci…ed tolerance. E¢ cient convergence requires a good choice of steplength : One method uses the following quadratic approximation. Set 0 = 0; 1 = 12 and 2 = 1: For p = 0; 1; 2; set p = j Rp = R ( ; p (R ( ; j )) 1 R ( ; j )) p) A quadratic function can be …t exactly through these three points. The value of this quadratic is ^ = R2 + 3R0 4R1 : 4R2 + 4R0 8R1 yielding the steplength to be plugged into (15.19). which minimizes CHAPTER 15. EMPIRICAL LIKELIHOOD A complication is that 264 must be constrained so that 0 n 1+ 0 gi ( ) pi 1 which holds if 1 (15.20) for all i: If (15.20) fails, the stepsize needs to be decreased. Outer Loop The outer loop is the minimization (15.6). This can be done by the modi…ed Newton method described in the previous section. The gradient for (15.6) is R = since R ( ; ) = 0 at @ @ R( ) = R( ; ) = R + @ @ 0 R =R = ( ); where = @ ( )= @ 0 1 R R ; the second equality following from the implicit function theorem applied to R ( ; ( )) = 0: The Hessian for (15.6) is R = @ R( ) @ @ 0 @ R ( ; ( )) + @ 0 R ( ; ( )) + R0 = R 0 = = R 1 R 0 R ( ; ( )) + 0 R It is not guaranteed that R > 0: If not, the eigenvalues of R are positive. The Newton iteration rule is where = 0 R : R j+1 + j R 1 should be adjusted so that all R is a scalar stepsize, and the rule is iterated until convergence. Chapter 16 Endogeneity We say that there is endogeneity in the linear model y = x0i + ei if is the parameter of interest and E(xi ei ) 6= 0: This cannot happen if is de…ned by linear projection, so requires a structural interpretation. The coe¢ cient must have meaning separately from the de…nition of a conditional mean or linear projection. Example: Measurement error in the regressor. Suppose that (yi ; xi ) are joint random variables, E(yi j xi ) = xi 0 is linear, is the parameter of interest, and xi is not observed. Instead we observe xi = xi + ui where ui is an k 1 measurement error, independent of yi and xi : Then yi = xi 0 + ei ui )0 = (xi = x0i + ei + vi where u0i : vi = ei The problem is that u0i E (xi vi ) = E (xi + ui ) ei if = E ui u0i 6= 0 6= 0 and E (ui u0i ) 6= 0: It follows that if ^ is the OLS estimator, then ^ p! E xi x0i = 1 E ui u0i 6= : This is called measurement error bias. Example: Supply and Demand. The variables qi and pi (quantity and price) are determined jointly by the demand equation qi = 1 pi + e1i and the supply equation qi = 2 pi + e2i : e1i is iid, Eei = 0; 1 + 2 = 1 and Eei e0i = I 2 (the latter for simplicity). e2i The question is, if we regress qi on pi ; what happens? It is helpful to solve for qi and pi in terms of the errors. In matrix notation, Assume that ei = 1 1 1 2 qi pi 265 = e1i e2i CHAPTER 16. ENDOGENEITY 266 so qi pi 1 1 1 = 2 2 = e1i e2i 1 1 1 2 e1i = e1i e2i 1 + (e1i 1 e2i e2i ) : The projection of qi on pi yields qi = pi + "i E (pi "i ) = 0 where = Hence if it is estimated by OLS, ^ simultaneous equations bias. 16.1 E (pi qi ) = E p2i p ! 2 1 2 ; which does not equal either 1 or 2: This is called Instrumental Variables Let the equation of interest be yi = x0i + ei (16.1) where xi is k 1; and assume that E(xi ei ) 6= 0 so there is endogeneity. We call (16.1) the structural equation. In matrix notation, this can be written as y = X + e: (16.2) Any solution to the problem of endogeneity requires additional information which we call instruments. De…nition 16.1.1 The ` 1 random vector z i is an instrumental variable for (16.1) if E (z i ei ) = 0: In a typical set-up, some regressors in xi will be uncorrelated with ei (for example, at least the intercept). Thus we make the partition xi = x1i x2i k1 k2 (16.3) where E(x1i ei ) = 0 yet E(x2i ei ) 6= 0: We call x1i exogenous and x2i endogenous. By the above de…nition, x1i is an instrumental variable for (16.1); so should be included in z i : So we have the partition x1i k1 zi = (16.4) z 2i `2 where x1i = z 1i are the included exogenous variables, and z 2i are the excluded exogenous variables. That is z 2i are variables which could be included in the equation for yi (in the sense that they are uncorrelated with ei ) yet can be excluded, as they would have true zero coe¢ cients in the equation. The model is just-identi…ed if ` = k (i.e., if `2 = k2 ) and over-identi…ed if ` > k (i.e., if `2 > k2 ): We have noted that any solution to the problem of endogeneity requires instruments. This does not mean that valid instruments actually exist. CHAPTER 16. ENDOGENEITY 16.2 267 Reduced Form The reduced form relationship between the variables or “regressors” xi and the instruments z i is found by linear projection. Let 1 = E z i z 0i be the ` E z i x0i k matrix of coe¢ cients from a projection of xi on z i ; and de…ne 0 ui = xi zi as the projection error. Then the reduced form linear relationship between xi and z i is 0 xi = z i + ui : (16.5) In matrix notation, we can write (16.5) as X =Z +U (16.6) where U is n k: By construction, E(z i u0i ) = 0; so (16.5) is a projection and can be estimated by OLS: 0 xi = b z i + u ^i : or where b X = Zb + U b = Z 0Z 1 Z 0X : Substituting (16:6) into (16.2), we …nd y = (Z + U ) +e = Z + v; (16.7) where = (16.8) and v=U + e: Observe that E (z i vi ) = E z i u0i + E (z i ei ) = 0: Thus (16.7) is a projection equation and may be estimated by OLS. This is y = Z^ + v ^; 0 ^ = Z Z 1 Z 0y The equation (16.7) is the reduced form for y: (16.6) and (16.7) together are the reduced form equations for the system y = Z +v X = Z + U: As we showed above, OLS yields the reduced-form estimates ^ ; ^ CHAPTER 16. ENDOGENEITY 16.3 268 Identi…cation The structural parameter relates to ( ; ) through (16.8). The parameter meaning that it can be recovered from the reduced form, if rank ( ) = k: is identi…ed, (16.9) 1 Assume that (16.9) holds. If ` = k; then = : If ` > k; then for any W > 0; = 1 0 0 ( W ) W : If (16.9) is not satis…ed, then cannot be recovered from ( ; ) : Note that a necessary (although not su¢ cient) condition for (16.9) is ` k: Since Z and X have the common variables X 1 ; we can rewrite some of the expressions. Using (16.3) and (16.4) to make the matrix partitions Z = [Z 1 ; Z 2 ] and X = [Z 1 ; X 2 ] ; we can partition as = 11 12 21 22 I 0 = 12 22 (16.6) can be rewritten as X1 = Z1 X2 = Z1 12 + Z2 22 + U 2: (16.10) is identi…ed if rank( ) = k; which is true if and only if rank( 22 ) = k2 (by the upper-diagonal structure of ): Thus the key to identi…cation of the model rests on the `2 k2 matrix 22 in (16.10). 16.4 Estimation The model can be written as yi = x0i + ei E (z i ei ) = 0 or Eg i ( ) = 0 x0i g i ( ) = z i yi : This is a moment condition model. Appropriate estimators include GMM and EL. The estimators and distribution theory developed in those Chapter 8 and 9 directly apply. Recall that the GMM estimator, for given weight matrix W n ; is 1 ^ = X 0 ZW n Z 0 X 16.5 X 0 ZW n Z 0 y: Special Cases: IV and 2SLS If the model is just-identi…ed, so that k = `; then the formula for GMM simpli…es. We …nd that b = X 0 ZW n Z 0 X 1 X 0 ZW n Z 0 y = Z 0X 1 W n 1 X 0Z = Z 0X 1 Z 0y 1 X 0 ZW n Z 0 y CHAPTER 16. ENDOGENEITY 269 This estimator is often called the instrumental variables estimator (IV) of ; where Z is used as an instrument for X: Observe that the weight matrix W n has disappeared. In the just-identi…ed case, the weight matrix places no role. This is also the method of moments estimator of ; and the 1 EL estimator. Another interpretation stems from the fact that since = ; we can construct the Indirect Least Squares (ILS) estimator: b = b = 1 b 1 Z 0Z 1 Z 0X = Z 0X 1 Z 0Z = Z 0X 1 Z 0y : Z 0Z 1 Z 0Z 1 Z 0y Z 0y which again is the IV estimator. Recall that the optimal weight matrix is an estimate of the inverse of = E z i z 0i e2i : In the 2 2 0 2 special case that E ei j z i = (homoskedasticity), then = E (z i z i ) / E (z i z 0i ) suggesting 1 the weight matrix W n = (Z 0 Z) : Using this choice, the GMM estimator equals b 2SLS 1 = X 0Z Z 0Z 1 Z 0X X 0Z Z 0Z 1 Z 0y This is called the two-stage-least squares (2SLS) estimator. It was originally proposed by Theil (1953) and Basmann (1957), and is the classic estimator for linear equations with instruments. Under the homoskedasticity assumption, the 2SLS estimator is e¢ cient GMM, but otherwise it is ine¢ cient. It is useful to observe that writing 1 then the 2SLS estimator is P = Z Z 0Z Z0 c = PX = Zb X b = 1 X 0P X c0 X c X = 1 X 0P y c0 y: X The source of the “two-stage” name is since it can be computed as follows First regress X on Z; vis., b = (Z 0 Z) 1 c vis., b = X c0 X c Second, regress y on X; c = Z b = P X: (Z 0 X) and X 1 c0 y: X c Recall, X = [X 1 ; X 2 ] and Z = [X 1 ; Z 2 ]: Then It is useful to scrutinize the projection X: h i c = X c1 ; X c2 X = [P X 1 ; P X 2 ] = [X 1 ; P X 2 ] h i c2 ; = X 1; X c2 : So only the since X 1 lies in the span of Z: Thus in the second stage, we regress y on X 1 and X endogenous variables X 2 are replaced by their …tted values: c2 = Z 1 b 12 + Z 2 b 22 : X CHAPTER 16. ENDOGENEITY 16.6 270 Bekker Asymptotics Bekker (1994) used an alternative asymptotic framework to analyze the …nite-sample bias in the 2SLS estimator. Here we present a simpli…ed version of one of his results. In our notation, the model is y = X +e (16.11) X = Z +U (16.12) = (e; U ) E E ( j Z) = 0 0 jZ = S As before, Z is n l so there are l instruments. First, let’s analyze the approximate bias of OLS applied to (16.11). Using (16.12), 1 0 Xe n E = E (xi ei ) = 0 E (z i ei ) + E (ui ei ) = s21 and E 1 0 XX n = E xi x0i = 0 E z i z 0i + E ui z 0i = 0 Q + S 22 0 + E z i u0i + E ui u0i where Q = E (z i z 0i ) : Hence by a …rst-order approximation E ^ OLS 0 = 1 1 0 XX n E Q + S 22 E 1 1 0 Xe n s21 (16.13) which is zero only when s21 = 0 (when X is exogenous). We now derive a similar result for the 2SLS estimator. ^ 2SLS = X 0 P X 1 X 0P y : 1 Let P = Z (Z 0 Z) Z 0 . By the spectral decomposition of an idempotent matrix, P = H H 0 where = diag (I l ; 0) : Let Q = H 0 S 1=2 which satis…es EQ0 Q = I n and partition Q = (q 01 Q02 ) where q 1 is l 1: Hence E 1 0 P jZ n 1 1=20 S E Q0 Q j Z S 1=2 n 1 1=20 1 0 S E q q S 1=2 n n 1 1 l 1=20 1=2 S S n S = = = = where = l : n Using (16.12) and this result, 1 1 E X 0P e = E n n 0 Z 0e + 1 E U 0 P e = s21 ; n CHAPTER 16. ENDOGENEITY 271 and 1 E X 0P X n = 0 E z i z 0i = 0 Q + S 22 : 0 + E (z i ui ) + E ui z 0i + 1 E U 0P U n Together E ^ 2SLS 0 = 1 1 0 X PX n E E Q + S 22 1 1 0 X Pe n s21 : (16.14) In general this is non-zero, except when s21 = 0 (when X is exogenous). It is also close to zero when = 0. Bekker (1994) pointed out that it also has the reverse implication –that when = l=n is large, the bias in the 2SLS estimator will be large. Indeed as ! 1; the expression in (16.14) approaches that in (16.13), indicating that the bias in 2SLS approaches that of OLS as the number of instruments increases. Bekker (1994) showed further that under the alternative asymptotic approximation that is …xed as n ! 1 (so that the number of instruments goes to in…nity proportionately with sample size) then the expression in (16.14) is the probability limit of ^ 2SLS 16.7 Identi…cation Failure Recall the reduced form equation X2 = Z1 12 + Z2 22 + U 2: The parameter fails to be identi…ed if 22 has de…cient rank. The consequences of identi…cation failure for inference are quite severe. Take the simplest case where k = l = 1 (so there is no Z 1 ): Then the model may be written as yi = xi + ei xi = zi + ui and 22 = = E (zi xi ) =Ezi2 : We see that is identi…ed if and only if 6= 0; which occurs when E (xi zi ) 6= 0. Thus identi…cation hinges on the existence of correlation between the excluded exogenous variable and the included endogenous variable. Suppose this condition fails, so E (xi zi ) = 0: Then by the CLT n 1 X d p zi ei ! N1 n N 0; E zi2 e2i (16.15) i=1 n n 1 X 1 X d p zi x i = p zi ui ! N2 n n i=1 therefore ^ N 0; E zi2 u2i (16.16) i=1 = p1 n p1 n Pn i=1 zi ei Pn i=1 zi xi d ! N1 N2 Cauchy; since the ratio of two normals is Cauchy. This is particularly nasty, as the Cauchy distribution does not have a …nite mean. This result carries over to more general settings, and was examined by Phillips (1989) and Choi and Phillips (1992). CHAPTER 16. ENDOGENEITY 272 Suppose that identi…cation does not completely fail, but is weak. This occurs when 22 is full rank, but small. This can be handled in an asymptotic analysis by modeling it as local-to-zero, viz 22 1=2 =n C; where C is a full rank matrix. The n 1=2 is picked because it provides just the right balancing to allow a rich distribution theory. To see the consequences, once again take the simple case k = l = 1: Here, the instrument xi is weak for zi if = n 1=2 c: Then (16.15) is una¤ected, but (16.16) instead takes the form n n n i=1 i=1 1 X 1 X 2 1 X p z i xi = p zi + p zi ui n n n i=1 n n 1X 2 1 X p = zi c + zi ui n n i=1 i=1 d ! Qc + N2 therefore ^ d ! N1 : Qc + N2 As in the case of complete identi…cation failure, we …nd that ^ is inconsistent for and the ^ asymptotic distribution of is non-normal. In addition, standard test statistics have non-standard distributions, meaning that inferences about parameters of interest can be misleading. The distribution theory for this model was developed by Staiger and Stock (1997) and extended to nonlinear GMM estimation by Stock and Wright (2000). Further results on testing were obtained by Wang and Zivot (1998). The bottom line is that it is highly desirable to avoid identi…cation failure. Once again, the equation to focus on is the reduced form X2 = Z1 12 + Z2 22 + U2 and identi…cation requires rank( 22 ) = k2 : If k2 = 1; this requires 22 6= 0; which is straightforward to assess using a hypothesis test on the reduced form. Therefore in the case of k2 = 1 (one RHS endogenous variable), one constructive recommendation is to explicitly estimate the reduced form equation for X 2 ; construct the test of 22 = 0, and at a minimum check that the test rejects H0 : 22 = 0. When k2 > 1; 22 6= 0 is not su¢ cient for identi…cation. It is not even su¢ cient that each column of 22 is non-zero (each column corresponds to a distinct endogenous variable in Z 2 ): So while a minimal check is to test that each columns of 22 is non-zero, this cannot be interpreted as de…nitive proof that 22 has full rank. Unfortunately, tests of de…cient rank are di¢ cult to implement. In any event, it appears reasonable to explicitly estimate and report the reduced form equations for Z 2 ; and attempt to assess the likelihood that 22 has de…cient rank. CHAPTER 16. ENDOGENEITY 273 Exercises 1. Consider the single equation model yi = z i + e i ; where yi and zi are both real-valued (1 1). Let ^ denote the IV estimator of using as an instrument a dummy variable di (takes only the values 0 and 1). Find a simple expression for the IV estimator in this context. 