Quantile

The generalized Pareto distribution is a two-parameter distribution that contains uniform, exponential, and Pareto distributions as special cases. It has applications in a number of fields, including reliability studies and the analysis of environmental extreme events. Maximum likelihood estimation of the generalized Pareto distribution has previously been considered in the literature, but we show, using computer simulation, that, unless the sample size is 500 or more, estimators derived by the method of moments or the method of probability-weighted moments are more reliable. We also use computer simulation to assess the accuracy of confidence intervals for the parameters and quantiles of the generalized Pareto distribution.

A re lone inventors more or less likely to invent breakthroughs? Recent research has attempted to resolve this question by considering the variance of creative outcome distributions. It has implicitly assumed a symmetric thickening or thinning of both tails, i.e., that a greater probability of breakthroughs comes at the cost of a greater probability of failures. In contrast, we propose that collaboration can have opposite effects at the two extremes: it reduces the probability of very poor outcomes-because of more rigorous selection processes-while simultaneously increasing the probability of extremely successful outcomes-because of greater recombinant opportunity in creative search. Analysis of over half a million patented inventions supports these arguments: Individuals working alone, especially those without affiliation to organizations, are less likely to achieve breakthroughs and more likely to invent particularly poor outcomes. Quantile regressions demonstrate that the effect is more than an upward mean shift. We find partial mediation of the effect of collaboration on extreme outcomes by the diversity of technical experience of team members and by the size of team members' external collaboration networks. Supporting our meta-argument for the importance of examining each tail of the distribution separately, experience diversity helps trim poor outcomes significantly more than it helps create breakthroughs, relative to the effect of external networks.

Quantile regression (QR) fits a linear model for conditional quantiles, just as ordinary least squares (OLS) fits a linear model for conditional means. An attractive feature of OLS is that it gives the minimum mean square error linear approximation to the conditional expectation function even when the linear model is misspecified. Empirical research using quantile regression with discrete covariates suggests that QR may have a similar property, but the exact nature of the linear approximation has remained elusive. In this paper, we show that QR can be interpreted as minimizing a weighted mean-squared error loss function for specification error. The weighting function is an average density of the dependent variable near the true conditional quantile. The weighted least squares interpretation of QR is used to derive an omitted variables bias formula and a partial quantile correlation concept, similar to the relationship between partial correlation and OLS. We also derive general asymptotic results for QR processes allowing for misspecification of the conditional quantile function, extending earlier results from a single quantile to the entire process. The approximation properties of QR are illustrated through an analysis of the wage structure and residual inequality in US census data for 1980, 1990, and 2000. The results suggest continued residual inequality growth in the 1990s, primarily in the upper half of the wage distribution and for college graduates. Econometrics Workshop, the University of Toronto, the University of Illinois at Urbana-Champaign, and the 2004 Winter Econometric Society Meetings for comments.

Some of the most powerful techniques currently available to test the goodness of fit of a hypothesized continuous cumulative distribution function (CDF) use statistics based on the empirical distribution function (EDF), such as those of Kolmogorov, Cramer-von Mises and Anderson-Darling, among others. The use of EDF statistics was analyzed for estimation purposes. In this approach, maximum goodness-of-fit estimators (also called minimum distance estimators) of the parameters of the CDF can be obtained by minimizing any of the EDF statistics with respect to the unknown parameters. The results showed that there is no unique EDF statistic that can be considered most efficient for all situations. Consequently, the possibility of defining new EDF statistics is entertained; in particular, an Anderson-Darling statistic of degree two and one-sided Anderson-Darling statistics of degree one and two appear to be notable in some situations. The procedure is shown to be able to deal successfully with the estimation of the parameters of homogeneous and heterogeneous generalized Pareto distributions, even when maximum likelihood and other estimation methods fail.

We propose a Bayesian semiparametric methodology for quantile regression modelling. In particular, working with parametric quantile regression functions, we develop Dirichlet process mixture models for the error distribution in an additive quantile regression formulation. The proposed non-parametric prior probability models allow the shape of the error density to adapt to the data and thus provide more reliable predictive inference than models based on parametric error distributions. We consider extensions to quantile regression for data sets that include censored observations. Moreover, we employ dependent Dirichlet processes to develop quantile regression models that allow the error distribution to change non-parametrically with the covariates. Posterior inference is implemented using Markov chain Monte Carlo methods. We assess and compare the performance of our models using both simulated and real data sets.

