Conditional Heteroscedasticity

This study investigates the application of learning-based and simulation-based Approximate Dynamic Programming (ADP) approaches to an inventory problem under the Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model. Specifically, we explore the robustness of a learning-based ADP method, Sarsa, with a GARCH(1,1) demand model, and provide empirical comparison between Sarsa and two simulation-based ADP methods: Rollout and Hindsight Optimization (HO). Our findings assuage a concern regarding the effect of GARCH(1,1) latent state variables on learning-based ADP and provide practical strategies to design an appropriate ADP method for inventory problems. In addition, we expose a relationship between ADP parameters and conservative behavior. Our empirical results are based on a variety of problem settings, including demand correlations, demand variances, and cost structures.

Forecasting stock exchange rates is an important financial problem that is receiving increasing attention. During the last few years, a number of neural network models and hybrid models have been proposed for obtaining accurate prediction results, in an attempt to outperform the traditional linear and nonlinear approaches. This paper evaluates the effectiveness of neural network models; recurrent neural network (RNN), dynamic artificial neural network (DAN2) and the hybrid neural networks which use generalized autoregressive conditional heteroscedasticity (GARCH) and exponential generalized autoregressive conditional heteroscedasticity (EGARCH) to extract new input variables. The comparison for each model is done in two view points: MSE and MAD using real exchange daily rate values of Istanbul Stock Exchange (ISE) index XU10).

The estimation of inflation volatility is important to Central Banks as it guides their policy initiatives for achieving and maintaining price stability. This paper employs three models from the Generalized Autoregressive Conditional Heteroscedasticity (GARCH) family with a view to providing a parsimonious approximation to the dynamics of Nigeria’s inflation volatility between 1996 and 2011. Of the competing models, the asymmetric TGARCH (1,1) provides an appropriate paradigm for explaining the dynamics of headline and core CPI volatilities in Nigeria, while the symmetric GARCH (1,1) was found to be adequate for food CPI. The results are quite revealing. Firstly, model outcomes indicate high persistence parameters for the core and food CPI, implying that the impacts of inflation shocks on their volatilities die away very slowly. However, the impact of inflation shocks on headline volatility die out rather quickly. Secondly, substantial evidence of asymmetric effect was found for bot...

Using the Investors' Intelligence sentiment index, we employ a generalized autoregressive conditional heteroscedasticity-in-mean specification to test the impact of noise trader risk on both the formation of conditional volatility and expected return as suggested by De Long et al. [Journal of Political Economy 98 (1990) 703]. Our empirical results show that sentiment is a systematic risk that is priced. Excess returns are contemporaneously positively correlated with shifts in sentiment. Moreover, the magnitude of bullish (bearish) changes in sentiment leads to downward (upward) revisions in volatility and higher (lower) future excess returns.

El presente Trabajo Fin de Master plantea como objetivo la aplicación práctica del modelo GARCH y distribuciones de colas anchas en el cálculo del riesgo de mercado, mediante metodología VeR Paramétrica, en una cartera de renta variable en el periodo de 1/1/2014 al 31/12/2018 para los principales títulos del sector bancario y financiero que cotizan en el mercado bursátil español y que pertenecen al IBEX-35. Se escogen los valores cotizados de Santander, BBVA, CaixaBank, Sabadell y Bankia. Se realiza metodología empírica mediante técnicas matemáticas, estadísticas y econométricas. Se comparan resultados obtenidos entre distribución normal y distribución T-Student (colas anchas), y según la forma de cálculo de la volatilidad (Estándar o GARCH). Finalmente se realiza el Backtesting en el periodo de 1/1/2019 al 30/6/2019 para los resultados de la cartera y se establecen las conclusiones finales del estudio.

This paper considers adaptive estimation in nonstationary autoregressive moving average models with the noise sequence satisfying a generalised autoregressive conditional heteroscedastic process. The locally asymptotic quadratic form of the log-likelihood ratio for the model is obtained. It is shown that the limit experiment is neither LAN nor LAMN, but is instead LABF. Adaptivity is discussed and it is found that the parameters in the model are generally not adaptively estimable if the density of the rescaled error is asymmetric. For the model with symmetric density of the rescaled error, a new efficiency criterion is established for a class of defined M ν -estimators. It is shown that such efficient estimators can be constructed when the density is known. Using the kernel estimator for the score function, adaptive estimators are constructed when the density of the rescaled error is symmetric, and it is shown that the adaptive procedure for the parameters in the conditional mean part uses the full sample without splitting. These estimators are demonstrated to be asymptotically efficient in the class of M ν -estimators. The paper includes the results that the stationary ARMA-GARCH model is LAN, and that the parameters in the model with symmetric density of the rescaled error are adaptively estimable after a reparameterisation of the GARCH process.

In this paper, we use recent advances in the financial econometrics literature to model the time-varying conditional variance in five energy markets -crude oil, gasoline, heating oil, propane, and natural gas -using daily data over the period from January 3, 1994 to September 23, 2008. We estimate autoregressive conditional heteroscedasticity (ARCH) and generalised ARCH (GARCH) models using a variety of error densities (the normal, Student-t, and generalised error distribution) and diagnostic checks. We use the models to perform static and dynamic forecasts over different horizons and compare their performance to that of a random walk model.