2. In the linear model yi = x0i + ei E (ei j xi ) = 0 suppose 2i = E e2i j xi is known. Show that the GLS estimator of IV estimator using some instrument z i : (Find an expression for z i :) can be written as an 3. Take the linear model y = X + e: be ^ and the OLS residual be e ^ = y X ^. Let the IV estimator for using some instrument Z be ~ and the IV residual be e ~ = y X ~. If X is indeed endogeneous, will IV “…t” better than OLS, in the sense that e ~0 e ~<e ^0 e ^; at least in large samples? Let the OLS estimator for 4. The reduced form between the regressors xi and instruments z i takes the form xi = 0 z i + ui or X =Z +U where xi is k 1; z i is l 1; X is n k; Z is n l; U is n is de…ned by the population moment condition k; and is l k: The parameter E z i u0i = 0 Show that the method of moments estimator for is ^ = (Z 0 Z) 1 (Z 0 X) : 5. In the structural model y = X +e X = Z +U with l k; l k; we claim that is identi…ed (can be recovered from the reduced form) if rank( ) = k: Explain why this is true. That is, show that if rank( ) < k then cannot be identi…ed. 6. Take the linear model yi = xi + ei E (ei j xi ) = 0: where xi and are 1 1: CHAPTER 16. ENDOGENEITY 274 (a) Show that E (xi ei ) = 0 and E x2i ei = 0: Is z i = (xi for estimation of ? x2i )0 a valid instrumental variable (b) De…ne the 2SLS estimator of ; using z i as an instrument for xi : How does this di¤er from OLS? (c) Find the e¢ cient GMM estimator of based on the moment condition E (z i (yi xi )) = 0: Does this di¤er from 2SLS and/or OLS? 7. Suppose that price and quantity are determined by the intersection of the linear demand and supply curves Demand : Q = a0 + a1 P + a2 Y + e1 Supply : Q = b0 + b1 P + b2 W + e2 where income (Y ) and wage (W ) are determined outside the market. In this model, are the parameters identi…ed? 8. The data …le card.dat is taken from Card (1995). There are 2215 observations with 29 variables, listed in card.pdf. We want to estimate a wage equation log(W age) = 0 + 1 Educ + 2 Exper + 3 Exper 2 + 4 South + 5 Black +e where Educ = Eduation (Years) Exper = Experience (Years), and South and Black are regional and racial dummy variables. (a) Estimate the model by OLS. Report estimates and standard errors. (b) Now treat Education as endogenous, and the remaining variables as exogenous. Estimate the model by 2SLS, using the instrument near4, a dummy indicating that the observation lives near a 4-year college. Report estimates and standard errors. (c) Re-estimate by 2SLS (report estimates and standard errors) adding three additional instruments: near2 (a dummy indicating that the observation lives near a 2-year college), f atheduc (the education, in years, of the father) and motheduc (the education, in years, of the mother). (d) Re-estimate the model by e¢ cient GMM. I suggest that you use the 2SLS estimates as the …rst-step to get the weight matrix, and then calculate the GMM estimator from this weight matrix without further iteration. Report the estimates and standard errors. (e) Calculate and report the J statistic for overidenti…cation. (f) Discuss your …ndings. Chapter 17 Univariate Time Series A time series yt is a process observed in sequence over time, t = 1; :::; T . To indicate the dependence on time, we adopt new notation, and use the subscript t to denote the individual observation, and T to denote the number of observations. Because of the sequential nature of time series, we expect that yt and yt 1 are not independent, so classical assumptions are not valid. We can separate time series into two categories: univariate (yt 2 R is scalar); and multivariate (yt 2 Rm is vector-valued). The primary model for univariate time series is autoregressions (ARs). The primary model for multivariate time series is vector autoregressions (VARs). 17.1 Stationarity and Ergodicity De…nition 17.1.1 fyt g is covariance (weakly) stationary if E(yt ) = is independent of t; and cov (yt ; yt k) = (k) is independent of t for all k: (k) is called the autocovariance function. (k) = (k)= (0) = corr(yt ; yt k) is the autocorrelation function. De…nition 17.1.2 fyt g is strictly stationary if the joint distribution of (yt ; :::; yt k ) is independent of t for all k: De…nition 17.1.3 A stationary time series is ergodic if k ! 1. 275 (k) ! 0 as CHAPTER 17. UNIVARIATE TIME SERIES 276 The following two theorems are essential to the analysis of stationary time series. There proofs are rather di¢ cult, however. Theorem 17.1.1 If yt is strictly stationary and ergodic and xt = f (yt ; yt 1 ; :::) is a random variable, then xt is strictly stationary and ergodic. Theorem 17.1.2 (Ergodic Theorem). If yt is strictly stationary and ergodic and E jyt j < 1; then as T ! 1; T 1X p yt ! E(yt ): T t=1 This allows us to consistently estimate parameters using time-series moments: The sample mean: T 1X ^= yt T t=1 The sample autocovariance ^ (k) = T 1X (yt T ^ ) (yt k ^) : t=1 The sample autocorrelation ^(k) = ^ (k) : ^ (0) Theorem 17.1.3 If yt is strictly stationary and ergodic and Eyt2 < 1; then as T ! 1; p 1. ^ ! E(yt ); p 2. ^ (k) ! (k); p 3. ^(k) ! (k): Proof of Theorem 17.1.3. Part (1) is a direct consequence of the Ergodic theorem. For Part (2), note that ^ (k) = = T 1X (yt T 1 T t=1 T X t=1 yt yt ^ ) (yt k k ^) T 1X yt ^ T t=1 T 1X yt T t=1 k^ + ^2: CHAPTER 17. UNIVARIATE TIME SERIES 277 By Theorem 17.1.1 above, the sequence yt yt k is strictly stationary and ergodic, and it has a …nite mean by the assumption that Eyt2 < 1: Thus an application of the Ergodic Theorem yields T 1X yt yt T p ! E(yt yt k t=1 Thus p ^ (k) ! E(yt yt k) 2 2 + 2 k ): = E(yt yt k) 2 = (k): p Part (3) follows by the continuous mapping theorem: ^(k) = ^ (k)=^ (0) ! (k)= (0) = (k): 17.2 Autoregressions In time-series, the series f:::; y1 ; y2 ; :::; yT ; :::g are jointly random. We consider the conditional expectation E (yt j Ft 1 ) where Ft 1 = fyt 1 ; yt 2 ; :::g is the past history of the series. An autoregressive (AR) model speci…es that only a …nite number of past lags matter: E (yt j Ft 1) = E (yt j yt 1 ; :::; yt k ) : A linear AR model (the most common type used in practice) speci…es linearity: E (yt j Ft 1) = + 1 yt 1 + 2 yt 1 + + k yt k : Letting et = yt E (yt j Ft 1) ; then we have the autoregressive model yt = E (et j Ft 1) + 1 yt 1 + 2 yt 1 + + k yt k + et = 0: The last property de…nes a special time-series process. De…nition 17.2.1 et is a martingale di¤ erence sequence (MDS) if E (et j Ft 1 ) = 0: Regression errors are naturally a MDS. Some time-series processes may be a MDS as a consequence of optimizing behavior. For example, some versions of the life-cycle hypothesis imply that either changes in consumption, or consumption growth rates, should be a MDS. Most asset pricing models imply that asset returns should be the sum of a constant plus a MDS. The MDS property for the regression error plays the same role in a time-series regression as does the conditional mean-zero property for the regression error in a cross-section regression. In fact, it is even more important in the time-series context, as it is di¢ cult to derive distribution theories without this property. A useful property of a MDS is that et is uncorrelated with any function of the lagged information Ft 1 : Thus for k > 0; E (yt k et ) = 0: CHAPTER 17. UNIVARIATE TIME SERIES 17.3 278 Stationarity of AR(1) Process A mean-zero AR(1) is yt = yt Assume that et is iid, E(et ) = 0 and By back-substitution, we …nd Ee2t 2 = + et : 1 < 1: yt = et + et 1 X k = et 2 + 1 et 2 + ::: k: k=0 ke Loosely speaking, this series converges if the sequence when j j < 1: t k gets small as k ! 1: This occurs Theorem 17.3.1 If and only if j j < 1 then yt is strictly stationary and ergodic. We can compute the moments of yt using the in…nite sum: Eyt = 1 X k E (et k) =0 var (et k) = k=0 var(yt ) = 1 X 2 2k k=0 1 2 : If the equation for yt has an intercept, the above results are unchanged, except that the mean of yt can be computed from the relationship Eyt = and solving for Eyt = Eyt 17.4 1 + Eyt we …nd Eyt = =(1 1; ): Lag Operator An algebraic construct which is useful for the analysis of autoregressive models is the lag operator. De…nition 17.4.1 The lag operator L satis…es Lyt = yt De…ning L2 = LL; we see that L2 yt = Lyt 1 = yt The AR(1) model can be written in the format 2: In general, Lk yt = yt yt yt (1 L) yt = et : 1 1: k: = et or The operator (L) = (1 L) is a polynomial in the operator L: We say that the root of the polynomial is 1= ; since (z) = 0 when z = 1= : We call (L) the autoregressive polynomial of yt . From Theorem 17.3.1, an AR(1) is stationary i¤ j j < 1: Note that an equivalent way to say this is that an AR(1) is stationary i¤ the root of the autoregressive polynomial is larger than one (in absolute value). CHAPTER 17. UNIVARIATE TIME SERIES 17.5 279 Stationarity of AR(k) The AR(k) model is yt = 1 yt 1 yt 1 Lyt + 2 yt 2 + + k yt k + et : k k L yt = et ; Using the lag operator, 2 2 L yt or (L)yt = et where (L) = 1 2 2L 1L kL k : We call (L) the autoregressive polynomial of yt : The Fundamental Theorem of Algebra says that any polynomial can be factored as (z) = 1 1 1 z 1 1 2 z 1 1 k z where the 1 ; :::; k are the complex roots of (z); which satisfy ( j ) = 0: We know that an AR(1) is stationary i¤ the absolute value of the root of its autoregressive polynomial is larger than one. For an AR(k), the requirement is that all roots are larger than one. Let j j denote the modulus of a complex number : Theorem 17.5.1 The AR(k) is strictly stationary and ergodic if and only if j j j > 1 for all j: One way of stating this is that “All roots lie outside the unit circle.” If one of the roots equals 1, we say that (L); and hence yt ; “has a unit root”. This is a special case of non-stationarity, and is of great interest in applied time series. 17.6 Estimation Let 1 yt xt = = 1 yt 1 yt 2 k 2 0 k 0 : Then the model can be written as yt = x0t + et : The OLS estimator is ^ = X 0X 1 X 0 y: To study ^ ; it is helpful to de…ne the process ut = xt et : Note that ut is a MDS, since E (ut j Ft 1) = E (xt et j Ft 1) = xt E (et j Ft 1) = 0: By Theorem 17.1.1, it is also strictly stationary and ergodic. Thus T T 1X 1X p xt et = ut ! E (ut ) = 0: T T t=1 t=1 (17.1) CHAPTER 17. UNIVARIATE TIME SERIES 280 The vector xt is strictly stationary and ergodic, and by Theorem 17.1.1, so is xt x0t : Thus by the Ergodic Theorem, T 1X p xt x0t ! E xt x0t = Q: T t=1 Combined with (17.1) and the continuous mapping theorem, we see that ^= + T 1X xt x0t T t=1 ! 1 T 1X xt et T t=1 ! p !Q 1 0 = 0: We have shown the following: Theorem 17.6.1 If the AR(k) process yt is strictly stationary and ergodic p and Eyt2 < 1; then ^ ! as T ! 1: 17.7 Asymptotic Distribution Theorem 17.7.1 MDS CLT. If ut is a strictly stationary and ergodic MDS and E (ut u0t ) = < 1; then as T ! 1; T 1 X d p ut ! N (0; ) : T t=1 Since xt et is a MDS, we can apply Theorem 17.7.1 to see that T 1 X d p xt et ! N (0; ) ; T t=1 where = E(xt x0t e2t ): Theorem 17.7.2 If the AR(k) process yt is strictly stationary and ergodic and Eyt4 < 1; then as T ! 1; p T ^ d ! N 0; Q 1 Q 1 : This is identical in form to the asymptotic distribution of OLS in cross-section regression. The implication is that asymptotic inference is the same. In particular, the asymptotic covariance matrix is estimated just as in the cross-section case. CHAPTER 17. UNIVARIATE TIME SERIES 17.8 281 Bootstrap for Autoregressions In the non-parametric bootstrap, we constructed the bootstrap sample by randomly resampling from the data values fyt ; xt g: This creates an iid bootstrap sample. Clearly, this cannot work in a time-series application, as this imposes inappropriate independence. Brie‡y, there are two popular methods to implement bootstrap resampling for time-series data. Method 1: Model-Based (Parametric) Bootstrap. 1. Estimate ^ and residuals e^t : 2. Fix an initial condition (y k+1 ; y k+2 ; :::; y0 ): 3. Simulate iid draws ei from the empirical distribution of the residuals f^ e1 ; :::; e^T g: 4. Create the bootstrap series yt by the recursive formula yt = ^ + ^ 1 yt 1 + ^ 2 yt 2 + + ^ k yt k + et : This construction imposes homoskedasticity on the errors ei ; which may be di¤erent than the properties of the actual ei : It also presumes that the AR(k) structure is the truth. Method 2: Block Resampling 1. Divide the sample into T =m blocks of length m: 2. Resample complete blocks. For each simulated sample, draw T =m blocks. 3. Paste the blocks together to create the bootstrap time-series yt : 4. This allows for arbitrary stationary serial correlation, heteroskedasticity, and for modelmisspeci…cation. 5. The results may be sensitive to the block length, and the way that the data are partitioned into blocks. 6. May not work well in small samples. 17.9 Trend Stationarity yt = 0 + 1t St = 1 St 1 + St + (17.2) 2 St 2 + + k St l + et ; (17.3) or yt = 0 + 1t + 1 yt 1 + 2 yt 1 + + k yt k + et : (17.4) There are two essentially equivalent ways to estimate the autoregressive parameters ( 1 ; :::; k ): You can estimate (17.4) by OLS. You can estimate (17.2)-(17.3) sequentially by OLS. That is, …rst estimate (17.2), get the residual S^t ; and then perform regression (17.3) replacing St with S^t : This procedure is sometimes called Detrending. CHAPTER 17. UNIVARIATE TIME SERIES 282 The reason why these two procedures are (essentially) the same is the Frisch-Waugh-Lovell theorem. Seasonal E¤ects There are three popular methods to deal with seasonal data. Include dummy variables for each season. This presumes that “seasonality” does not change over the sample. Use “seasonally adjusted” data. The seasonal factor is typically estimated by a two-sided weighted average of the data for that season in neighboring years. Thus the seasonally adjusted data is a “…ltered”series. This is a ‡exible approach which can extract a wide range of seasonal factors. The seasonal adjustment, however, also alters the time-series correlations of the data. First apply a seasonal di¤erencing operator. If s is the number of seasons (typically s = 4 or s = 12); yt s ; s yt = yt or the season-to-season change. The series s yt is clearly free of seasonality. But the long-run trend is also eliminated, and perhaps this was of relevance. 17.10 Testing for Omitted Serial Correlation For simplicity, let the null hypothesis be an AR(1): yt = + yt 1 + ut : (17.5) We are interested in the question if the error ut is serially correlated. We model this as an AR(1): ut = ut 1 + et (17.6) with et a MDS. The hypothesis of no omitted serial correlation is H0 : =0 H1 : 6= 0: We want to test H0 against H1 : To combine (17.5) and (17.6), we take (17.5) and lag the equation once: yt We then multiply this by yt 1 = + yt 2 + ut 1: and subtract from (17.5), to …nd yt 1 = + yt 1 yt 1 + ut ut 1; or yt = (1 ) + ( + ) yt 1 yt 2 + et = AR(2): Thus under H0 ; yt is an AR(1), and under H1 it is an AR(2). H0 may be expressed as the restriction that the coe¢ cient on yt 2 is zero. An appropriate test of H0 against H1 is therefore a Wald test that the coe¢ cient on yt 2 is zero. (A simple exclusion test). In general, if the null hypothesis is that yt is an AR(k), and the alternative is that the error is an AR(m), this is the same as saying that under the alternative yt is an AR(k+m), and this is equivalent to the restriction that the coe¢ cients on yt k 1 ; :::; yt k m are jointly zero. An appropriate test is the Wald test of this restriction. CHAPTER 17. UNIVARIATE TIME SERIES 17.11 283 Model Selection What is the appropriate choice of k in practice? This is a problem of model selection. One approach to model selection is to choose k based on a Wald tests. Another is to minimize the AIC or BIC information criterion, e.g. AIC(k) = log ^ 2 (k) + 2k ; T where ^ 2 (k) is the estimated residual variance from an AR(k) One ambiguity in de…ning the AIC criterion is that the sample available for estimation changes as k changes. (If you increase k; you need more initial conditions.) This can induce strange behavior in the AIC. The best remedy is to …x a upper value k; and then reserve the …rst k as initial conditions, and then estimate the models AR(1), AR(2), ..., AR(k) on this (uni…ed) sample. 