This article discusses a family of distributions which would seem not to receive proper attention in the literature. The twoparameter distribution is introduced with an application in the financial engineering domain. Special cases of this family include the triangular distribution, the standard power function distribution, and the uniform distribution. Properties of the distribution are investigated and the maximum likelihood estimation procedure for its two parameters is derived. The flexibility of the family as compared to that of the beta family is discussed.

Single equation instrumental variable models for discrete outcomes are shown to be set identifying, not point identifying, for the structural functions that deliver the values of the discrete outcome. Bounds on identified sets are derived for a general nonpara-metric model and ...

This article studies the problem of optimally dividing individuals into peer groups to maximize social gains from heterogeneous peer effects. The specific setting analyzed here concerns efficient ways of allocating roommates in college dormitories. Using confidential data on a sample ...

The properties of the sample quantiles of ranked-set samples are dealt with in this article. For any ÿxed set size in the ranked-set sampling the strong consistency and the asymptotic normality of the ranked-set sample quantiles for large samples are established. Bahadur representation of the ranked-set sample quantiles are also obtained. These properties are used to develop procedures for the inference on population quantiles such as conÿdence intervals and hypotheses testings. The e ciency of these procedures relative to their counterpart in simple random sampling is investigated. The ranked-set sampling is generally more e cient than the simple random sampling in this context. The gain in e ciency by using ranked-set sampling is quite substantial for the inference on middle quantiles and is the largest on the median. However, the relative e ciency damps away as the quantiles move away from the median on both directions. The gain in e ciency becomes negligible for extreme quantiles.

Ano. Inst. Statist. Math, Vol. 53, No. 2, 380-403 (2001) ©2001 The Institute of Statistical Mathematics ON AFFINE EQUIVARIANT MULTIVARIATE QUANTILES BlMAN CHAKRABORTY Department of Statistics and Applied Probability, The National University of Singapore, 10, Kent Ridge ...

Accelerated Destructive Degradation Tests (ADDTs) provide reliability information quickly. An ADDT plan specifies factor level combinations of an accelerating variable (e.g., temperature) and evaluation time and the allocations of test units to these combinations. This paper describes methods to find good ADDT plans for an important class of destructive degradation models. First, a collection of optimum plans is derived. These plans minimize the large sample approximate variance of the maximum likelihood (ML) estimator of a specified failure-time quantile. The General Equivalence Theorem (GET) is used to verify the optimality of these plans. Because an optimum plan is not robust to the model specification and the planning information used in deriving the plan, a more robust and useful compromise plan is proposed. Sensitivity analyses show the effects that changes in sample size, time duration of the experiment, levels of the accelerating variable, and misspecification of the planning information have on the precision of the ML estimator of a quantile of the failure-time distribution. Monte Carlo simulations are used to evaluate the statistical characteristics of the ADDT plans. The methods are illustrated with an application for an adhesive bond.

The purpose of this paper is to provide a strategic collaborative approach to risk and quality control in a cooperative supply chain by using a Neyman-Pearson quantile risk framework for the statistical control of risks. The paper is focused on the statistical quality control of a supplier and a producer, applying the traditional Neyman-Pearson theory to the control of quality in a supply chain environment. In our framework, the risks assumed by the parties in the supply chain depend on the organizational structure, the motivations and the power relationships that exist between members of the supply chain.

A method to estimate an extreme quantile that requires no distributional assumptions is presented. The approach is based on transformed kernel estimation of the cumulative distribution function (cdf). The proposed method consists of a double transformation kernel estimation. We derive optimal bandwidth selection methods that have a direct expression for the smoothing parameter. The bandwidth can accommodate to the given quantile level. The procedure is useful for large data sets and improves quantile estimation compared to other methods in heavy tailed distributions. Implementation is straightforward and R programs are available.