17.12 Autoregressive Unit Roots The AR(k) model is (L)yt = + et (L) = 1 k kL : 1L As we discussed before, yt has a unit root when (1) = 0; or 1 + 2 + + k = 1: In this case, yt is non-stationary. The ergodic theorem and MDS CLT do not apply, and test statistics are asymptotically non-normal. A helpful way to write the equation is the so-called Dickey-Fuller reparameterization: yt = + 0 yt 1 + yt 1 1 + + k 1 yt (k 1) + et : (17.7) These models are equivalent linear transformations of one another. The DF parameterization is convenient because the parameter 0 summarizes the information about the unit root, since (1) = 0 : To see this, observe that the lag polynomial for the yt computed from (17.7) is (1 L) 0L 1 (L k 1 k 1 (L L2 ) Lk ) But this must equal (L); as the models are equivalent. Thus (1) = (1 1) (1 0 1) (1 1) = 0: Hence, the hypothesis of a unit root in yt can be stated as H0 : Note that the model is stationary if 0 0 = 0: < 0: So the natural alternative is H1 : 0 < 0: Under H0 ; the model for yt is yt = + 1 yt 1 + + k 1 yt (k 1) + et ; which is an AR(k-1) in the …rst-di¤erence yt : Thus if yt has a (single) unit root, then yt is a stationary AR process. Because of this property, we say that if yt is non-stationary but d yt is stationary, then yt is “integrated of order d”; or I(d): Thus a time series with unit root is I(1): CHAPTER 17. UNIVARIATE TIME SERIES 284 Since 0 is the parameter of a linear regression, the natural test statistic is the t-statistic for H0 from OLS estimation of (17.7). Indeed, this is the most popular unit root test, and is called the Augmented Dickey-Fuller (ADF) test for a unit root. It would seem natural to assess the signi…cance of the ADF statistic using the normal table. However, under H0 ; yt is non-stationary, so conventional normal asymptotics are invalid. An alternative asymptotic framework has been developed to deal with non-stationary data. We do not have the time to develop this theory in detail, but simply assert the main results. Theorem 17.12.1 Dickey-Fuller Theorem. Assume 0 = 0: As T ! 1; d T ^ 0 ! (1 1 ADF = k 1 ) DF 2 ^0 ! DFt : s(^ 0 ) The limit distributions DF and DFt are non-normal. They are skewed to the left, and have negative means. The …rst result states that ^ 0 converges to its true value (of zero) at rate T; rather than the conventional rate of T 1=2 : This is called a “super-consistent” rate of convergence. The second result states that the t-statistic for ^ 0 converges to a limit distribution which is non-normal, but does not depend on the parameters : This distribution has been extensively tabulated, and may be used for testing the hypothesis H0 : Note: The standard error s(^ 0 ) is the conventional (“homoskedastic”) standard error. But the theorem does not require an assumption of homoskedasticity. Thus the Dickey-Fuller test is robust to heteroskedasticity. Since the alternative hypothesis is one-sided, the ADF test rejects H0 in favor of H1 when ADF < c; where c is the critical value from the ADF table. If the test rejects H0 ; this means that the evidence points to yt being stationary. If the test does not reject H0 ; a common conclusion is that the data suggests that yt is non-stationary. This is not really a correct conclusion, however. All we can say is that there is insu¢ cient evidence to conclude whether the data are stationary or not. We have described the test for the setting of with an intercept. Another popular setting includes as well a linear time trend. This model is yt = 1 + 2t + 0 yt 1 + 1 yt 1 + + k 1 yt (k 1) + et : (17.8) This is natural when the alternative hypothesis is that the series is stationary about a linear time trend. If the series has a linear trend (e.g. GDP, Stock Prices), then the series itself is nonstationary, but it may be stationary around the linear time trend. In this context, it is a silly waste of time to …t an AR model to the level of the series without a time trend, as the AR model cannot conceivably describe this data. The natural solution is to include a time trend in the …tted OLS equation. When conducting the ADF test, this means that it is computed as the t-ratio for 0 from OLS estimation of (17.8). If a time trend is included, the test procedure is the same, but di¤erent critical values are required. The ADF test has a di¤erent distribution when the time trend has been included, and a di¤erent table should be consulted. Most texts include as well the critical values for the extreme polar case where the intercept has been omitted from the model. These are included for completeness (from a pedagogical perspective) but have no relevance for empirical practice where intercepts are always included. Chapter 18 Multivariate Time Series A multivariate time series y t is a vector process m 1. Let Ft 1 = (y t 1 ; y t 2 ; :::) be all lagged information at time t: The typical goal is to …nd the conditional expectation E (y t j Ft 1 ) : Note that since y t is a vector, this conditional expectation is also a vector. 18.1 Vector Autoregressions (VARs) A VAR model speci…es that the conditional mean is a function of only a …nite number of lags: E (y t j Ft 1) = E yt j yt 1 ; :::; y t k : A linear VAR speci…es that this conditional mean is linear in the arguments: E yt j yt 1 ; :::; y t k = a0 + A1 y t 1 + A2 y t Observe that a0 is m 1,and each of A1 through Ak are m De…ning the m 1 regression error et = y t E (y t j Ft 2 + Ak y t k: m matrices. 1) ; we have the VAR model y t = a0 + A1 y t E (et j Ft 1) 1 + A2 y t 2 + Ak y t k + et = 0: Alternatively, de…ning the mk + 1 vector 0 and the m 1 B yt B B xt = B y t B .. @ . yt (mk + 1) matrix A= 1 2 k 1 C C C C C A a0 A1 A2 Ak ; then y t = Axt + et : The VAR model is a system of m equations. One way to write this is to let a0j be the jth row of A. Then the VAR system can be written as the equations Yjt = a0j xt + ejt : Unrestricted VARs were introduced to econometrics by Sims (1980). 285 CHAPTER 18. MULTIVARIATE TIME SERIES 18.2 286 Estimation Consider the moment conditions E (xt ejt ) = 0; j = 1; :::; m: These are implied by the VAR model, either as a regression, or as a linear projection. The GMM estimator corresponding to these moment conditions is equation-by-equation OLS a ^j = (X 0 X) 1 X 0yj : An alternative way to compute this is as follows. Note that a ^0j = y 0j X(X 0 X) 1 : ^ we …nd And if we stack these to create the estimate A; 1 0 y 01 B y0 C 2 C B ^ A = B . C X(X 0 X) @ .. A 1 y 0m+1 = Y 0 X(X 0 X) 1 ; where Y = y1 y2 ym the T m matrix of the stacked y 0t : This (system) estimator is known as the SUR (Seemingly Unrelated Regressions) estimator, and was originally derived by Zellner (1962) 18.3 Restricted VARs The unrestricted VAR is a system of m equations, each with the same set of regressors. A restricted VAR imposes restrictions on the system. For example, some regressors may be excluded from some of the equations. Restrictions may be imposed on individual equations, or across equations. The GMM framework gives a convenient method to impose such restrictions on estimation. 18.4 Single Equation from a VAR Often, we are only interested in a single equation out of a VAR system. This takes the form yjt = a0j xt + et ; and xt consists of lagged values of yjt and the other ylt0 s: In this case, it is convenient to re-de…ne the variables. Let yt = yjt ; and z t be the other variables. Let et = ejt and = aj : Then the single equation takes the form yt = x0t + et ; (18.1) and xt = h 1 yt 1 yt k z 0t 1 This is just a conventional regression with time series data. z 0t k 0 i : CHAPTER 18. MULTIVARIATE TIME SERIES 18.5 287 Testing for Omitted Serial Correlation Consider the problem of testing for omitted serial correlation in equation (18.1). Suppose that et is an AR(1). Then yt = x0t + et et = E (ut j Ft 1) et 1 + ut (18.2) = 0: Then the null and alternative are H0 : =0 H1 : 6= 0: Take the equation yt = x0t + et ; and subtract o¤ the equation once lagged multiplied by ; to get yt yt 1 = = x0t + et x0t xt x0t 1 1 + et + et et 1 1; or yt = yt 1 + x0t + x0t 1 + ut ; (18.3) which is a valid regression model. So testing H0 versus H1 is equivalent to testing for the signi…cance of adding (yt 1 ; xt 1 ) to the regression. This can be done by a Wald test. We see that an appropriate, general, and simple way to test for omitted serial correlation is to test the signi…cance of extra lagged values of the dependent variable and regressors. You may have heard of the Durbin-Watson test for omitted serial correlation, which once was very popular, and is still routinely reported by conventional regression packages. The DW test is appropriate only when regression yt = x0t + et is not dynamic (has no lagged values on the RHS), and et is iid N(0; 2 ): Otherwise it is invalid. Another interesting fact is that (18.2) is a special case of (18.3), under the restriction = : This restriction, which is called a common factor restriction, may be tested if desired. If valid, the model (18.2) may be estimated by iterated GLS. (A simple version of this estimator is called Cochrane-Orcutt.) Since the common factor restriction appears arbitrary, and is typically rejected empirically, direct estimation of (18.2) is uncommon in recent applications. 18.6 Selection of Lag Length in an VAR If you want a data-dependent rule to pick the lag length k in a VAR, you may either use a testingbased approach (using, for example, the Wald statistic), or an information criterion approach. The formula for the AIC and BIC are p AIC(k) = log det ^ (k) + 2 T p log(T ) BIC(k) = log det ^ (k) + T T X ^ (k) = 1 e ^t (k)^ et (k)0 T t=1 p = m(km + 1) where p is the number of parameters in the model, and e ^t (k) is the OLS residual vector from the model with k lags. The log determinant is the criterion from the multivariate normal likelihood. CHAPTER 18. MULTIVARIATE TIME SERIES 18.7 288 Granger Causality Partition the data vector into (y t ; z t ): De…ne the two information sets F1t = yt; yt F2t = 1 ; y t 2 ; ::: yt; zt; yt 1 ; z t 1 ; y t 2 ; z t 2 ; ; ::: The information set F1t is generated only by the history of y t ; and the information set F2t is generated by both y t and z t : The latter has more information. We say that z t does not Granger-cause y t if E (y t j F1;t 1) = E (y t j F2;t 1) : That is, conditional on information in lagged y t ; lagged z t does not help to forecast y t : If this condition does not hold, then we say that z t Granger-causes y t : The reason why we call this “Granger Causality” rather than “causality” is because this is not a physical or structure de…nition of causality. If z t is some sort of forecast of the future, such as a futures price, then z t may help to forecast y t even though it does not “cause” y t : This de…nition of causality was developed by Granger (1969) and Sims (1972). In a linear VAR, the equation for y t is yt = + 1yt 1 + + k yt k + z 0t 1 1 + + z 0t k k + et : In this equation, z t does not Granger-cause y t if and only if H0 : 1 = 2 = = k = 0: This may be tested using an exclusion (Wald) test. This idea can be applied to blocks of variables. That is, y t and/or z t can be vectors. The hypothesis can be tested by using the appropriate multivariate Wald test. If it is found that z t does not Granger-cause y t ; then we deduce that our time-series model of E (y t j Ft 1 ) does not require the use of z t : Note, however, that z t may still be useful to explain other features of y t ; such as the conditional variance. Clive W. J. Granger Clive Granger (1934-2009) of England was one of the leading …gures in timeseries econometrics, and co-winner in 2003 of the Nobel Memorial Prize in Economic Sciences (along with Robert Engle). In addition to formalizing the de…nition of causality known as Granger causality, he invented the concept of cointegration, introduced spectral methods into econometrics, and formalized methods for the combination of forecasts. 18.8 Cointegration The idea of cointegration is due to Granger (1981), and was articulated in detail by Engle and Granger (1987). CHAPTER 18. MULTIVARIATE TIME SERIES 289 De…nition 18.8.1 The m 1 series y t is cointegrated if y t is I(1) yet there exists ; m r, of rank r; such that z t = 0 y t is I(0): The r vectors in are called the cointegrating vectors. If the series y t is not cointegrated, then r = 0: If r = m; then y t is I(0): For 0 < r < m; y t is I(1) and cointegrated. In some cases, it may be believed that is known a priori. Often, = (1 1)0 : For example, if 0 y t is a pair of interest rates, then = (1 1) speci…es that the spread (the di¤erence in returns) 0 is stationary. If y = (log(C) log(I)) ; then = (1 1)0 speci…es that log(C=I) is stationary. In other cases, may not be known. If y t is cointegrated with a single cointegrating vector (r = 1); then it turns out that can be consistently estimated by an OLS regression of one component of y t on the others. Thus y t = p (Y1t ; Y2t ) and = ( 1 2 ) and normalize 1 = 1: Then ^ 2 = (y 02 y 2 ) 1 y 02 y 1 ! 2 : Furthermore d this estimation is super-consistent: T ( ^ ) ! Limit; as …rst shown by Stock (1987). This 2 2 is not, in general, a good method to estimate ; but it is useful in the construction of alternative estimators and tests. We are often interested in testing the hypothesis of no cointegration: H0 : r = 0 H1 : r > 0: Suppose that is known, so z t = 0 y t is known. Then under H0 z t is I(1); yet under H1 z t is I(0): Thus H0 can be tested using a univariate ADF test on z t : When is unknown, Engle and Granger (1987) suggested using an ADF test on the estimated 0 residual z^t = ^ y t ; from OLS of y1t on y2t : Their justi…cation was Stock’s result that ^ is superconsistent under H1 : Under H0 ; however, ^ is not consistent, so the ADF critical values are not appropriate. The asymptotic distribution was worked out by Phillips and Ouliaris (1990). When the data have time trends, it may be necessary to include a time trend in the estimated cointegrating regression. Whether or not the time trend is included, the asymptotic distribution of the test is a¤ected by the presence of the time trend. The asymptotic distribution was worked out in B. Hansen (1992). 18.9 Cointegrated VARs We can write a VAR as A(L)y t = et A(L) = I A1 L Ak Lk A2 L2 or alternatively as yt = yt 1 + D(L) y t 1 + et where = A(1) = I + A1 + A2 + + Ak : CHAPTER 18. MULTIVARIATE TIME SERIES 290 Theorem 18.9.1 Granger Representation Theorem y t is cointegrated with m r if and only if rank( ) = r and where is m r, rank ( ) = r: = 0 Thus cointegration imposes a restriction upon the parameters of a VAR. The restricted model can be written as yt = 0 yt yt = zt 1 1 + D(L) y t + D(L) y t 1 1 + et + et : If is known, this can be estimated by OLS of y t on z t 1 and the lags of y t : If is unknown, then estimation is done by “reduced rank regression”, which is least-squares subject to the stated restriction. Equivalently, this is the MLE of the restricted parameters under the assumption that et is iid N(0; ): One di¢ culty is that is not identi…ed without normalization. When r = 1; we typically just normalize one element to equal unity. When r > 1; this does not work, and di¤erent authors have adopted di¤erent identi…cation schemes. In the context of a cointegrated VAR estimated by reduced rank regression, it is simple to test for cointegration by testing the rank of : These tests are constructed as likelihood ratio (LR) tests. As they were discovered by Johansen (1988, 1991, 1995), they are typically called the “Johansen Max and Trace” tests. Their asymptotic distributions are non-standard, and are similar to the Dickey-Fuller distributions. Chapter 19 Limited Dependent Variables A “limited dependent variable”y is one which takes a “limited”set of values. The most common cases are Binary: y 2 f0; 1g Multinomial: y 2 f0; 1; 2; :::; kg Integer: y 2 f0; 1; 2; :::g Censored: y 2 R+ The traditional approach to the estimation of limited dependent variable (LDV) models is parametric maximum likelihood. A parametric model is constructed, allowing the construction of the likelihood function. A more modern approach is semi-parametric, eliminating the dependence on a parametric distributional assumption. We will discuss only the …rst (parametric) approach, due to time constraints. They still constitute the majority of LDV applications. If, however, you were to write a thesis involving LDV estimation, you would be advised to consider employing a semi-parametric estimation approach. For the parametric approach, estimation is by MLE. A major practical issue is construction of the likelihood function. 19.1 Binary Choice The dependent variable yi 2 f0; 1g: This represents a Yes/No outcome. Given some regressors xi ; the goal is to describe Pr (yi = 1 j xi ) ; as this is the full conditional distribution. The linear probability model speci…es that Pr (yi = 1 j xi ) = x0i : As Pr (yi = 1 j xi ) = E (yi j xi ) ; this yields the regression: yi = x0i + ei which can be estimated by OLS. However, the linear probability model does not impose the restriction that 0 Pr (yi j xi ) 1: Even so estimation of a linear probability model is a useful starting point for subsequent analysis. The standard alternative is to use a function of the form Pr (yi = 1 j xi ) = F x0i where F ( ) is a known CDF, typically assumed to be symmetric about zero, so that F (u) = 1 F ( u): The two standard choices for F are Logistic: F (u) = (1 + e u) 1 : 291 CHAPTER 19. LIMITED DEPENDENT VARIABLES Normal: F (u) = 292 (u): If F is logistic, we call this the logit model, and if F is normal, we call this the probit model. This model is identical to the latent variable model yi = x0i + ei ei F() 1 = 0 yi if yi > 0 : otherwise For then Pr (yi = 1 j xi ) = Pr (yi > 0 j xi ) = Pr x0i + ei > 0 j xi x0i = Pr ei > = 1 j xi x0i F = F x0i : Estimation is by maximum likelihood. To construct the likelihood, we need the conditional distribution of an individual observation. Recall that if y is Bernoulli, such that Pr(y = 1) = p and Pr(y = 0) = 1 p, then we can write the density of y as f (y) = py (1 y p)1 ; y = 0; 1: In the Binary choice model, yi is conditionally Bernoulli with Pr (yi = 1 j xi ) = pi = F (x0i ) : Thus the conditional density is f (yi j xi ) = pyi i (1 p i )1 yi = F x0i yi F x0i (1 )1 yi : Hence the log-likelihood function is log L( ) = = n X i=1 n X log f (yi j xi ) log F x0i yi (1 F x0i )1 yi i=1 = n X yi log F x0i + (1 i=1 = X log F x0i + yi =1 X yi ) log(1 log(1 F x0i F x0i ) ): yi =0 The MLE ^ is the value of which maximizes log L( ): Standard errors and test statistics are computed by asymptotic approximations. Details of such calculations are left to more advanced courses. 19.2 Count Data If y 2 f0; 1; 2; :::g; a typical approach is to employ Poisson regression. This model speci…es that Pr (yi = k j xi ) = i exp ( i) k! = exp(x0i ): k i ; k = 0; 1; 2; ::: CHAPTER 19. LIMITED DEPENDENT VARIABLES The conditional density is the Poisson with parameter picked to ensure that i > 0. The log-likelihood function is log L( ) = n X i=1 log f (yi j xi ) = n X 293 i: The functional form for exp(x0i ) + yi x0i i has been log(yi !) : i=1 The MLE is the value ^ which maximizes log L( ): Since E (yi j xi ) = i = exp(x0i ) is the conditional mean, this motivates the label Poisson “regression.” Also observe that the model implies that var (yi j xi ) = i = exp(x0i ); so the model imposes the restriction that the conditional mean and variance of yi are the same. This may be considered restrictive. A generalization is the negative binomial. 19.3 Censored Data The idea of “censoring” is that some data above or below a threshold are mis-reported at the threshold. Thus the model is that there is some latent process yi with unbounded support, but we observe only yi if yi 0 : (19.1) yi = 0 if yi < 0 (This is written for the case of the threshold being zero, any known value can substitute.) The observed data yi therefore come from a mixed continuous/discrete distribution. Censored models are typically applied when the data set has a meaningful proportion (say 5% or higher) of data at the boundary of the sample support. The censoring process may be explicit in data collection, or it may be a by-product of economic constraints. An example of a data collection censoring is top-coding of income. In surveys, incomes above a threshold are typically reported at the threshold. The …rst censored regression model was developed by Tobin (1958) to explain consumption of durable goods. Tobin observed that for many households, the consumption level (purchases) in a particular period was zero. He proposed the latent variable model yi = x0i + ei ei iid N(0; 2 ) with the observed variable yi generated by the censoring equation (19.1). This model (now called the Tobit) speci…es that the latent (or ideal) value of consumption may be negative (the household would prefer to sell than buy). All that is reported is that the household purchased zero units of the good. The naive approach to estimate is to regress yi on xi . This does not work because regression estimates E (yi j xi ) ; not E (yi j xi ) = x0i ; and the latter is of interest. Thus OLS will be biased for the parameter of interest : [Note: it is still possible to estimate E (yi j xi ) by LS techniques. The Tobit framework postulates that this is not inherently interesting, that the parameter of is de…ned by an alternative statistical structure.] CHAPTER 19. LIMITED DEPENDENT VARIABLES 294 Consistent estimation will be achieved by the MLE. To construct the likelihood, observe that the probability of being censored is Pr (yi = 0 j xi ) = Pr (yi < 0 j xi ) = Pr x0i + ei < 0 j xi ei x0i = Pr < j xi x0i = : The conditional density function above zero is normal: y 1 Therefore, the density function for y x0i ; y > 0: 0 can be written as x0i f (y j xi ) = 1(y=0) z 1 x0i 1(y>0) ; where 1 ( ) is the indicator function. Hence the log-likelihood is a mixture of the probit and the normal: log L( ) = n X i=1 = X log f (yi j xi ) log x0i + X log 1 yi x0i : yi >0 yi =0 The MLE is the value ^ which maximizes log L( ): 19.4 Sample Selection The problem of sample selection arises when the sample is a non-random selection of potential observations. This occurs when the observed data is systematically di¤erent from the population of interest. For example, if you ask for volunteers for an experiment, and they wish to extrapolate the e¤ects of the experiment on a general population, you should worry that the people who volunteer may be systematically di¤erent from the general population. This has great relevance for the evaluation of anti-poverty and job-training programs, where the goal is to assess the e¤ect of “training” on the general population, not just on the volunteers. A simple sample selection model can be written as the latent model yi = x0i + e1i Ti = 1 z 0i + e0i > 0 where 1 ( ) is the indicator function. The dependent variable yi is observed if (and only if) Ti = 1: Else it is unobserved. For example, yi could be a wage, which can be observed only if a person is employed. The equation for Ti is an equation specifying the probability that the person is employed. The model is often completed by specifying that the errors are jointly normal e0i e1i N 0; 1 2 : CHAPTER 19. LIMITED DEPENDENT VARIABLES 295 It is presumed that we observe fxi ; z i ; Ti g for all observations. Under the normality assumption, e1i = e0i + vi ; where vi is independent of e0i that N(0; 1): A useful fact about the standard normal distribution is E (e0i j e0i > x) = (x) = (x) ; (x) and the function (x) is called the inverse Mills ratio. The naive estimator of is OLS regression of yi on xi for those observations for which yi is available. The problem is that this is equivalent to conditioning on the event fTi = 1g: However, E (e1i j Ti = 1; z i ) = E e1i j fe0i > = = E e0i j fe0i > z 0i z 0i g; z i ; z 0i g; z i + E vi j fe0i > z 0i g; z i which is non-zero. Thus e1i = z 0i + ui ; where E (ui j Ti = 1; z i ) = 0: Hence yi = x0i + z 0i + ui (19.2) is a valid regression equation for the observations for which Ti = 1: Heckman (1979) observed that we could consistently estimate and from this equation, if were known. It is unknown, but also can be consistently estimated by a Probit model for selection. The “Heckit” estimator is thus calculated as follows Estimate ^ from a Probit, using regressors z i : The binary dependent variable is Ti : Estimate ^ ; ^ from OLS of yi on xi and (z 0i ^ ): The OLS standard errors will be incorrect, as this is a two-step estimator. They can be corrected using a more complicated formula. Or, alternatively, by viewing the Probit/OLS estimation equations as a large joint GMM problem. The Heckit estimator is frequently used to deal with problems of sample selection. However, the estimator is built on the assumption of normality, and the estimator can be quite sensitive to this assumption. Some modern econometric research is exploring how to relax the normality assumption. The estimator can also work quite poorly if (z 0i ^ ) does not have much in-sample variation. This can happen if the Probit equation does not “explain”much about the selection choice. Another potential problem is that if z i = xi ; then (z 0i ^ ) can be highly collinear with xi ; so the second step OLS estimator will not be able to precisely estimate : Based this observation, it is typically recommended to …nd a valid exclusion restriction: a variable should be in z i which is not in xi : If this is valid, it will ensure that (z 0i ^ ) is not collinear with xi ; and hence improve the second stage estimator’s precision. Chapter 20 Panel Data A panel is a set of observations on individuals, collected over time. An observation is the pair fyit ; xit g; where the i subscript denotes the individual, and the t subscript denotes time. A panel may be balanced : fyit ; xit g : t = 1; :::; T ; i = 1; :::; n; or unbalanced : fyit ; xit g : For i = 1; :::; n; 20.1 t = ti ; :::; ti : Individual-E¤ects Model The standard panel data speci…cation is that there is an individual-speci…c e¤ect which enters linearly in the regression yit = x0it + ui + eit : The typical maintained assumptions are that the individuals i are mutually independent, that ui and eit are independent, that eit is iid across individuals and time, and that eit is uncorrelated with xit : OLS of yit on xit is called pooled estimation. It is consistent if E (xit ui ) = 0 (20.1) If this condition fails, then OLS is inconsistent. (20.1) fails if the individual-speci…c unobserved e¤ect ui is correlated with the observed explanatory variables xit : This is often believed to be plausible if ui is an omitted variable. If (20.1) is true, however, OLS can be improved upon via a GLS technique. In either event, OLS appears a poor estimation choice. Condition (20.1) is called the random e¤ ects hypothesis. It is a strong assumption, and most applied researchers try to avoid its use. 20.2 Fixed E¤ects This is the most common technique for estimation of non-dynamic linear panel regressions. The motivation is to allow ui to be arbitrary, and have arbitrary correlated with xi : The goal is to eliminate ui from the estimator, and thus achieve invariance. There are several derivations of the estimator. First, let 8 if i = j < 1 dij = ; : 0 else 296 CHAPTER 20. PANEL DATA 297 and 0 1 di1 B C di = @ ... A ; din 1 dummy vector with a “1” in the i0 th place. 0 u1 B .. u=@ . an n un Then note that Let 1 C A: ui = d0i u; and yit = x0it + d0i u + eit : (20.2) Observe that E (eit j xit ; di ) = 0; so (20.2) is a valid regression, with di as a regressor along with xi : OLS on (20.2) yields estimator ^ ; u ^ : Conventional inference applies. Observe that This is generally consistent. If xit contains an intercept, it will be collinear with di ; so the intercept is typically omitted from xit : Any regressor in xit which is constant over time for all individuals (e.g., their gender) will be collinear with di ; so will have to be omitted. There are n + k regression parameters, which is quite large as typically n is very large. Computationally, you do not want to actually implement conventional OLS estimation, as the parameter space is too large. OLS estimation of proceeds by the FWL theorem. Stacking the observations together: y = X + Du + e; then by the FWL theorem, ^ = = X 0 (I P D) X 1 X 0X X 0y 1 X 0 (I P D) y ; where y = y X = X D(D 0 D) D(D 0 D) 1 D0 y 1 D 0 X: Since the regression of yit on di is a regression onto individual-speci…c dummies, the predicted value from these regressions is the individual speci…c mean y i ; and the residual is the demean value yit = yit yi: The …xed e¤ects estimator ^ is OLS of yit on xit , the dependent variable and regressors in deviationfrom-mean form. CHAPTER 20. PANEL DATA 298 Another derivation of the estimator is to take the equation yit = x0it + ui + eit ; and then take individual-speci…c means by taking the average for the i0 th individual: ti ti ti 1 X 1 X 1 X yit = x0it + ui + eit Ti t=t Ti t=t Ti t=t i i i or y i = x0i + ui + ei : Subtracting, we …nd yit = xit0 + eit ; which is free of the individual-e¤ect ui : 20.3 Dynamic Panel Regression A dynamic panel regression has a lagged dependent variable yit = yit 1 + x0it + ui + eit : (20.3) This is a model suitable for studying dynamic behavior of individual agents. Unfortunately, the …xed e¤ects estimator is inconsistent, at least if T is held …nite as n ! 1: This is because the sample mean of yit 1 is correlated with that of eit : The standard approach to estimate a dynamic panel is to combine …rst-di¤erencing with IV or GMM. Taking …rst-di¤erences of (20.3) eliminates the individual-speci…c e¤ect: yit = yit 1 + However, if eit is iid, then it will be correlated with E ( yit 1 eit ) = E ((yit 1 yit 2 ) (eit x0it + yit eit 1 eit : (20.4) : 1 )) = E (yit 1 eit 1 ) = 2 e: So OLS on (20.4) will be inconsistent. But if there are valid instruments, then IV or GMM can be used to estimate the equation. Typically, we use lags of the dependent variable, two periods back, as yt 2 is uncorrelated with eit : Thus values of yit k ; k 2, are valid instruments. Hence a valid estimator of and is to estimate (20.4) by IV using yt 2 as an instrument for yt 1 (which is just identi…ed). Alternatively, GMM using yt 2 and yt 3 as instruments (which is overidenti…ed, but loses a time-series observation). A more sophisticated GMM estimator recognizes that for time-periods later in the sample, there are more instruments available, so the instrument list should be di¤erent for each equation. This is conveniently organized by the GMM principle, as this enables the moments from the di¤erent timeperiods to be stacked together to create a list of all the moment conditions. A simple application of GMM yields the parameter estimates and standard errors. Chapter 21 Nonparametric Density Estimation 21.1 Kernel Density Estimation d Let X be a random variable with continuous distribution F (x) and density f (x) = dx F (x): The goal is to estimate f (x) from a random sample (X ; :::; X g While F (x) can be estimated by 1 n P d ^ F (x) since F^ (x) is a step function. The the EDF F^ (x) = n 1 ni=1 1 (Xi x) ; we cannot de…ne dx standard nonparametric method to estimate f (x) is based on smoothing using a kernel. While we are typically interested in estimating the entire function f (x); we can simply focus on the problem where x is a speci…c …xed number, and then see how the method generalizes to estimating the entire function. De…nition 21.1.1 K(u) is a second-order kernel function if it is a symmetric zero-mean density function. Three common choices for kernels include the Normal u2 2 1 K(u) = p exp 2 the Epanechnikov K(u) = 3 4 1 15 16 1 u2 ; juj 1 0 juj > 1 and the Biweight or Quartic K(u) = u2 0 2 ; juj 1 juj > 1 In practice, the choice between these three rarely makes a meaningful di¤erence in the estimates. The kernel functions are used to smooth the data. The amount of smoothing is controlled by the bandwidth h > 0. Let 1 u Kh (u) = K : h h be the kernel K rescaled by the bandwidth h: The kernel density estimator of f (x) is n 1X f^(x) = Kh (Xi n i=1 299 x) : CHAPTER 21. NONPARAMETRIC DENSITY ESTIMATION 300 This estimator is the average of a set of weights. If a large number of the observations Xi are near x; then the weights are relatively large and f^(x) is larger. Conversely, if only a few Xi are near x; then the weights are small and f^(x) is small. The bandwidth h controls the meaning of “near”. Interestingly, f^(x) is a valid density. That is, f^(x) 0 for all x; and Z 1 Z 1 X n 1 ^ f (x)dx = Kh (Xi x) dx 1 1 n i=1 n Z 1X 1 = Kh (Xi x) dx n 1 i=1 n Z 1X 1 K (u) du = 1 = n 1 i=1 where the second-to-last equality makes the change-of-variables u = (Xi x)=h: We can also calculate the moments of the density f^(x): The mean is Z 1 n Z 1X 1 ^ xf (x)dx = xKh (Xi x) dx n 1 1 i=1 n Z 1X 1 = (Xi + uh) K (u) du n 1 i=1 Z 1 Z 1 n n 1X 1X = Xi K (u) du + h uK (u) du n n 1 1 1 n = i=1 n X i=1 Xi i=1 the sample mean of the Xi ; where the second-to-last equality used the change-of-variables u = (Xi x)=h which has Jacobian h: The second moment of the estimated density is Z 1 n Z 1X 1 2 2^ x f (x)dx = x Kh (Xi x) dx n 1 1 i=1 n Z 1X 1 = (Xi + uh)2 K (u) du n 1 i=1 Z 1 Z n n n X 2X 1X 2 1 2 1 2 K(u)du + u K (u) du = Xi + Xi h h n n n 1 1 i=1 = i=1 n 1X 2 Xi + h2 n i=1 2 K i=1 where 2 K = Z 1 u2 K (u) du 1 is the variance of the kernel. It follows that the variance of the density f^(x) is Z 1 1 x f^(x)dx 2 Z 1 1 n 2 xf^(x)dx = 1X 2 Xi + h2 n n 2 K i=1 = ^ 2 + h2 2 K Thus the variance of the estimated density is in‡ated by the factor h2 moment. 