A statistical analysis of a bank's credit card database is presented. The database is a snapshot of accounts whose holders have missed a payment on a given month but who do not subsequently default. The variables on which there is information are observable measures on the account (such as profit and activity), and whether actions that are available to the bank (such as letters and telephone calls) have been taken. A primary objective for the bank is to gain insight into the effect that collections activity has on ongoing account usage. A neglog transformation that highlights features that are hidden on the original scale and improves the joint distribution of the covariates is introduced. Quantile regression, a novel methodology to the credit scoring industry, is used as it is relatively assumption free, and it is suspected that different relationships may be manifest in different parts of the response distribution. The large size is handled by selecting relatively small subsamples for training and then building empirical distributions from repeated samples for validation. In the application to the database of clients who have missed a single payment a substantive finding is that the predictor of the median of the target variable contains different variables from those of the predictor of the 30% quantile. This suggests that different mechanisms may be at play in different parts of the distribution.

Vincentizing (quantile averaging) is a popular means of pooling response time distributions across individuals to produce a group average. The benefit of Vincentizing is that the resulting histogram "looks like" an average of the individuals. In this article, we competitively test Vincentizing against the more mundane approach of averaging parameter estimates from fits to individuals. We simulate data from three distributions: the ex-Gaussian, the Weibull, and the shifted-Wald. For the ex-Gaussian and the shifted-Wald, parameter averaging outperforms Vincentizing. There is only an advantage of Vincentizing for the Weibull and only when there are few observations per participant. Overall, we recommend that researchers use Vincentizing only in select circumstances and with the knowledge that Vincentized estimates are often inconsistent estimators of averaged parameters.

A new approach is presented here to better take into account inter-individual growth variability in age-structured models used for stock assessment. Cohort analysis requires knowledge of the age structure of the catch, generally derived from an age-length key and length-structure information. Age distribution at length is estimated by applying conditional quantile regression to a data set of lengths and ages estimated from calcareous parts. A ''stochastic'' age-length key that describes the probability of age-at-length is applied to the English Channel squid Loligo forbesi. Age distribution at length from quantile regression proved to be considerably less biased than that resulting from both polymodal decomposition (PD) and two separate slicing methods. Both catch-at-age and stock size were underestimated using classical methods. Estimations of fishing mortalities from classical methods were higher causing underestimation in yield simulations. Quantile regression offers a more complete statistical analysis of the stochastic relationships among random variables than mean regression and PD. Keywords Age-length key Á Quantile regression Á Stochastic relationship Á Major axis Á Age-structured population model Communicated by Wolf Mooij.

A statistical problem that arises in several Þelds is that of estimating the features of an unknown distribution, which may be conditioned on covariates, using a sample of binomial observations on whether draws from this distribution exceed threshold levels set by experimental design. One application is destructive duration analysis, where the process is censored at an observation test time. Another is referendum contingent valuation in resource economics, where one is interested in features of the distribution of values placed by consumers on a public good such as endangered species. Sample consumers are asked whether they would vote for a referendum that would provide the good at a cost speciÞed by experimental design. This paper provides estimators for moments and quantiles of the unknown distribution in this problem.

Importance measures have been widely studied and applied in reliability and safety engineering. This paper presents a general formulation of moment-independent importance measures and several commonly discussed importance measures are unified based on Minkowski distance (MD). Moment-independent importance measures can be categorized into three classes of MD importance measures, i.e. probability density function based MD importance measure, cumulative distribution function based MD importance measure and quantile based MD importance measure. Some properties of the proposed MD importance measures are investigated. Several new importance measures are also derived as special cases of the generalized MD importance measures and illustrated with some case studies.

Since the pioneering work of Koenker and Bassett, median-restricted models have attracted considerable interest. Attention in these models, so far, has focused on least absolute deviation (auto-)regression quantile estimation and the corresponding sign tests. These methods use a pseudolikelihood that is based on a double-exponential reference density and enjoy quite attractive properties of root n consistency (for estimators) and distribution freeness (for tests). The paper extends these results to general, i.e. not necessarily double-exponential, reference densities. Using residual signs and ranks (not signed ranks) and a general reference density f , we construct estimators that remain root n consistent, irrespective of the true underlying density g (i.e. also for g = D f). However, instead of reaching semiparametric efficiency bounds under double-exponential g, they reach these bounds when g coincides with the chosen reference density f. Moreover, we show that choosing reference densities other than the double-exponential in applications can lead to sizable gains in efficiency. The particular case of median regression is treated in detail; extensions to general quantile regression, heteroscedastic errors and time series models are briefly described. The performance of the method is also assessed by simulation and illustrated on financial data.