1X Xi n i=1 2 K !2 relative to the sample CHAPTER 21. NONPARAMETRIC DENSITY ESTIMATION 21.2 301 Asymptotic MSE for Kernel Estimates For …xed x and bandwidth h observe that Z EKh (X x) = Z = Z = 1 1 1 1 1 Kh (z x) f (z)dz Kh (uh) f (x + hu)hdu K (u) f (x + hu)du 1 The second equality uses the change-of variables u = (z x)=h: The last expression shows that the expected value is an average of f (z) locally about x: This integral (typically) is not analytically solvable, so we approximate it using a second order Taylor expansion of f (x + hu) in the argument hu about hu = 0; which is valid as h ! 0: Thus 1 f (x + hu) ' f (x) + f 0 (x)hu + f 00 (x)h2 u2 2 and therefore EKh (X Z 1 1 K (u) f (x) + f 0 (x)hu + f 00 (x)h2 u2 du 2 1 Z 1 Z 1 = f (x) K (u) du + f 0 (x)h K (u) udu 1 1 Z 1 1 K (u) u2 du + f 00 (x)h2 2 1 1 = f (x) + f 00 (x)h2 2K : 2 x) ' The bias of f^(x) is then n Bias(x) = Ef^(x) f (x) = 1X EKh (Xi n i=1 x) 1 f (x) = f 00 (x)h2 2 2 K: We see that the bias of f^(x) at x depends on the second derivative f 00 (x): The sharper the derivative, the greater the bias. Intuitively, the estimator f^(x) smooths data local to Xi = x; so is estimating a smoothed version of f (x): The bias results from this smoothing, and is larger the greater the curvature in f (x): We now examine the variance of f^(x): Since it is an average of iid random variables, using …rst-order Taylor approximations and the fact that n 1 is of smaller order than (nh) 1 var (x) = = ' = ' = 1 var (Kh (Xi x)) n 1 1 EKh (Xi x)2 (EKh (Xi x))2 n n Z 1 1 z x 2 1 f (z)dz K f (x)2 2 nh h n 1 Z 1 1 K (u)2 f (x + hu) du nh 1 Z f (x) 1 K (u)2 du nh 1 f (x) R(K) : nh CHAPTER 21. NONPARAMETRIC DENSITY ESTIMATION 302 R1 where R(K) = 1 K (u)2 du is called the roughness of K: Together, the asymptotic mean-squared error (AMSE) for …xed x is the sum of the approximate squared bias and approximate variance 1 AM SEh (x) = f 00 (x)2 h4 4 4 K + f (x) R(K) : nh A global measure of precision is the asymptotic mean integrated squared error (AMISE) AM ISEh = Z AM SEh (x)dx = h4 4 R(f 00 ) K 4 + R(K) : nh (21.1) R where R(f 00 ) = (f 00 (x))2 dx is the roughness of f 00 : Notice that the …rst term (the squared bias) is increasing in h and the second term (the variance) is decreasing in nh: Thus for the AMISE to decline with n; we need h ! 0 but nh ! 1: That is, h must tend to zero, but at a slower rate than n 1 : Equation (21.1) is an asymptotic approximation to the MSE. We de…ne the asymptotically optimal bandwidth h0 as the value which minimizes this approximate MSE. That is, h0 = argmin AM ISEh h It can be found by solving the …rst order condition d AM ISEh = h3 dh R(K) =0 nh2 4 00 K R(f ) yielding R(K) 4 R(f 00 ) K h0 = 1=5 n 1=2 : (21.2) This solution takes the form h0 = cn 1=5 where c is a function of K and f; but not of n: We thus say that the optimal bandwidth is of order O(n 1=5 ): Note that this h declines to zero, but at a very slow rate. In practice, how should the bandwidth be selected? This is a di¢ cult problem, and there is a large and continuing literature on the subject. The asymptotically optimal choice given in (21.2) depends on R(K); 2K ; and R(f 00 ): The …rst two are determined by the kernel function. Their values for the three functions introduced in the previous section are given here. K Gaussian Epanechnikov Biweight 2 K = R1 2 1u K 1 1=5 1=7 (u) du R1 R(K) = 1 K (u)2 du p 1/(2 ) 1=5 5=7 An obvious di¢ culty is that R(f 00 ) is unknown. A classic simple solution proposed by Silverman (1986) has come to be known as the reference bandwidth or Silverman’s Rule-of-Thumb. It uses formula (21.2) but replaces R(f 00 ) with ^ 5 R( 00 ); where is the N(0; 1) distribution and ^ 2 is an estimate of 2 = var(X): This choice for h gives an optimal rule when f (x) is normal, and gives a nearly optimal rule when f (x) is close to normal. The downside is that if the density is very far p from normal, the rule-of-thumb h can be quite ine¢ cient. We can calculate that R( 00 ) = 3= (8 ) : Together with the above table, we …nd the reference rules for the three kernel functions introduced earlier. Gaussian Kernel: hrule = 1:06^ n 1=5 Epanechnikov Kernel: hrule = 2:34^ n 1=5 Biweight (Quartic) Kernel: hrule = 2:78^ n 1=5 CHAPTER 21. NONPARAMETRIC DENSITY ESTIMATION 303 Unless you delve more deeply into kernel estimation methods the rule-of-thumb bandwidth is a good practical bandwidth choice, perhaps adjusted by visual inspection of the resulting estimate f^(x): There are other approaches, but implementation can be delicate. I now discuss some of these choices. The plug-in approach is to estimate R(f 00 ) in a …rst step, and then plug this estimate into the formula (21.2). This is more treacherous than may …rst appear, as the optimal h for estimation of the roughness R(f 00 ) is quite di¤erent than the optimal h for estimation of f (x): However, there are modern versions of this estimator work well, in particular the iterative method of Sheather and Jones (1991). Another popular choice for selection of h is cross-validation. This works by constructing an estimate of the MISE using leave-one-out estimators. There are some desirable properties of cross-validation bandwidths, but they are also known to converge very slowly to the optimal values. They are also quite ill-behaved when the data has some discretization (as is common in economics), in which case the cross-validation rule can sometimes select very small bandwidths leading to dramatically undersmoothed estimates. Fortunately there are remedies, which are known as smoothed cross-validation which is a close cousin of the bootstrap. Appendix A Matrix Algebra A.1 Notation A scalar a is a single number. A vector a is a k 1 list of numbers, typically arranged in a column. We write this as 0 1 a1 B a2 C B C a=B . C . @ . A ak Equivalently, a vector a is an element of Euclidean k space, written as a 2 Rk : If k = 1 then a is a scalar. A matrix A is a k r rectangular array of numbers, written as 2 3 a11 a12 a1r 6 a21 a22 a2r 7 6 7 A=6 . .. .. 7 4 .. . . 5 ak1 ak2 akr By convention aij refers to the element in the i0 th row and j 0 th column of A: If r = 1 then A is a column vector. If k = 1 then A is a row vector. If r = k = 1; then A is a scalar. A standard convention (which we will follow in this text whenever possible) is to denote scalars by lower-case italics (a); vectors by lower-case bold italics (a); and matrices by upper-case bold italics (A): Sometimes a matrix A is denoted by the symbol (aij ): A matrix can be written as a set of column vectors or as a set of row vectors. That is, 2 3 1 A= a1 a2 ar 6 2 7 6 7 =6 . 7 . 4 . 5 k where 2 6 6 ai = 6 4 are column vectors and j = a1i a2i .. . aki aj1 aj2 304 3 7 7 7 5 ajr APPENDIX A. MATRIX ALGEBRA 305 are row vectors. The transpose of a matrix, denoted A0 ; is obtained by Thus 2 a11 a21 ak1 6 a12 a22 ak2 6 A0 = 6 . . .. .. 4 .. . a1r a2r akr ‡ipping the matrix on its diagonal. 3 7 7 7 5 Alternatively, letting B = A0 ; then bij = aji . Note that if A is k r, then A0 is r k: If a is a k 1 vector, then a0 is a 1 k row vector. An alternative notation for the transpose of A is A> : A matrix is square if k = r: A square matrix is symmetric if A = A0 ; which requires aij = aji : A square matrix is diagonal if the o¤-diagonal elements are all zero, so that aij = 0 if i 6= j: A square matrix is upper (lower) diagonal if all elements below (above) the diagonal equal zero. An important diagonal matrix is the identity matrix, which has ones on the diagonal. The k k identity matrix is denoted as 3 2 1 0 0 6 0 1 0 7 7 6 Ik = 6 . . .. 7 : . . 4 . . . 5 0 0 1 A partitioned matrix takes the form 2 A11 A12 6 A21 A22 6 A=6 . .. 4 .. . Ak1 Ak2 A1r A2r .. . Akr where the Aij denote matrices, vectors and/or scalars. A.2 3 7 7 7 5 Matrix Addition If the matrices A = (aij ) and B = (bij ) are of the same order, we de…ne the sum A + B = (aij + bij ) : Matrix addition follows the communtative and associative laws: A+B = B+A A + (B + C) = (A + B) + C: A.3 Matrix Multiplication If A is k r and c is real, we de…ne their product as Ac = cA = (aij c) : If a and b are both k 1; then their inner product is a0 b = a1 b1 + a2 b2 + + ak bk = k X aj bj : j=1 Note that a0 b = b0 a: We say that two vectors a and b are orthogonal if a0 b = 0: APPENDIX A. MATRIX ALGEBRA 306 If A is k r and B is r s; so that the number of columns of A equals the number of rows of B; we say that A and B are conformable. In this event the matrix product AB is de…ned. Writing A as a set of row vectors and B as a set of column vectors (each of length r); then the matrix product is de…ned as 2 0 3 a1 6 a0 7 6 2 7 bs AB = 6 . 7 b1 b2 4 .. 5 a0k 3 2 0 a1 b1 a01 b2 a01 bs 6 a0 b1 a0 b2 a02 bs 7 2 7 6 2 = 6 . .. .. 7 : . 4 . . . 5 a0k b1 a0k b2 a0k bs Matrix multiplication is not commutative: in general AB 6= BA: However, it is associative and distributive: A (BC) = (AB) C A (B + C) = AB + AC An alternative way to write the matrix product is to use matrix partitions. For example, AB = = B 11 B 12 B 21 B 22 A11 A12 A21 A22 A11 B 11 + A12 B 21 A11 B 12 + A12 B 22 A21 B 11 + A22 B 21 A21 B 12 + A22 B 22 As another example, AB = A1 A2 Ar = A1 B 1 + A2 B 2 + r X = Aj B j 2 6 6 6 4 B1 B2 .. . Br + Ar B r : 3 7 7 7 5 j=1 An important property of the identity matrix is that if A is k r; then AI r = A and I k A = A: The k r matrix A, r k, is called orthogonal if A0 A = I r : A.4 Trace The trace of a k k square matrix A is the sum of its diagonal elements tr (A) = k X aii : i=1 Some straightforward properties for square matrices A and B and real c are tr (cA) = c tr (A) tr A0 = tr (A) tr (A + B) = tr (A) + tr (B) tr (I k ) = k: APPENDIX A. MATRIX ALGEBRA Also, for k r A and r 307 k B we have tr (AB) = tr (BA) : (A.1) Indeed, 2 6 6 tr (AB) = tr 6 4 k X = i=1 k X = a01 b1 a01 b2 a02 b1 a02 b2 .. .. . . 0 0 ak b1 ak b2 a01 bk a02 bk .. . a0k bk 3 7 7 7 5 a0i bi b0i ai i=1 = tr (BA) : A.5 Rank and Inverse The rank of the k r matrix (r k) A= a1 a2 ar is the number of linearly independent columns aj ; and is written as rank (A) : We say that A has full rank if rank (A) = r: A square k k matrix A is said to be nonsingular if it is has full rank, e.g. rank (A) = k: This means that there is no k 1 c 6= 0 such that Ac = 0: If a square k k matrix A is nonsingular then there exists a unique matrix k k matrix A 1 called the inverse of A which satis…es 1 AA =A 1 A = Ik: For non-singular A and C; some important properties include AA A A 1 1 1 0 = A 1 = 0 A A = Ik 1 (AC) 1 = C 1 (A + C) 1 = A 1 A 1 +C 1 1 (A + C) 1 = A 1 A 1 +C 1 1 A 1 C 1 A 1 Also, if A is an orthogonal matrix, then A 1 = A0 : Another useful result for non-singular A is known as the Woodbury matrix identity (A + BCD) In particular, for C = Morrison formula 1 =A 1 A 1 BC C + CDA 1 BC 1 CDA 1 : (A.2) 1; B = b and D = b0 for vector b we …nd what is known as the Sherman– A bb0 1 =A 1 + 1 b0 A 1 b 1 A 1 bb0 A 1 : (A.3) APPENDIX A. MATRIX ALGEBRA 308 The following fact about inverting partitioned matrices is quite useful. A11 A12 A21 A22 1 = A11 A12 A21 A22 = A1112 A2211 A21 A111 A1112 A12 A221 A2211 (A.4) where A11 2 = A11 A12 A221 A21 and A22 1 = A22 A21 A111 A12 : There are alternative algebraic representations for the components. For example, using the Woodbury matrix identity you can show the following alternative expressions A11 = A111 + A111 A12 A2211 A21 A111 A22 = A221 + A221 A21 A1112 A12 A221 A12 = A111 A12 A2211 A21 = 1 A221 A21 A112 Even if a matrix A does not possess an inverse, we can still de…ne the Moore-Penrose generalized inverse A as the matrix which satis…es AA A = A A AA = A AA is symmetric A A is symmetric For any matrix A; the Moore-Penrose generalized inverse A exists and is unique. For example, if A11 0 A= 0 0 then A = A.6 A11 0 0 0 : Determinant The determinant is a measure of the volume of a square matrix. While the determinant is widely used, its precise de…nition is rarely needed. However, we present the de…nition here for completeness. Let A = (aij ) be a general k k matrix . Let = (j1 ; :::; jk ) denote a permutation of (1; :::; k) : There are k! such permutations. There is a unique count of the number of inversions of the indices of such permutations (relative to the natural order (1; :::; k) ; and let " = +1 if this count is even and " = 1 if the count is odd. Then the determinant of A is de…ned as X det A = " a1j1 a2j2 akjk : For example, if A is 2 2; then the two permutations of (1; 2) are (1; 2) and (2; 1) ; for which "(1;2) = 1 and "(2;1) = 1. Thus det A = "(1;2) a11 a22 + "(2;1) a21 a12 = a11 a22 Some properties include det (A) = det (A0 ) det (cA) = ck det A a12 a21 : APPENDIX A. MATRIX ALGEBRA 309 det (AB) = (det A) (det B) det A 1 = (det A) A B C D det 1 = (det D) det A 1 BD C if det D 6= 0 det A 6= 0 if and only if A is nonsingular. If A is triangular (upper or lower), then det A = If A is orthogonal, then det A = A.7 1 Qk i=1 aii Eigenvalues The characteristic equation of a k k square matrix A is det (A I k ) = 0: The left side is a polynomial of degree k in so it has exactly k roots, which are not necessarily distinct and may be real or complex. They are called the latent roots or characteristic roots or eigenvalues of A. If i is an eigenvalue of A; then A i I k is singular so there exists a non-zero vector hi such that (A i I k ) hi = 0: The vector hi is called a latent vector or characteristic vector or eigenvector of A corresponding to i : We now state some useful properties. Let i and hi , i = 1; :::; k denote the k eigenvalues and eigenvectors of a square matrix A: Let be a diagonal matrix with the characteristic roots in the diagonal, and let H = [h1 hk ]: Q det(A) = ki=1 i P tr(A) = ki=1 i A is non-singular if and only if all its characteristic roots are non-zero. If A has distinct characteristic roots, there exists a nonsingular matrix P such that A = P 1 P and P AP 1 = . If A is symmetric, then A = H H 0 and H 0 AH = ; and the characteristic roots are all real. A = H H 0 is called the spectral decomposition of a matrix. When the eigenvalues of k k A are real they are written in decending order k : We also write min (A) = k = minf ` g and max (A) = 1 = maxf ` g. The characteristic roots of A 1 are 1 1 ; 1 2 ; ..., 1 k : The matrix H has the orthonormal properties H 0 H = I and HH 0 = I. H 1 = H 0 and (H 0 ) 1 =H 1 2 APPENDIX A. MATRIX ALGEBRA A.8 310 Positive De…niteness We say that a k k symmetric square matrix A is positive semi-de…nite if for all c 6= 0; 0: This is written as A 0: We say that A is positive de…nite if for all c 6= 0; c0 Ac > 0: This is written as A > 0: Some properties include: c0 Ac If A = G0 G for some matrix G; then A is positive semi-de…nite. (For any c 6= 0; c0 Ac = 0 0 where = Gc:) If G has full rank, then A is positive de…nite. If A is positive de…nite, then A is non-singular and A 1 exists. Furthermore, A 1 > 0: A > 0 if and only if it is symmetric and all its characteristic roots are positive. By the spectral decomposition, A = H H 0 where H 0 H = I and is diagonal with nonnegative diagonal elements. All diagonal elements of are strictly positive if (and only if) A > 0: If A > 0 then A 1 =H 1 H 0: If A 0 and rank (A) = r < k then A = H H 0 where A 1 1 1 generalized inverse, and = diag 1 ; 2 ; :::; k ; 0; :::; 0 is the Moore-Penrose If A 0 we can …nd a matrix B such that A = BB 0 : We call B a matrix square root of A: The matrix B need not be unique. One way to construct B is to use the spectral decomposition A = H H 0 where is diagonal, and then set B = H 1=2 : There is a unique root root B which is also positive semi-de…nite B 0: A square matrix A is idempotent if AA = A: If A is idempotent and symmetric then all its characteristic roots equal either zero or one and is thus positive semi-de…nite. To see this, note that we can write A = H H 0 where H is orthogonal and contains the r (real) characteristic roots. Then A = AA = H H 0 H H 0 = H 2 H 0 : By the uniqueness of the characteristic roots, we deduce that 2 = and 2i = i for i = 1; :::; r: Hence they must equal either 0 or 1. It follows that the spectral decomposition of idempotent A takes the form Ik r 0 H0 (A.5) A=H 0 0 with H 0 H = I k . Additionally, tr(A) = rank(A): If A is idempotent then I A is also idempotent. One useful fact is that A is idempotent then for any conformable vector c, 0 c (I c0 Ac c0 c (A.6) A) c 0 (A.7) cc To see this, note that c0 c = c0 Ac + c0 (I A) c: Since A and I A are idempotent, they are both positive semi-de…nite, so both c0 Ac and 0 c (I A) c are negative. Thus they must satisfy (A.6)-(A.7),. APPENDIX A. MATRIX ALGEBRA A.9 311 Matrix Calculus 1 and g(x) = g(x1 ; :::; xk ) : Rk ! R: The vector derivative is 1 0 @ @x1 g (x) @ C B .. g (x) = @ A . @x @ @xk g (x) Let x = (x1 ; :::; xk ) be k and @ g (x) = @x0 Some properties are now summarized. @ @x @ @x0 @ @x (a0 x) = @ @xk g (x) : (x0 a) = a (Ax) = A (x0 Ax) = (A + A0 ) x @2 @x@x0 A.10 @ @x @ @x1 g (x) (x0 Ax) = A + A0 Kronecker Products and the Vec Operator Let A = [a1 a2 an ] be m n: The vec of A; denoted by vec (A) ; is the mn 0 1 a1 B a2 C B C vec (A) = B . C : @ .. A 1 vector an Let A = (aij ) be an m n matrix and let B be any matrix. The Kronecker product of A and B; denoted A B; is the matrix 2 3 a11 B a12 B a1n B 6 a21 B a22 B a2n B 7 6 7 A B=6 7: .. .. .. 4 5 . . . am1 B am2 B amn B Some important properties are now summarized. These results hold for matrices for which all matrix multiplications are conformable. (A + B) (A A C=A B) (C (B (A tr (A D) = AC C) = (A B)0 = A0 C BD B) C B0 B) = tr (A) tr (B) If A is m (A C +B B) m and B is n 1 =A 1 B) = (det (A))n (det (B))m 1 B If A > 0 and B > 0 then A vec (ABC) = (C 0 n; det(A B>0 A) vec (B) 0 tr (ABCD) = vec (D 0 ) (C 0 A) vec (B) APPENDIX A. MATRIX ALGEBRA A.11 312 Vector and Matrix Norms The Euclidean norm of an m 1 vector a is 1=2 a0 a kak = m X = a2i i=1 The Frobenius norm of an m !1=2 : n matrix A is kAk = kvec (A)k 1=2 tr A0 A 0 11=2 m X n X = @ a2ij A : = i=1 j=1 If an m m matrix A is symmetric with eigenvalues ` ; ` = 1; :::; m; then !1=2 m X 2 : kAk = ` `=1 To see this, by the spectral decomposition A = H H 0 with H 0 H = I and = diagf so !1=2 m X 1=2 0 0 1=2 2 : kAk = tr H H H H = (tr ( )) = ` 1 ; :::; m g; (A.8) `=1 A useful calculation is for any m 1 vectors a and b, using (A.1), 0 1=2 ab0 = tr ba0 ab 1=2 = b0 ba0 a = kak kbk and in particular aa0 = kak2 : (A.9) The are other matrix norms. Another norm of frequent use is the spectral norm kAkS = where A.12 max (B) max A0 A 1=2 denotes the largest eigenvalue of the matrix B. Matrix Inequalities Schwarz Inequality: For any m 1 vectors a and b, a0 b Schwarz Matrix Inequality: For any m kak kbk : n matrices A and B; A0 B Triangle Inequality: For any m (A.10) kAk kBk : (A.11) n matrices A and B; kA + Bk kAk + kBk : (A.12) APPENDIX A. MATRIX ALGEBRA Trace Inequality. For any m 313 m matrices A and B such that A is symmetric and B tr (AB) max (A) tr (B) where max (A) is the largest eigenvalue of A. Quadratic Inequality. For any m 1 b and m b0 Ab Norm Inequality. For any m 0 (A.14) b m matrices A and B such that A min (B) ` (AB) 0 and B 0 max (A) kBk where max (A) is the largest eigenvalue of A. Eigenvalue Product Inequaly. For any k ` (AB) are real and satisfy min (A) (A.13) m symmetric matrix A max (A) b kABk = ` 0 (A.15) k matrices A 0 and B A1=2 BA1=2 max (A) 0; the eigenvalues max (B) (A.16) (Zhang and Zhang, 2006, Corollary 11) Jensen’ s Inequality. If g( ) : R ! R is convex, then for any non-negative weights aj such that Pm j=1 aj = 1; and any real numbers xj 0 1 m m X X g@ aj xj A aj g (xj ) : (A.17) j=1 In particular, setting aj = 1=m; then 0 j=1 1 m X 1 g@ xj A m m 1 X g (xj ) : m j=1 (A.18) j=1 Loève’s cr Inequality. For r > 0; m X r aj cr j=1 m X jaj jr (A.19) 2a0 a + 2b0 b (A.20) j=1 where cr = 1 when r 1 and cr = mr 1 when r c2 Inequality. For any m 1 vectors a and b, 1: (a + b)0 (a + b) Proof of Schwarz Inequality: First, suppose that kbk = 0: Then b = 0 and both ja0 bj = 0 and 1 0 kak kbk = 0 so the inequality is true. Second, suppose that kbk > 0 and de…ne c = a b b0 b b a: 0 Since c is a vector, c c 0: Thus 0 c0 c = a0 a Rearranging, this implies that a0 b 2 a0 b a0 a Taking the square root of each side yields the result. 2 = b0 b : b0 b : APPENDIX A. MATRIX ALGEBRA 314 Proof of Schwarz Matrix Inequality: Partition A = [a1 ; :::; an ] and B = [b1 ; :::; bn ]. Then by partitioned matrix multiplication, the de…nition of the matrix Euclidean norm and the Schwarz inequality A0 B a01 b1 a01 b2 a02 b1 a02 b2 .. .. . . = .. . ka1 k kb1 k ka1 k kb2 k ka2 k kb1 k ka2 k kb2 k .. .. .. . . . 0 11=2 n X n X @ = kai k2 kbj k2 A i=1 j=1 = n X i=1 2 kai k !1=2 n X i=1 2 kbi k !1=2 0 11=2 0 11=2 n X m n X m X X = @ a2ji A @ kbji k2 A i=1 j=1 i=1 j=1 = kAk kBk Proof of Triangle Inequality: Let a = vec (A) and b = vec (B) . Then by the de…nition of the matrix norm and the Schwarz Inequality kA + Bk2 = ka + bk2 = a0 a + 2a0 b + b0 b a0 a + 2 a0 b + b0 b kak2 + 2 kak kbk + kbk2 = (kak + kbk)2 = (kAk + kBk)2 Proof of Trace Inequality. By the spectral decomposition for symmetric matices, A = H H 0 where has the eigenvalues j of A on the diagonal and H is orthonormal. De…ne C = H 0 BH which has non-negative diagonal elements Cjj since B is positive semi-de…nite. Then tr (AB) = tr ( C) = m X j Cjj j=1 where the inequality uses the fact that Cjj max j j m X Cjj = max (A) tr (C) j=1 0: But note that tr (C) = tr H 0 BH = tr HH 0 B = tr (B) since H is orthonormal. Thus tr (AB) max (A) tr (B) as stated. Proof of Quadratic Inequality: In the Trace Inequality set B = bb0 and note tr (AB) = b0 Ab and tr (B) = b0 b: APPENDIX A. MATRIX ALGEBRA 315 Proof of Norm Inequality. Using the Trace Inequality kABk = (tr (BAAB))1=2 = (tr (AABB))1=2 1=2 max (AA) tr (BB)) ( = max (A) kBk : The …nal equality holds since A 0 implies that max (AA) = max (A)2 : Proof of Jensen’s Inequality (A.17). By the de…nition of convexity, for any g ( x1 + (1 This implies ) x2 ) g (x1 ) + (1 0 1 0 m X g@ aj xj A = g @a1 g (x1 ) + (1 j=1 a1 g (x1 )+(1 a1 ) and 0 Pm j=2 bj a1 ) @b2 g(x2 ) + (1 where cj = bj =(1 a1 ) 0 a1 ) g @ a1 g (x1 ) + (1 where bj = aj =(1 ) g (x2 ) : m X j=2 m X j=2 aj 1 a1 1 2 [0; 1] (A.21) 1 xj A bj xj A : = 1: By another application of (A.21) this is bounded by 0 11 0 1 m m X X b2 )g @ cj xj AA = a1 g (x1 )+a2 g(x2 )+(1 a1 ) (1 b2 )g @ cj xj A j=2 j=2 b2 ): By repeated application of (A.21) we obtain (A.17). Proof of Loève’s cr Inequality. For r 1 this is simply a rewriting of the …nite form Jensen’s Pm inequality (A.18) with g(u) = ur : For r < 1; de…ne bj = jaj j = bj 1 j=1 jaj j : The facts that 0 r and r < 1 imply bj bj and thus m m X X 1= bj brj j=1 which implies j=1 0 1r m X @ jaj jA m X j=1 The proof is completed by observing that 0 1r m X @ aj A j=1 0 @ j=1 m X j=1 jaj jr : 1r jaj jA : Proof of c2 Inequality. By the cr inequality, (aj + bj )2 (a + b)0 (a + b) = m X 2a2j + 2b2j . Thus (aj + bj )2 j=1 m X 2 j=1 0 a2j +2 m X j=1 0 = 2a a + 2b b b2j Appendix B Probability B.1 Foundations The set S of all possible outcomes of an experiment is called the sample space for the experiment. Take the simple example of tossing a coin. There are two outcomes, heads and tails, so we can write S = fH; T g: If two coins are tossed in sequence, we can write the four outcomes as S = fHH; HT; T H; T T g: An event A is any collection of possible outcomes of an experiment. An event is a subset of S; including S itself and the null set ;: Continuing the two coin example, one event is A = fHH; HT g; the event that the …rst coin is heads. We say that A and B are disjoint or mutually exclusive if A \ B = ;: For example, the sets fHH; HT g and fT Hg are disjoint. Furthermore, if the sets A1 ; A2 ; ::: are pairwise disjoint and [1 i=1 Ai = S; then the collection A1 ; A2 ; ::: is called a partition of S: The following are elementary set operations: Union: A [ B = fx : x 2 A or x 2 Bg: Intersection: A \ B = fx : x 2 A and x 2 Bg: Complement: Ac = fx : x 2 = Ag: The following are useful properties of set operations. Communtatitivity: A [ B = B [ A; A \ B = B \ A: Associativity: A [ (B [ C) = (A [ B) [ C; A \ (B \ C) = (A \ B) \ C: Distributive Laws: A \ (B [ C) = (A \ B) [ (A \ C) ; A [ (B \ C) = (A [ B) \ (A [ C) : c c c c DeMorgan’s Laws: (A [ B) = A \ B ; (A \ B) = Ac [ B c : A probability function assigns probabilities (numbers between 0 and 1) to events A in S: This is straightforward when S is countable; when S is uncountable we must be somewhat more careful: A set B is called a sigma algebra (or Borel …eld) if ; 2 B , A 2 B implies Ac 2 B, and A1 ; A2 ; ::: 2 B implies [1 i=1 Ai 2 B. A simple example is f;; Sg which is known as the trivial sigma algebra. For any sample space S; let B be the smallest sigma algebra which contains all of the open sets in S: When S is countable, B is simply the collection of all subsets of S; including ; and S: When S is the real line, then B is the collection of all open and closed intervals. We call B the sigma algebra associated with S: We only de…ne probabilities for events contained in B. We now can give the axiomatic de…nition of probability. Given S and B, a probability function Pr satis…es Pr(S) 0 for all A 2 B, and if A1 ; A2 ; ::: 2 B are pairwise disjoint, then P1= 1; Pr(A) Pr ([1 A ) = Pr(A ): i i i=1 i=1 Some important properties of the probability function include the following Pr (;) = 0 Pr(A) 1 Pr (Ac ) = 1 Pr(A) 316 APPENDIX B. PROBABILITY Pr (B \ Ac ) = Pr(B) 317 Pr(A \ B) Pr (A [ B) = Pr(A) + Pr(B) If A B then Pr(A) Pr(A \ B) Pr(B) Bonferroni’s Inequality: Pr(A \ B) Boole’s Inequality: Pr (A [ B) Pr(A) + Pr(B) 1 Pr(A) + Pr(B) For some elementary probability models, it is useful to have simple rules to count the number of objects in a set. These counting rules are facilitated by using the binomial coe¢ cients which are de…ned for nonnegative integers n and r; n r; as n r = n! : r! (n r)! When counting the number of objects in a set, there are two important distinctions. Counting may be with replacement or without replacement. Counting may be ordered or unordered. For example, consider a lottery where you pick six numbers from the set 1, 2, ..., 49. This selection is without replacement if you are not allowed to select the same number twice, and is with replacement if this is allowed. Counting is ordered or not depending on whether the sequential order of the numbers is relevant to winning the lottery. Depending on these two distinctions, we have four expressions for the number of objects (possible arrangements) of size r from n objects. Without Replacement n! (n r)! n r Ordered Unordered With Replacement nr n+r 1 r In the lottery example, if counting is unordered and without replacement, the number of potential combinations is 49 6 = 13; 983; 816. If Pr(B) > 0 the conditional probability of the event A given the event B is Pr (A j B) = Pr (A \ B) : Pr(B) For any B; the conditional probability function is a valid probability function where S has been replaced by B: Rearranging the de…nition, we can write Pr(A \ B) = Pr (A j B) Pr(B) which is often quite useful. We can say that the occurrence of B has no information about the likelihood of event A when Pr (A j B) = Pr(A); in which case we …nd Pr(A \ B) = Pr (A) Pr(B) (B.1) We say that the events A and B are statistically independent when (B.1) holds. Furthermore, we say that the collection of events A1 ; :::; Ak are mutually independent when for any subset fAi : i 2 Ig; ! \ Y Pr Ai = Pr (Ai ) : i2I i2I Theorem 3 (Bayes’ Rule). For any set B and any partition A1 ; A2 ; ::: of the sample space, then for each i = 1; 2; ::: Pr (B j Ai ) Pr(Ai ) Pr (Ai j B) = P1 j=1 Pr (B j Aj ) Pr(Aj ) APPENDIX B. PROBABILITY B.2 318 Random Variables A random variable X is a function from a sample space S into the real line. This induces a new sample space – the real line – and a new probability function on the real line. Typically, we denote random variables by uppercase letters such as X; and use lower case letters such as x for potential values and realized values. (This is in contrast to the notation adopted for most of the textbook.) For a random variable X we de…ne its cumulative distribution function (CDF) as F (x) = Pr (X x) : (B.2) Sometimes we write this as FX (x) to denote that it is the CDF of X: A function F (x) is a CDF if and only if the following three properties hold: 1. limx! 1 F (x) = 0 and limx!1 F (x) = 1 2. F (x) is nondecreasing in x 3. F (x) is right-continuous We say that the random variable X is discrete if F (x) is a step function. In the latter case, the range of X consists of a countable set of real numbers 1 ; :::; r : The probability function for X takes the form Pr (X = j ) = j ; j = 1; :::; r (B.3) Pr where 0 1 and j=1 j = 1. j We say that the random variable X is continuous if F (x) is continuous in x: In this case Pr(X = ) = 0 for all 2 R so the representation (B.3) is unavailable. Instead, we represent the relative probabilities by the probability density function (PDF) f (x) = so that F (x) = and Pr (a X Z d F (x) dx x f (u)du 1 b) = Z b f (u)du: a These expressions only make sense if F (x) is di¤erentiable. While there are examples of continuous random variables which do not possess a PDF, these cases are unusualRand are typically ignored. 