In this paper we assess accuracy of some commonly used estimators of upper quantiles of a right skewed distribution under both parameter and model uncertainty. In particular, for each of lognormal, log-logistic, and log-double exponential distributions, we study the bias and mean squared error of the maximum likelihood estimator (MLE) of the upper quantiles under both the correct and incorrect model specifications. We also consider two data dependent or adaptive estimators.

The problem of missing values commonly arises in data sets, and imputation is usually employed to compensate for non-response. We propose a novel imputation method based on quantiles, which can be implemented with or without the presence of auxiliary information. The proposed method is extended to unequal sampling designs and nonuniform response mechanisms. Iterative algorithms to compute the proposed imputation methods are presented. Monte Carlo simulations are conducted to assess the performance of the proposed imputation methods with respect to alternative imputation methods. Simulation results indicate that the proposed methods perform competitively in terms of relative bias and relative root mean square error.

Statistical criteria used to evaluate the best distribution fit give large weights to the center of distributions. This is, however, not consistent with the objective of frequency analysis which is to estimate the quantiles with large return periods. In this study, the usefulness of a recently proposed decision support system ͑DSS͒, which defines the class of distributions prior to a model selection practice with respect to tail behavior of sample data, was investigated using three large hydroclimatic databases ͓Reference Hydrometric Basin Network ͑RHBN͒, precipitation, and UNESCO͔. According to the DSS, although a considerable majority of RHBN flood sample data belonged to Class C ͑regularly varying distributions͒, a slight and great majority of UNESCO discharge as well as annual precipitation sample data, respectively, belonged to Class D ͑subexponential distributions͒. This difference in classification is attributed to the nature of studied variables: RHBN sample data represent extreme events with heavy tails ͑Class C͒, whereas UNESCO and especially precipitation sample data come from relatively lighter tailed processes and therefore belong mostly to Class D distributions. The impact of classification on model selection was the largest for the RHBN and the smallest for precipitation sample data. This confirms that discriminating between classes of competing models prior to model selection is critical when the sample data come from extreme events. Observed inconsistency in model selection for the RHBN database resulted in an underestimation of quantiles in more than 2/3 of the cases regardless of class of distributions. For the UNESCO and precipitation data, however, inappropriate model selection resulted equally in over-and underestimation in Class C, whereas it resulted in underestimation of quantiles in Class D in the majority of observed inconsistencies. It can be concluded that an inappropriate model selection due to choosing a wrong class of distributions leads, in the majority of cases, to an underestimation of the quantity of the variable under study which is associated with a higher socioeconomy risk compared to that corresponding to an overestimation of a specific quantile. It was also observed that model selection using Bayesian Information Criterion ͑compared to Akaike Information Criterion͒ is more consistent with tail behavior of the natural processes.

Crop yield can be decreased by many limiting factors such as water stress, nitrogen stress and disease. The agronomic diagnosis method was developed by agronomists to understand the origin of crop yield variability, identify important limiting factors, and define new cropping systems. The rigorous implementation of this method requires the determination of boundary lines giving the maximum value of a yield component in relation to the value of another yield component; for example, grain weight versus grain number per square metre. Such boundary lines are used by agronomists to adjust cropping practices to environmental characteristics and, thus, to reduce the risk of pollution due to agricultural activities. We describe here a new method based on quantile regression to estimate boundary line parameters from experimental data. First, quantile values were computed from models describing the effect of limiting factors on yield components and accounting for measurement errors. Then, bou...