1 A function f (x) is a PDF if and only if f (x) 0 for all x 2 R and 1 f (x)dx: B.3 Expectation For any measurable real function g; we de…ne the mean or expectation Eg(X) as follows. If X is discrete, r X Eg(X) = g( j ) j ; j=1 and if X is continuous Eg(X) = The latter is well de…ned and …nite if Z 1 1 Z 1 g(x)f (x)dx: 1 jg(x)j f (x)dx < 1: (B.4) APPENDIX B. PROBABILITY 319 If (B.4) does not hold, evaluate I1 = Z g(x)f (x)dx g(x)>0 I2 = Z g(x)f (x)dx g(x)<0 If I1 = 1 and I2 < 1 then we de…ne Eg(X) = 1: If I1 < 1 and I2 = 1 then we de…ne Eg(X) = 1: If both I1 = 1 and I2 = 1 then Eg(X) is unde…ned. Since E (a + bX) = a + bEX; we say that expectation is a linear operator. For m > 0; we de…ne the m0 th moment of X as EX m and the m0 th central moment as E (X EX)m : 2 : We Two special moments are the mean = EX and variance 2 = E (X )2 = EX 2 p 2 2 call = the standard deviation of X: We can also write = var(X). For example, this 2 allows the convenient expression var(a + bX) = b var(X): The moment generating function (MGF) of X is M ( ) = E exp ( X) : The MGF does not necessarily exist. However, when it does and E jXjm < 1 then dm M( ) d m = E (X m ) =0 which is why it is called the moment generating function. More generally, the characteristic function (CF) of X is where i = p C( ) = E exp (i X) 1 is the imaginary unit. The CF always exists, and when E jXjm < 1 dm C( ) d m The Lp norm, p = im E (X m ) : =0 1; of the random variable X is kXkp = (E jXjp )1=p : B.4 Gamma Function The gamma function is de…ned for > 0 as Z 1 ( )= x 1 exp ( x) : 0 It satis…es the property (1 + ) = ( ) so for positive integers n; (n) = (n 1)! Special values include (1) = 1 and 1 = 1=2 : 2 Sterling’s formula is an expansion for the its logarithm log ( ) = 1 log(2 ) + 2 1 2 log z+ 1 12 1 360 3 + 1 1260 5 + APPENDIX B. PROBABILITY B.5 320 Common Distributions For reference, we now list some important discrete distribution function. Bernoulli Pr (X = x) = px (1 p)1 x ; x = 0; 1; x ; x = 0; 1; :::; n; 0 p 1 EX = p var(X) = p(1 p) Binomial n x p (1 x EX = np p)n Pr (X = x) = var(X) = np(1 0 p p 1 1 p) Geometric Pr (X = x) = p(1 p)x 1 EX = p 1 p var(X) = p2 1 ; x = 1; 2; :::; 0 Multinomial n! px1 px2 x1 !x2 ! xm ! 1 2 = n; pxmm ; Pr (X1 = x1 ; X2 = x2 ; :::; Xm = xm ) = x1 + + xm p1 + + pm = 1 EXi = pi var(Xi ) = npi (1 cov (Xi ; Xj ) = pi ) npi pj Negative Binomial Pr (X = x) = EX = var(X) = (r + x) r p (1 x! (r) r (1 p) p r (1 p) p2 p)x 1 ; x = 0; 1; 2; :::; 0 Poisson Pr (X = x) = exp ( ) x! x ; x = 0; 1; 2; :::; EX = var(X) = We now list some important continuous distributions. >0 p 1 APPENDIX B. PROBABILITY 321 Beta ( + ) x ( ) ( ) f (x) = = 1 (1 1 x) ; 0 x 1; > 0; >0 + var(X) = + 1) ( + )2 ( + Cauchy 1 ; (1 + x2 ) EX = 1 f (x) = 1<x<1 var(X) = 1 Exponential 1 f (x) = x exp ; 0 x < 1; >0 EX = 2 var(X) = Logistic exp ( x) ; (1 + exp ( x))2 EX = 0 1 < x < 1; f (x) = 2 var(X) = 3 Lognormal f (x) = (log x 2 1 p exp 2 x 2 =2 var(X) = exp 2 + 2 2 EX = exp + )2 2 ! exp 2 + ; 0 x < 1; >0 2 Pareto f (x) = x EX = +1 ; 1 x < 1; ; > 0; >1 2 var(X) = ( 1)2 ( 2) ; >2 Uniform f (x) = EX = var(X) = 1 ; b a a+b 2 (b a)2 12 a x b >0 APPENDIX B. PROBABILITY 322 Weibull f (x) = x EX = 1= var(X) = 2= 1 x exp ; 0 x < 1; > 0; >0 1 1+ 1+ 2 2 1+ ; 0 1 Gamma f (x) = 1 ( ) 1 x x exp x < 1; > 0; >0 EX = 2 var(X) = Chi-Square 1 xr=2 (r=2)2r=2 f (x) = 1 exp x ; 2 0 x < 1; r>0 EX = r var(X) = 2r Normal f (x) = 1 p 2 exp )2 (x 2 2 ! ; 1 < x < 1; 1< < 1; 2 >0 EX = var(X) = 2 Student t f (x) = p r+1 2 x2 1+ r ( r+1 2 ) ; r r 2 EX = 0 if r > 1 r var(X) = if r > 2 r 2 B.6 1 < x < 1; r>0 Multivariate Random Variables A pair of bivariate random variables (X; Y ) is a function from the sample space into R2 : The joint CDF of (X; Y ) is F (x; y) = Pr (X x; Y y) : If F is continuous, the joint probability density function is f (x; y) = @2 F (x; y): @x@y For a Borel measurable set A 2 R2 ; Pr ((X; Y ) 2 A) = Z Z f (x; y)dxdy A APPENDIX B. PROBABILITY 323 For any measurable function g(x; y); Eg(X; Y ) = Z 1 1 Z 1 g(x; y)f (x; y)dxdy: 1 The marginal distribution of X is FX (x) = Pr(X = = x) lim F (x; y) y!1 Z x Z 1 1 f (x; y)dydx 1 so the marginal density of X is fX (x) = d FX (x) = dx Z 1 f (x; y)dy: 1 Similarly, the marginal density of Y is fY (y) = Z 1 f (x; y)dx: 1 The random variables X and Y are de…ned to be independent if f (x; y) = fX (x)fY (y): Furthermore, X and Y are independent if and only if there exist functions g(x) and h(y) such that f (x; y) = g(x)h(y): If X and Y are independent, then Z Z E (g(X)h(Y )) = g(x)h(y)f (y; x)dydx Z Z = g(x)h(y)fY (y)fX (x)dydx Z Z = g(x)fX (x)dx h(y)fY (y)dy = Eg (X) Eh (Y ) : (B.5) if the expectations exist. For example, if X and Y are independent then E(XY ) = EXEY: Another implication of (B.5) is that if X and Y are independent and Z = X + Y; then MZ ( ) = E exp ( (X + Y )) = E (exp ( X) exp ( Y )) = E exp 0 0 X E exp Y = MX ( )MY ( ): (B.6) The covariance between X and Y is cov(X; Y ) = XY = E ((X EX) (Y EY )) = EXY The correlation between X and Y is corr (X; Y ) = XY = XY X Y : EXEY: APPENDIX B. PROBABILITY 324 The Cauchy-Schwarz Inequality implies that j XY j 1: (B.7) The correlation is a measure of linear dependence, free of units of measurement. If X and Y are independent, then XY = 0 and XY = 0: The reverse, however, is not true. For example, if EX = 0 and EX 3 = 0, then cov(X; X 2 ) = 0: A useful fact is that var (X + Y ) = var(X) + var(Y ) + 2 cov(X; Y ): An implication is that if X and Y are independent, then var (X + Y ) = var(X) + var(Y ); the variance of the sum is the sum of the variances. A k 1 random vector X = (X1 ; :::; Xk )0 is a function from S to Rk : Let x = (x1 ; :::; xk )0 denote a vector in Rk : (In this Appendix, we use bold to denote vectors. Bold capitals X are random vectors and bold lower case x are nonrandom vectors. Again, this is in distinction to the notation used in the bulk of the text) The vector X has the distribution and density functions F (x) = Pr(X x) @k F (x): f (x) = @x1 @xk For a measurable function g : Rk ! Rs ; we de…ne the expectation Z Eg(X) = g(x)f (x)dx Rk where the symbol dx denotes dx1 dxk : In particular, we have the k 1 multivariate mean = EX and k k covariance matrix = E (X ) (X = EXX 0 0 )0 If the elements of X are mutually independent, then is a diagonal matrix and ! k k X X var Xi = var (X i ) i=1 B.7 i=1 Conditional Distributions and Expectation The conditional density of Y given X = x is de…ned as fY jX (y j x) = f (x; y) fX (x) APPENDIX B. PROBABILITY 325 if fX (x) > 0: One way to derive this expression from the de…nition of conditional probability is fY jX (y j x) = = = = = = @ lim Pr (Y y j x X x + ") @y "!0 @ Pr (fY yg \ fx X x + "g) lim "!0 @y Pr(x X x + ") @ F (x + "; y) F (x; y) lim @y "!0 FX (x + ") FX (x) @ lim @y "!0 @ @x F (x + "; y) fX (x + ") @2 @x@y F (x; y) fX (x) f (x; y) : fX (x) The conditional mean or conditional expectation is the function Z 1 m(x) = E (Y j X = x) = yfY jX (y j x) dy: 1 The conditional mean m(x) is a function, meaning that when X equals x; then the expected value of Y is m(x): Similarly, we de…ne the conditional variance of Y given X = x as 2 (x) = var (Y j X = x) = E (Y m(x))2 j X = x = E Y2 jX =x m(x)2 : Evaluated at x = X; the conditional mean m(X) and conditional variance 2 (X) are random variables, functions of X: We write this as E(Y j X) = m(X) and var (Y j X) = 2 (X): For example, if E (Y j X = x) = + 0 x; then E (Y j X) = + 0 X; a transformation of X: The following are important facts about conditional expectations. Simple Law of Iterated Expectations: E (E (Y j X)) = E (Y ) (B.8) Proof : E (E (Y j X)) = E (m(X)) Z 1 = m(x)fX (x)dx 1 Z 1Z 1 = yfY jX (y j x) fX (x)dydx 1 1 Z 1 Z 1 = yf (y; x) dydx 1 = E(Y ): 1 Law of Iterated Expectations: E (E (Y j X; Z) j X) = E (Y j X) (B.9) APPENDIX B. PROBABILITY 326 Conditioning Theorem. For any function g(x); E (g(X)Y j X) = g (X) E (Y j X) (B.10) Proof : Let h(x) = E (g(X)Y j X = x) Z 1 g(x)yfY jX (y j x) dy = 1 Z 1 = g(x) yfY jX (y j x) dy 1 = g(x)m(x) where m(x) = E (Y j X = x) : Thus h(X) = g(X)m(X), which is the same as E (g(X)Y j X) = g (X) E (Y j X) : B.8 Transformations Suppose that X 2 Rk with continuous distribution function FX (x) and density fX (x): Let Y = g(X) where g(x) : Rk ! Rk is one-to-one, di¤erentiable, and invertible. Let h(y) denote the inverse of g(x). The Jacobian is @ h(y) : @y 0 J(y) = det Consider the univariate case k = 1: If g(x) is an increasing function, then g(X) if X h(Y ); so the distribution function of Y is FY (y) = Pr (g(X) = Pr (X Y if and only y) h(Y )) = FX (h(Y )) : Taking the derivative, the density of Y is fY (y) = d d FY (y) = fX (h(Y )) h(y): dy dy If g(x) is a decreasing function, then g(X) Y if and only if X FY (y) = Pr (g(X) h(Y ); so y) = 1 Pr (X h(Y )) = 1 FX (h(Y )) and the density of Y is fY (y) = fX (h(Y )) d h(y): dy We can write these two cases jointly as fY (y) = fX (h(Y )) jJ(y)j : (B.11) This is known as the change-of-variables formula. This same formula (B.11) holds for k > 1; but its justi…cation requires deeper results from analysis. As one example, take the case X U [0; 1] and Y = log(X). Here, g(x) = log(x) and h(y) = exp( y) so the Jacobian is J(y) = exp(y): As the range of X is [0; 1]; that for Y is [0,1): Since fX (x) = 1 for 0 x 1 (B.11) shows that fY (y) = exp( y); an exponential density. 0 y 1; APPENDIX B. PROBABILITY B.9 327 Normal and Related Distributions The standard normal density is x2 2 1 (x) = p exp 2 ; 1 < x < 1: It is conventional to write X N (0; 1) ; and to denote the standard normal density function by (x) and its distribution function by (x): The latter has no closed-form solution. The normal density has all moments …nite. Since it is symmetric about zero all odd moments are zero. By iterated integration by parts, we can also show that EX 2 = 1 and EX 4 = 3: In fact, for any positive integer m, EX 2m = (2m 1)!! = (2m 1) (2m 3) 1: Thus EX 4 = 3; EX 6 = 15; EX 8 = 105; 10 and EX = 945: If Z is standard normal and X = + Z; then using the change-of-variables formula, X has density ! 1 (x )2 f (x) = p exp ; 1 < x < 1: 2 2 2 which is the univariate normal density. The mean and variance of the distribution are 2 ; and it is conventional to write X N ; 2 . For x 2 Rk ; the multivariate normal density is f (x) = 1 (2 ) k=2 1=2 det ( ) (x exp )0 1 (x ) 2 and x 2 Rk : ; The mean and covariance matrix of the distribution are and ; and it is conventional to write X N ( ; ). 0 The MGF and CF of the multivariate normal are exp 0 + 0 =2 and exp i 0 =2 ; respectively. If X 2 Rk is multivariate normal and the elements of X are mutually uncorrelated, then = diagf 2j g is a diagonal matrix. In this case the density function can be written as f (x) = = 1 (2 )k=2 k Y j=1 exp 1 1=2 exp j 2 2 1) = 1 + + (xk 2 2 k) = k 2 k 1 (2 ) (x1 xj j 2 j 2 ! 2 !! which is the product of marginal univariate normal densities. This shows that if X is multivariate normal with uncorrelated elements, then they are mutually independent. Theorem B.9.1 If X N (a + B ; B B 0 ) : Theorem B.9.2 Let X freedom, written 2r . Theorem B.9.3 If Z N ( ; ) and Y = a + BX with B an invertible matrix, then Y N (0; I r ) : Then Q = X 0 X is distributed chi-square with r degrees of N (0; A) with A > 0; q Theorem B.9.4 Let Z N (0; 1) and Q as student’s t with r degrees of freedom. 2 r q; then Z 0 A 1 Z 2: q be independent. Then Tr = Z= p Q=r is distributed APPENDIX B. PROBABILITY 328 Proof of Theorem B.9.1. By the change-of-variables formula, the density of Y = a + BX is ! 0 1 (y 1 Y) Y) Y (y ; y 2 Rk : f (y) = exp k=2 1=2 2 (2 ) det ( Y ) where Y = a+B and Y 1=2 = B B 0 ; where we used the fact that det (B B 0 ) = det ( )1=2 det (B) : Proof of Theorem B.9.2. First, suppose a random variable Q is distributed chi-square with r degrees of freedom. It has the MGF Z 1 1 E exp (tQ) = xr=2 1 exp (tx) exp ( x=2) dy = (1 2t) r=2 r r=2 2 0 2 R1 where the second equality uses the fact that 0 y a 1 exp ( by) dy = b a (a); which can be found by applying change-of-variables to the gamma function. Our goal is to calculate the MGF of r=2 2. Q = X 0 X and show that it equals (1 2t) P ; which will establish that Q r r 0 2 Note that we can write Q = X X = j=1 Zj where the Zj are independent N (0; 1) : The distribution of each of the Zj2 is Pr Zj2 p = 2 Pr (0 Zj y) Z py x2 1 p exp = 2 dx 2 2 0 Z y 1 s = ds s 1=2 exp 1 1=2 2 0 2 2 y using the change–of-variables s = x2 and the fact 1 f1 (x) = which is the h Since (1 2t) 2 1 1 2 21=2 x p 1 2 = 1=2 exp : Thus the density of Zj2 is x 2 and by our above calculation has the MGF of E exp tZj2 = (1 the Zj2 ir 1=2 2t) are mutually independent, (B.6) implies that the MGF of Q = = (1 2t) r=2 2 r ; which is the MGF of the density as desired: 1=2 : Pr 2 j=1 Zj is Proof of Theorem B.9.3. The fact that A > 0 means that we can write A = CC 0 where C is non-singular. Then A 1 = C 10 C 1 and C 1 Z N 0; C 1 AC 10 = N 0; C C 1 1 CC 0 C Thus Z 0A 1 Z = Z 0C 10 Z= C 1 Z 0 C 10 1 Z = N (0; I q ) : 2 q: Proof of Theorem B.9.4. Using the simple law of iterated expectations, Tr has distribution APPENDIX B. PROBABILITY 329 function Z p Q=r ( r F (x) = Pr = E Z x " = E Pr Z = E x r x ! ) Q r r x Q r ! Q jQ r !