In the context of Bayesian statistical analysis, elicitation is the process of formulating a prior density f (•) about one or more uncertain quantities to represent a person's knowledge and beliefs. Several different methods of eliciting prior distributions for one unknown parameter have been proposed. However, there are relatively few methods for specifying a multivariate prior distribution and most are just applicable to specific classes of problems and/or based on restrictive conditions, such as independence of variables. Besides, many of these procedures require the elicitation of variances and correlations, and sometimes elicitation of hyperparameters which are difficult for experts to specify in practice. Garthwaite, Kadane and O'Hagan (2005) discuss the different methods proposed in the literature and the difficulties of eliciting multivariate prior distributions. We describe a flexible method of eliciting multivariate prior distributions applicable to a wide class of practical problems. Our approach does not assume a parametric form for the unknown prior density f (•), instead we use nonparametric Bayesian inference, modelling f (•) by a Gaussian process prior distribution. The expert is then asked to specify certain summaries of his/her distribution, such as the mean, mode, marginal quantiles and a small number of joint probabilities. The analyst receives that information, treating it as a data set D with which to update his/her prior beliefs to obtain the posterior distribution for f (•). Theoretical properties of joint and marginal priors are derived and numerical illustrations to demonstrate our approach are given.

A copula de…nes the probability that observations from two time series lie below given quantiles. It is proposed that stationarity tests constructed from indicator variables be used to test against the hypothesis that the copula is changing over time. Tests associated with di¤erent quantiles may point to changes in di¤erent parts of the copula, with the lower quantiles being of particular interest in …nancial applications concerned with risk. Tests located at the median provide an overall test of a changing relationship. The properties of vari-1 ous tests are compared and it is shown that they are still e¤ective if pre-…ltering is carried out to correct for changing volatility or, more generally, changing quantiles. Applying the tests to daily stock return indices in Korea and Thailand over the period 1995-9 indicates that the relationship between them is not constant over time.

This paper proposes a nonparametric test of Granger causality in quantile. Zheng (1998, Econometric Theory 14, 123–138) studied the idea to reduce the problem of testing a quantile restriction to a problem of testing a particular type of mean restriction in independent data. We extend Zheng’s approach to the case of dependent data, particularly to the test of Granger causality in quantile. Combining the results of Zheng (1998) and Fan and Li (1999, Journal of Nonparametric Statistics 10, 245–271), we establish the asymptotic normal distribution of the test statistic under a β-mixing process. The test is consistent against all fixed alternatives and detects local alternatives approaching the null at proper rates. Simulations are carried out to illustrate the behavior of the test under the null and also the power of the test under plausible alternatives. An economic application considers the causal relations between the crude oil price, the USD/GBP exchange rate, and the gold price in...

Using the concept of near-exact approximation to a distribution we developed two different near-exact approximations to the distribution of the product of an odd number of particular independent Beta random variables (r.v.'s). One of them is a particular generalized near-integer Gamma (GNIG) distribution and the other is a mixture of two GNIG distributions. These near-exact distributions are mostly adequate to be used as a basis for approximations of distributions of several statistics used in multivariate analysis. By factoring the characteristic function (c.f.) of the logarithm of the product of the Beta r.v.'s, and then replacing a suitably chosen factor of that c.f. by an adequate asymptotic result it is possible to obtain what we call a near-exact c.f., which gives rise to the near-exact approximation to the exact distribution. Depending on the asymptotic result used to replace the chosen parts of the c.f., one may obtain different near-exact approximations. Moments from the two near-exact approximations developed are compared with the exact ones. The two approximations are also compared with each other, namely in terms of moments and quantiles.

Exact nonparametric inference based on ordinary Type-II right censored samples has been extended here to the situation when there are multiple samples with Type-II censoring from a common continuous distribution. It is shown that marginally, the order statistics from the pooled sample are mixtures of the usual order statistics with multivariate hypergeometric weights. Relevant formulas are then derived for the construction of nonparametric confidence intervals for population quantiles, prediction intervals, and tolerance intervals in terms of these pooled order statistics. It is also shown that this pooled-sample approach assists in achieving higher confidence levels when estimating large quantiles as compared to a single Type-II censored sample with same number of observations from a sample of comparable size. We also present some examples to illustrate all the methods of inference developed here.