# Thus its density is ! Q x r r !r ! Q Q x r r r d f (x) = E dx = E = Z 1 0 = p qx2 2r 1 p exp 2 r+1 2 r r 2 x2 1+ r r q r 1 r 2 2r=2 q r=2 1 ! exp ( q=2) dq ( r+1 2 ) which is that of the student t with r degrees of freedom. B.10 Inequalities Jensen’s Inequality. If g( ) : Rm ! R is convex, then for any random vector x for which E kxk < 1 and E jg (x)j < 1; g(E(x)) E (g (x)) : (B.12) Conditional Jensen’s Inequality. If g( ) : Rm ! R is convex, then for any random vectors (y; x) for which E kyk < 1 and E kg (y)k < 1; g(E(y j x)) E (g (y) j x) : Conditional Expectation Inequality. For any r E jE(y j x)jr (B.13) such that E jyjr < 1; then E jyjr < 1: (B.14) Expectation Inequality. For any random matrix Y for which E kY k < 1; kE(Y )k E kY k : Hölder’s Inequality. If p > 1 and q > 1 and p1 + 1q = 1; then for any random m and Y; E X 0Y (E kXkp )1=p (E kY kq )1=q : (B.15) n matrices X (B.16) APPENDIX B. PROBABILITY 330 Cauchy-Schwarz Inequality. For any random m E kXk2 E X 0Y n matrices X and Y; 1=2 E kY k2 1=2 : (B.17) Matrix Cauchy-Schwarz Inequality. Tripathi (1999). For any random x 2 Rm and y 2 R` , Eyx0 Exx0 Minkowski’s Inequality. For any random m Exy 0 Eyy 0 (B.18) n matrices X and Y; (E kXkp )1=p + (E kY kp )1=p (E kX + Y kp )1=p Liapunov’s Inequality. For any random m (E kXkr )1=r n matrix X and 1 r (B.19) p; (E kXkp )1=p (B.20) Markov’s Inequality (standard form). For any random vector x and non-negative function g(x) 0; 1 Pr(g(x) > ) Eg(x): (B.21) Markov’s Inequality (strong form). For any random vector x and non-negative function g(x) 0; 1 Pr(g(x) > ) E (g (x) 1 (g(x) > )) : (B.22) Chebyshev’s Inequality. For any random variable x; Pr(jx Exj > ) var (x) 2 : (B.23) Proof of Jensen’s Inequality (B.12). Since g(u) is convex, at any point u there is a nonempty set of subderivatives (linear surfaces touching g(u) at u but lying below g(u) for all u). Let a + b0 u be a subderivative of g(u) at u = Ex: Then for all u; g(u) a + b0 u yet g(Ex) = a + b0 Ex: Applying expectations, Eg(x) a + b0 Ex = g(Ex); as stated. Proof of Conditional Jensen’s Inequality. The same as the proof of (B.12), but using conditional expectations. The conditional expectations exist since E kyk < 1 and E kg (y)k < 1: Proof of Conditional Expectation Inequality. As the function jujr is convex for r Conditional Jensen’s inequality implies jE(y j x)jr E (jyjr j x) : Taking unconditional expectations and the law of iterated expectations, we obtain E jE(y j x)jr EE (jyjr j x) = E jyjr < 1 as required. Proof of Expectation Inequality. By the Triangle inequality, for k U 1 + (1 )U 2 k kU 1 k + (1 2 [0; 1]; ) kU 2 k 1, the APPENDIX B. PROBABILITY 331 which shows that the matrix norm g(U ) = kU k is convex. Applying Jensen’s Inequality (B.12) we …nd (B.15). Proof of Hölder’s Inequality. Since shows that for any real a and b exp 1 p + 1 q = 1 an application of Jensen’s Inequality (A.17) 1 1 a+ b p q 1 1 exp (a) + exp (b) : p q Setting u = exp (a) and v = exp (b) this implies u1=p v 1=q u v + p q and this inequality holds for any u > 0 and v > 0: Set u = kXkp =E kXkp and v = kY kq =E kY kq : Note that Eu = Ev = 1: By the matrix Schwarz Inequality (A.11), kX 0 Y k kXk kY k. Thus E kX 0 Y k p 1=p (E kXk ) E (kXk kY k) q 1=q (E kY k ) (E kXkp )1=p (E kY kq )1=q = E u1=p v 1=q u v + p q 1 1 = + p q = 1; E which is (B.16). Proof of Cauchy-Schwarz Inequality. Special case of Hölder’s with p = q = 2: (Eyx0 ) (Exx0 ) x: Note that Proof of Matrix Cauchy-Schwarz Inequality. De…ne e = y Eee0 0 is positive semi-de…nite. We can calculate that Eee0 = Eyy 0 Eyx0 Exx0 Exy 0 : Since the left-hand-side is positive semi-de…nite, so is the right-hand-side, which means Eyy 0 (Eyx0 ) (Exx0 ) Exy 0 as stated. Proof of Liapunov’s Inequality. The function g(u) = up=r is convex for u > 0 since p u = kXkr : By Jensen’s inequality, g (Eu) Eg (u) or (E kXkr )p=r r: Set E (kXkr )p=r = E kXkp : Raising both sides to the power 1=p yields (E kXkr )1=r (E kXkp )1=p as claimed. Proof of Minkowski’s Inequality. Note that by rewriting, using the triangle inequality (A.12), and then Hölder’s Inequality to the two expectations E kX + Y kp = E kX + Y k kX + Y kp E kXk kX + Y kp 1 1 + E kY k kX + Y kp (E kXkp )1=p E kX + Y kq(p 1) + (E kY kp )1=p E kX + Y kq(p = 1 1=q 1) 1=q (E kXkp )1=p + (E kY kp )1=p E (kX + Y kp )(p 1)=p APPENDIX B. PROBABILITY 332 where the second equality picks q to satisfy 1=p+1=q = 1; and the …nal equality uses this fact to make the substitution q = p=(p 1) and then collects terms. Dividing both sides by E (kX + Y kp )(p 1)=p ; we obtain (B.19). Proof of Markov’s Inequality. Let F denote the distribution function of x: Then Z dF (u) Pr (g(x) ) = fg(u) g fg(u) g Z = 1 = 1 Z g(u) dF (u) 1 (g(u) > ) g(u)dF (u) E (g (x) 1 (g(x) > )) the inequality using the region of integration fg(u) > g: This establishes the strong form (B.22). 1 E (g(x)) ; establishing the standard Since 1 (g(x) > ) 1; the …nal expression is less than form (B.21). Proof of Chebyshev’s Inequality. De…ne y = (x Ex)2 and note that Ey = var (x) : The events fjx Exj > g and y > 2 are equal, so by an application Markov’s inequality we …nd Pr(jx Exj > ) = Pr(y > 2 2 ) E (y) = 2 var (x) as stated. B.11 Maximum Likelihood In this section we provide a brief review of the asymptotic theory of maximum likelihood estimation. When the density of y i is f (y j ) where F is a known distribution function and 2 is an unknown m 1 vector, we say that the distribution is parametric and that is the parameter of the distribution F: The space is the set of permissible value for : In this setting the method of maximum likelihood is an appropriate technique for estimation and inference on : We let denote a generic value of the parameter and let 0 denote its true value. The joint density of a random sample (y 1 ; :::; y n ) is fn (y 1 ; :::; y n j ) = n Y i=1 f (y i j ) : The likelihood of the sample is this joint density evaluated at the observed sample values, viewed as a function of . The log-likelihood function is its natural logarithm log L( ) = n X i=1 log f (y i j ) : The likelihood score is the derivative of the log-likelihood, evaluated at the true parameter value. @ Si = log f (y i j 0 ) : @ We also de…ne the Hessian @2 H= E log f (y i j 0 ) (B.24) @ @ 0 APPENDIX B. PROBABILITY 333 and the outer product matrix = E S i S 0i : (B.25) We now present three important features of the likelihood. Theorem B.11.1 @ E log f (y j ) @ =0 = (B.26) 0 ES i = 0 (B.27) and H= (B.28) I The matrix I is called the information, and the equality (B.28) is called the information matrix equality. The maximum likelihood estimator (MLE) ^ is the parameter value which maximizes the likelihood (equivalently, which maximizes the log-likelihood). We can write this as ^ = argmax log L( ): (B.29) 2 In some simple cases, we can …nd an explicit expression for ^ as a function of the data, but these cases are rare. More typically, the MLE ^ must be found by numerical methods. To understand why the MLE ^ is a natural estimator for the parameter observe that the standardized log-likelihood is a sample average and an estimator of E log f (y i j ) : n 1 1X p log L( ) = log f (y i j ) ! E log f (y i j ) : n n i=1 As the MLE ^ maximizes the left-hand-side, we can see that it is an estimator of the maximizer of the right-hand-side. The …rst-order condition for the latter problem is 0= @ E log f (y i j ) @ which holds at = 0 by (B.26). This suggests that ^ is an estimator of 0 : In. fact, under p conventional regularity conditions, ^ is consistent, ^ ! 0 as n ! 1: Furthermore, we can derive its asymptotic distribution. Theorem B.11.2 Under N 0; I 1 . regularity conditions, p n ^ d 0 ! We omit the regularity conditions for Theorem B.11.2, but the result holds quite broadly for models which are smooth functions of the parameters. Theorem B.11.2 gives the general form for the asymptotic distribution of the MLE. A famous result shows that the asymptotic variance is the smallest possible. APPENDIX B. PROBABILITY 334 Theorem B.11.3 Cramer-Rao Lower Bound. If e is an unbiased regular estimator of ; then var(e) (nI) : The Cramer-Rao Theorem shows that the …nite sample variance of an unbiased estimator is bounded below by (nI) 1 : This means that the asymptotic variance of the standardized estimator p e 1 n : In other words, the best possible asymptotic variance among 0 is bounded below by I all (regular) estimators is I 1 : An estimator is called asymptotically e¢ cient if its asymptotic variance equals this lower bound. Theorem B.11.2 shows that the MLE has this asymptotic variance, and is thus asymptotically e¢ cient. Theorem B.11.4 The MLE is asymptotically e¢ cient in the sense that its asymptotic variance equals the Cramer-Rao Lower Bound. Theorem B.11.4 gives a strong endorsement for the MLE in parametric models. Finally, consider functions of parameters. If = g( ) then the MLE of is b = g(b): This is because maximization (e.g. (B.29)) is una¤ected by parameterization and transformation. Applying the Delta Method to Theorem B.11.2 we conclude that p p ' G0 n b n b d ! N 0; G0 I 1 (B.30) G where G = @@ g( 0 )0 : By Theorem B.11.4, b is an asymptotically e¢ cient estimator for since it is the MLE. The asymptotic variance G0 I 1 G is the Cramer-Rao lower bound for estimation of . Theorem B.11.5 The Cramer-Rao lower bound for and the MLE b = g(b) is asymptotically e¢ cient. = g( ) is G0 I Proof of Theorem B.11.1. To see (B.26); @ E log f (y j ) @ @ @ Z = = 0 Z log f (y j ) f (y j @ f (y j 0 ) f (y j ) dy @ f (y j ) Z @ f (y j ) dy @ = 0 = = @ 1 @ = = 0: = 0 Equation (B.27) follows by exchanging integration and di¤erentiation E @ log f (y j @ 0 ) dy 0) = @ E log f (y j @ 0) = 0: = = 0 0 1 G, APPENDIX B. PROBABILITY 335 Similarly, we can show that @2 @ @ f (y j 0 ) f (y j 0 ) E 0 ! = 0: By direction computation, @2 @ @ 0 log f (y j @2 @ @ 0) = f (y j 0 ) f (y j 0 ) @ @ f (y j 0 ) f (y j 0 ) @ log f (y j @ 0 @2 @ @ = @ 0) @ f (y j f (y j 0 f (y j 0) 0) 0 0) 2 @ log f (y j @ 0) 0 : Taking expectations yields (B.28). Proof of Theorem B.11.2 Taking the …rst-order condition for maximization of log L( ), and making a …rst-order Taylor series expansion, @ log L( ) @ =^ n X @ log f y i j ^ = @ 0 = i=1 n X = i=1 @ log f (y i j @ 0) + n X i=1 @2 @ @ 0 log f (y i j ^ n) 0 ; where n lies on a line segment joining ^ and 0 : (Technically, the speci…c value of row in this expansion.) Rewriting this equation, we …nd ^ 0 n X = i=1 @2 @ @ 0 ! log f (y i j 1 n X n) Si i=1 n varies by ! where S i are the likelihood scores. Since the score S i is mean-zero (B.27) with covariance matrix (equation B.25) an application of the CLT yields n 1 X d p S i ! N (0; ) : n i=1 The analysis of the sample Hessian is somewhat more complicated due to the presence of n : 2 p Let H( ) = @ @@ 0 log f (y i ; ) : If it is continuous in ; then since n ! 0 it follows that H( n) p ! H and so n 1 X @2 n @ @ i=1 n 0 log f (y i ; n) = 1X n i=1 p @2 @ @ 0 log f (y i ; n) H( n) + H( n) !H by an application of a uniform WLLN. (By uniform, we mean that the WLLN holds uniformly over the parameter value. This requires the second derivative to be a smooth function of the parameter.) Together, p n ^ d 0 !H 1 N (0; ) = N 0; H the …nal equality using Theorem B.11.1 . 1 H 1 = N 0; I 1 ; APPENDIX B. PROBABILITY 336 Proof of Theorem B.11.3. Let Y = (y 1 ; :::; y n ) be the sample, and set @ log fn (Y ; @ S= 0) = n X Si i=1 which by Theorem (B.11.1) has mean zero and variance nI: Write the estimator e = e (Y ) as a function of the data. Since e is unbiased for any ; Z e = E = e (Y ) f (Y ; ) dY : Di¤erentiating with respect to and evaluating at 0 yields Z e (Y ) @ f (Y ; ) dY I = @ 0 Z e (Y ) @ log f (Y ; ) f (Y ; = @ 0 = E eS 0 e = E 0 S0 the …nal equality since E (S) = 0 By the matrix Cauchy-Schwarz inequality (B.18), E nI; var e = E E e e = E SS 0 = (nI) as stated. 0 0 0 ) dY e 0 e 0 S 0 = I; and var (S) = E (SS 0 ) = 0 S 0 E SS 0 E S e 0 0 Appendix C Numerical Optimization Many econometric estimators are de…ned by an optimization problem of the form ^ = argmin Q( ) (C.1) 2 where the parameter is 2 Rm and the criterion function is Q( ) : ! R: For example NLLS, GLS, MLE and GMM estimators take this form. In most cases, Q( ) can be computed for given ; but ^ is not available in closed form. In this case, numerical methods are required to obtain ^: C.1 Grid Search Many optimization problems are either one dimensional (m = 1) or involve one-dimensional optimization as a sub-problem (for example, a line search). In this context grid search may be employed. Grid Search. Let = [a; b] be an interval. Pick some " > 0 and set G = (b a)=" to be the number of gridpoints. Construct an equally spaced grid on the region [a; b] with G gridpoints, which is { (j) = a + j(b a)=G : j = 0; :::; Gg. At each point evaluate the criterion function and …nd the gridpoint which yields the smallest value of the criterion, which is (^ |) where |^ = ^ |) is the gridpoint estimate of : If the grid is su¢ ciently …ne to argmin0 j G Q( (j)): This value (^ capture small oscillations in Q( ); the approximation error is bounded by "; that is, (^ |) ^ ": Plots of Q( (j)) against (j) can help diagnose errors in grid selection. This method is quite robust but potentially costly: Two-Step Grid Search. The gridsearch method can be re…ned by a two-step execution. For an error bound of " pick G so that G2 = (b a)=" For the …rst step de…ne an equally spaced grid on the region [a; b] with G gridpoints, which is { (j) = a + j(b a)=G : j = 0; :::; Gg: At each point evaluate the criterion function and let |^ = argmin0 j G Q( (j)). For the second step de…ne an equally spaced grid on [ (^ | 1); (^ | + 1)] with G gridpoints, which is { 0 (k) = (^ | 1) + 2k(b a)=G2 : k = 0; :::; Gg: Let k^ = argmin0 k G Q( 0 (k)): The estimate of ^ is k^ . The advantage of the two-step method over a one-step grid search is that the number of p function evaluations has been reduced from (b a)=" to 2 (b a)=" which can be substantial. The disadvantage is that if the function Q( ) is irregular, the …rst-step grid may not bracket ^ which thus would be missed. C.2 Gradient Methods Gradient Methods are iterative methods which produce a sequence are designed to converge to ^: All require the choice of a starting value 337 i 1; : i = 1; 2; ::: which and all require the APPENDIX C. NUMERICAL OPTIMIZATION 338 computation of the gradient of Q( ) g( ) = @ Q( ) @ and some require the Hessian @2 Q( ): @ @ 0 If the functions g( ) and H( ) are not analytically available, they can be calculated numerically. Take the j 0 th element of g( ): Let j be the j 0 th unit vector (zeros everywhere except for a one in the j 0 th row). Then for " small H( ) = gj ( ) ' Q( + j ") Q( ) " : Similarly, Q( + k ") Q( + j ") + Q( ) "2 In many cases, numerical derivatives can work well but can be computationally costly relative to analytic derivatives. In some cases, however, numerical derivatives can be quite unstable. Most gradient methods are a variant of Newton’s method which is based on a quadratic approximation. By a Taylor’s expansion for close to ^ gjk ( ) ' Q( + j" + k ") 0 = g(^) ' g( ) + H( ) ^ which implies ^= H( ) 1 g( ): This suggests the iteration rule ^i+1 = i H( i ) 1 g( i ): where One problem with Newton’s method is that it will send the iterations in the wrong direction if H( i ) is not positive de…nite. One modi…cation to prevent this possibility is quadratic hill-climbing which sets ^i+1 = i (H( i ) + i I m ) 1 g( i ): where i is set just above the smallest eigenvalue of H( i ) if H( ) is not positive de…nite. Another productive modi…cation is to add a scalar steplength i : In this case the iteration rule takes the form Di gi i (C.2) i+1 = i where g i = g( i ) and D i = H( i ) 1 for Newton’s method and Di = (H( i ) + i I m ) 1 for quadratic hill-climbing. Allowing the steplength to be a free parameter allows for a line search, a one-dimensional optimization. To pick i write the criterion function as a function of Q( ) = Q( i + Di gi ) a one-dimensional optimization problem. There are two common methods to perform a line search. A quadratic approximation evaluates the …rst and second derivatives of Q( ) with respect to ; and picks i as the value minimizing this approximation. The half-step method considers the sequence = 1; 1/2, 1/4, 1/8, ... . Each value in the sequence is considered and the criterion Q( i + D i g i ) evaluated. If the criterion has improved over Q( i ), use this value, otherwise move to the next element in the sequence. APPENDIX C. NUMERICAL OPTIMIZATION 339 Newton’s method does not perform well if Q( ) is irregular, and it can be quite computationally costly if H( ) is not analytically available. These problems have motivated alternative choices for the weight matrix Di : These methods are called Quasi-Newton methods. Two popular methods are do to Davidson-Fletcher-Powell (DFP) and Broyden-Fletcher-Goldfarb-Shanno (BFGS). Let gi = gi = i gi i 1 i 1 and : The DFP method sets Di = Di 1 i 0 i + 0 i gi + Di g i g 0i D i g 0i D i 1 g i 1 1 : The BFGS methods sets Di = Di 1 + i 0 i 0 i gi i 0 i 0 i gi 2 gi0 D i 1 gi + i g 0i D i gi 0 i 1 + Di 1 0 i gi gi 0 i : For any of the gradient methods, the iterations continue until the sequence has converged in some sense. This can be de…ned by examining whether j i Q ( i 1 )j or jg( i )j i 1 j ; jQ ( i ) has become small. C.3 Derivative-Free Methods All gradient methods can be quite poor in locating the global minimum when Q( ) has several local minima. Furthermore, the methods are not well de…ned when Q( ) is non-di¤erentiable. In these cases, alternative optimization methods are required. One example is the simplex method of Nelder-Mead (1965). A more recent innovation is the method of simulated annealing (SA). For a review see Go¤e, Ferrier, and Rodgers (1994). The SA method is a sophisticated random search. 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