Quantile regression is applied in two retail credit risk assessment exercises exemplifying the power of the technique to account for the diverse distributions that arise in the financial service industry. The first application is to predict loss given default for secured loans, in particular retail mortgages. This is an asymmetric process since where the security (such as a property) value exceeds the loan balance the banks cannot retain the profit, whereas when the security does not cover the value of the defaulting loan then the bank realises a loss. In the light of this asymmetry it becomes apparent that estimating the low tail of the house value is much more relevant for estimating likely losses than estimates of the average value where in most cases no loss is realised. In our application quantile regression is used to estimate the distribution of property values realised on repossession that is then used to calculate loss given default estimates. An illustration is given for a mortgage portfolio from a European mortgage lender. A second application is to revenue modelling. While credit issuing organisations have access to large databases, they also build models to assess the likely effects of new strategies for which, by definition, there is no existing data. Certain strategies are aimed at increasing the revenue stream or decreasing the risk in specific market segments. Using a simple artificial revenue model, quantile regression is applied to elucidate the details of subsets of accounts, such as the least profitable, as predicted from their covariates. The application uses standard linear and kernel smoothed quantile regression.

After almost a decade of crisis, the housing market in Spain shows significant signs of recovery, with increases in both the average price and the number of sales transactions. Housing is the main asset for the majority of households, and it also has the most resources devoted to it, thus, when it comes to buying a residence, people do not only look at the asset’s intrinsic characteristics, but also consider other particularities such as the neighbourhood, accessibility to services, availability of public transport or adequate funding. The study aimed to analyse and quantify the relationship that exists between the asking price of second-hand housing on the market in Alicante and the attributes that characterise them. This was done using a multivariate analysis to estimate a hedonic pricing model by ordinary least squares and a quantile regression to analyse the impact of the characteristics in different price ranges. The results show the segmentation of the prices in the Alicante m...

Using panel data from a developing country on individuals aged 16 to 59 who reported their monthly wages, we estimated a relationship between health (nutrition) measures (i.e. height and BMI) and wages (which proxies productivity/growth). We controlled for endogeneity of BMI and found heterogeneous returns to different human capital indicators. Our findings indicate that productivity is positively and significantly affected

This paper applies quantile data analysis to input modeling in simulation. We introduce the use of QIQ plots to identify suitable distributions fitting the data and comparison distribution PP plots to test the fit. Two examples illustrate the utility of these quantile statistical methods for ...

Despite the widespread popularity of linear models for correlated outcomes (e.g. linear mixed modesl and time series models), distribution diagnostic methodology remains relatively underdeveloped in this context. In this paper we present an easy-to-implement approach that lends itself to graphical displays of model fit. Our approach involves multiplying the estimated marginal residual vector by the Cholesky decomposition of the inverse of the estimated marginal variance matrix. Linear functions or the resulting "rotated" residuals are used to construct an empirical cumulative distribution function (ECDF), whose stochastic limit is characterized. We describe a resampling technique that serves as a computationally efficient parametric bootstrap for generating representatives of the stochastic limit of the ECDF. Through functionals, such representatives are used to construct global tests for the hypothesis of normal margional errors. In addition, we demonstrate that the ECDF of the predicted random effects, as described by Lange and Ryan (1989), can be formulated as a special case of our approach. Thus, our method supports both omnibus and directed tests. Our method works well in a variety of circumstances, including models having independent units of sampling (clustered data) and models for which all observations are correlated (e.g., a single time series).

Are lone inventors more or less likely to invent breakthroughs? Recent research has attempted to resolve this question by considering the variance of creative outcome distributions. It has implicitly assumed a symmetric thickening or thinning of both tails, that a greater probability of breakthroughs comes at the cost of a greater probability of failures. In contrast, we propose that collaboration can have opposite effects at the two extremes: it reduces the probability of very poor outcomes -due to more rigorous selection processes -while simultaneously increasing the probability of extremely successful outcomes -due to greater recombinant opportunity in creative search. Analysis of over half a million patented inventions supports these arguments: individuals working alone, especially those without affiliation to organizations, are less likely to achieve breakthroughs and more likely to invent particularly poor outcomes. Quantile regressions demonstrate that the effect is more than an upward mean shift. We find partial mediation of the effect of collaboration on extreme outcomes by the diversity of technical experience of team members and by the size of team members' external collaboration networks. Supporting our meta-argument for the importance of examining each tail of the distribution separately, experience diversity helps trim poor outcomes significantly more than it helps create breakthroughs, relative to the effect of external networks. [212 words]

Various regional flood frequency analysis procedures are used in hydrology to estimate hydrological variables at ungauged or partially gauged sites. Relatively few studies have been conducted to evaluate the accuracy of these procedures and estimate the error induced in regional flood frequency estimation models. The objective of this paper is to assess the overall error induced in the residual kriging (RK) regional flood frequency estimation model. The two main error sources in specific flood quantile estimation using RK are the error induced in the quantiles local estimation procedure and the error resulting from the regional quantile estimation process. Therefore, for an overall error assessment, the corresponding errors associated with these two steps must be quantified. Results show that the main source of error in RK is the error induced into the regional quantile estimation method. Results also indicate that the accuracy of the regional estimates increases with decreasing return periods.

The issue of monetary policy rules in Kazakhstan is studied. It is established that at each stage of economic development the National Bank used a certain rule. In particular, it is revealed that during the post-crisis period after reaching macroeconomic stability the short term interest rate, rather than the monetary base, was used as a monetary policy tool.

Investigating the factors affecting CO2 emissions has always been a challenge. One problem with existing studies is that these studies have been relied on mean-based regression approaches, such as ordinary least squares (OLS) or instrumental variables, which implicitly assumes that the impact of variables along the distribution of CO2 emissions is the same. Unlike previous studies, the present study will use the quantile regression developed by Koenker & Bassett, which is not limited to the assumption. So that, the purpose of this study is to investigate the impacts of technological innovation and renewable energy on CO2 emissions in selected countries of the International Renewable Energy Agency (IRENA) using quantile regression over the period 1990-2016. The results of this study exhibited that the impact of renewable energy on CO2 emissions was negative and statistically significant. This impact is also enhanced in high quantiles (countries with high pollution). In all the studied quantiles, the impact of technological innovations on CO2 emissions was positive, significant and initially decreasing, while increasing again over time. The results of the symmetry test also indicated that by increasing in the volume of CO2 emissions, the variable impact of renewable energy upraised. However, no incremental trend was observed in innovation.

European volatility markets exhibit differences consistent with institutional clusters identified in the previous literature. The degree of integration between leading markets, however, is very high and shocks on the implied volatility spread die out within few days. Our Markov switching model distinguishes three volatility regimes. In the high volatility regime, level, slope, and curvature factors explain 97% of the complete volatility term structure. A shock to the second principal component has regime dependent influence on the slope of the volatility term structure. Our findings lend support to the behaviour explanation of the return-implied volatility relation and have implications for risk management and option pricing.

Настоящее эссе содержит краткое обозрение рафинирований высокого порядка, свойственных бутстрапу тестов, основанных на обобщенном методе моментов. Вопервых, мы вкратце описываем асимптотическое поведение двухшаговых ОММоценок. Во-вторых, мы эвристически поясняем, почему инференция, основанная на бутстраповских критических значениях, более точна, чем инференция, полагающаяся на асимптотическую нормальность. В-третьих, мы сжато описываем непараметрические методы ресэмплинга. В-четвертых, мы обрисовываем, как использование критических значений, основанных на блочном бутстрапе, приводит к уменьшению ошибки вероятности отклонения гипотез для основанных на ОММ-оценках t-тестов. Наконец, мы даем обзор некоторых альтернативных бутстраповских процедур, которые приводят к улучшению блочно-бутстраповского рафинирования. 1 Введение Данное эссе содержит краткое обозрение рафинирований высокого порядка, свойственных бутстрапу тестов, основанных на обобщенном методе моментов (ОММ). Из экспериментов Монте-Карло видно, что в умеренного размера выборках тесты, основанные на ОММ, такие как t-тесты и тесты на сверхидентифицирующие ограничения, часто страдают резкими искажениями уровня. Хорошо известно, что при определенных условиях регулярности инференция, основанная на бутстраповских критических значениях, приводит к уменьшению ошибки вероятности отклонения (ОВО) нулевой гипотезы, то есть к уменьшению разницы между номинальным и фактическим уровнями теста. Тем не менее, изучение свойств высокого порядка у бутстрапа ОММ-оценок для зависимых данных началось сравнительно недавно. Hall & Horowitz (1996) показали, что использование бутстраповских критических значений действительно дает уменьшения ОВО, а Andrews (2002) определил порядок величины уменьшение ОВО бутстрапом. Эссе организовано следующим образом. Во-первых, мы вкратце опишем асимптотическое поведение двухшаговых ОММ-оценок. Во-вторых, мы эвристически поясним, почему инференция, основанная на бутстраповских критических значениях, более точна, чем инференция, полагающаяся на асимптотическую нормальность. В-третьих, мы сжато опишем непараметрические методы ресэмплинга, подходящие для случая, когда распределение данных неизвестно. В-четвертых, следуя Andrews (2002), мы обрисуем бутстраповское рафинирование для t-тестов, основанных на ОММ, для случая зависимых наблюдений. В-пятых, мы обратимся к некоторым вопросам практической реализации бутстраповских оценок. Наконец, мы упомянем, каким образом блочно-блочный бутстрап (Andrews, 2004) и марковский бутстрап (Horowitz, 2003) приводят к улучшенному рафинированию в случае марковских или приблизительно марковских процессов.

Настоящее эссе-обзор двух сфер применения полупараметрической эконометрики: анализа цензурированных данных о продолжительности занятости и анализа данных о заявленной готовности платить за природные ресурсы. 1 Введение Многие эконометрические задачи можно рассматривать как один из вариантов следующей модели. Имеется случайный вектор (Y, X) ∈ R k × R m , такой, что X имеет (неизвестную) плотность распределения g(x), а Y почти наверное характеризуется (неизвестной) функцией условной плотности f (y|x). Также известно преобразование t(y, x) из R k × R m в множество действительных чисел R, и условное математическое ожидание этого преобразования, θ(x) = E[t(Y, x)|X = x], является объектом эконометрического исследования. Примерами подобных преобразований могут быть: (1) t(y, x) ≡ y, когда θ(x) = E[Y |X = x]-математическое ожидание Y при условии X = x, или функция регрессии Y на x; (2) t(y, x) = yy , когда θ(x) = E[Y Y |X = x]-матрица вторых условных моментов, а в комбинации с первым примеромусловная дисперсия E[Y Y |X = x] − (E[Y |X = x])(E[Y |X = x]) ; и (3) t(y, x) = I A (y), то есть индикатор-функция множества A, когда θ(x)-вероятность события A при условии X = x. Примерами из экономических приложений могут быть вектор потребительского спроса Y и вектор дохода и цен x, или вектор чистого выпуска фирмы Y и вектор уровней постоянных затрат и цен на переменные факторы x. Определим возмущение ε = ε(y, x) ≡ t(y, x) − θ(x). Тогда описанную выше постановку можно сформулировать в виде обобщенной регрессионной модели t(y, x) = θ(x) + ε, где E[ε|x] = 0. Эконометрические задачи, подходящие под эту модель, можно классифицировать как полностью параметрические, полупараметрические или непараметрические. Модель является полностью параметрической, если априори известно, что функция θ и распределение ошибки ε принадлежат семействам с конечным числом параметров. Модель является непараметрической, если о функциональных формах θ и ε ничего неизвестно, за исключением, возможно, некоторых свойств регулярности и формы, таких как непрерывная дифференцируемость или вогнутость. Модель является полупараметрической, если она содержит конечный вектор параметров, обычно представляющий первостепенный интерес, но части θ и/или распределение ε не ограничены семействами с конечным числом параметров. Это определение полупараметрической модели в довольно широком смысле, и оно включает, например, модель линейной регрессии при условиях Гаусса-Маркова, когда распределение ошибок не ограничено параметрическим семейством, и только первые два момента параметризованы. Некоторые эконометристы предпочитают применять термин полупараметрическая модель в тех ситуациях, когда задачу можно охарактеризовать с помощью