Generalized autoregressive conditional heteroscedasticity (GARCH) models have long been considered as one of the most successful families of approaches for volatility modeling in financial return series. In this paper, we propose an alternative approach based on methodologies widely used in the field of statistical machine learning. Specifically, we propose a novel nonparametric Bayesian mixture of Gaussian process regression models, each component of which models the noise variance process that contaminates the observed data as a separate latent Gaussian process driven by the observed data. This way, we essentially obtain a mixture Gaussian process conditional heteroscedasticity (MGPCH) model for volatility modeling in financial return series. We impose a nonparametric prior with power-law nature over the distribution of the model mixture components, namely the Pitman-Yor process prior, to allow for better capturing modeled data distributions with heavy tails and skewness. Finally, we provide a copulabased approach for obtaining a predictive posterior for the covariances over the asset returns modeled by means of a postulated MGPCH model. We evaluate the efficacy of our approach in a number of benchmark scenarios, and compare its performance to state-of-the-art methodologies.

This paper investigates the oil price -exchange rate nexus for Nigeria during the period 2007-2010 using daily data. The generalised autoregressive conditional heteroscedasticity (GARCH) and exponential GARCH (EGARCH) models are employed to examine the impact of oil price changes on the nominal exchange rate .The outcome of this research indicates that a rise in oil prices leads to a depreciation of the Nigerian Naira vis-à-vis the US dollar over the study period.

This paper considers adaptive estimation in nonstationary autoregressive moving average models with the noise sequence satisfying a generalized autoregressive conditional heteroscedastic process. The locally asymptotic quadratic form of the log-likelihood ratio for the model is obtained. It is shown that the limit experiment is neither LAN nor LAMN, but is instead LABF. For the model with symmetric density of the rescaled error, a new efficiency criterion is established for a class of defined M ν -estimators. It is shown that such efficient estimators can be constructed when the density is known. Using the kernel estimator for the score function, adaptive estimators are constructed when the density of the rescaled error is symmetric, and it is shown that the adaptive procedure for the parameters in the conditional mean part uses the full sample without splitting. These estimators are demonstrated to be asymptotically efficient in the class of M ν -estimators. The paper includes the results that the stationary ARMA-GARCH model is LAN, and that the parameters in the model with symmetric density of the rescaled error are adaptively estimable after a reparameterization of the GARCH process. This paper also establishes the locally asymptotic quadratic form of the loglikelihood ratio for nonlinear time series models with ARCH-type errors.

Climate change has been widely blamed to have caused an increase in unstable weather events such as storm, temperature extremes, severe precipitation and unseasonal weather. In the last decade, some notable weather events have occurred e.g. high temperature in 1995 and 2003 created significantly increased electricity demand compared with the long term average. Such events pose a very challenging task for energy forecasters who must provide high quality predictions to utility decision makers. This paper outlines a methodology using AutoRegressive Integrated Moving Average (ARIMA) model to predict daily load patterns. The purpose of this work serves as an initial step to investigate the impacts of climate change and weather extremes on electricity demand patterns and the electricity network. It is necessary for us not just to develop a robust methodology to predict the daily load accurately for planning purposes like most conventional models, but more importantly, to provide a reliable long range forecast. Our model takes account of four future climate change scenarios and socioeconomic scenarios to project 90 years (2011 -2100) of future load demand patterns. The forecasted load will be used as an input to our transmission network model to study security and grid reinforcement of the power network as the result of climate change.

This paper reviews the literature on GARCH-type models proposed to represent the dynamic evolution of conditional variances. Effects of level outliers on the diagnostic and estimation of GARCH models are also studied. Both outliers and conditional heteroscedasticity can generate time series with excess kurtosis and autocorrelated squared observations. Consequently, both phenomena can be confused. However, since outliers are generated by

We attempt empirical detection and characterization of power laws in financial time series. Fractional Brownian motion is defined. After testing for multifractality we calculate the multifractal spectrum of the series. The multifractal nature of stock prices leads to volatility clus- tering (conditional heteroscedasticity) and long memory (slowly decaying autocorrelation). Wavelet Transform Modulus Maxima approach to mul- tifractal spectrum estimation proved

Finansal piyasalarda volatilitenin tahmin edilebilir bir kavram olması, volatiliteyi modelleyen birçok yöntemin ortaya çıkması ile sonuçlanmıştır. ARCH/GARCH sınıfı koşullu değişen varyans modellemeleri, özellikle yüksek frekanslı finansal verilere gösterdiği uyum nedeniyle ön plana çıkan modelleme teknikleridir. Özellikle finansal kararların, risk ve getiri bağlamında değerlendirilmesinin gerekliliği, bu tür modellerin önem kazanmasında rol oynamıştır. BİST 100 endeksi ve altın fiyatları, hem yatırım aracı olarak hem de makroekonomik gelişmelerin bir barometresi olarak görev yapan göstergeler konumundadır. Bu çalışma, BİST 100 endeks getirisi ve altın getirisi volatilitesinin modellenmesi amacıyla en uygun koşullu değişen varyans modellerinin belirlenmesini amaçlamaktadır. Bu amaçla, çalışmada 3 Ocak 2003-11 Eylül 2013 tarihleri arasındaki kapanış fiyatlarına göre günlük BİST-100 endeksi serisi ve 3 Ocak 2005-10 Eylül 2013 tarihleri arasındaki günlük altın borsası işlemleri, ağırlıklı ortalama fiyat (TL/KG) verileri kullanılmıştır. Çalışmada, BİST 100 endeks volatilitesi için en uygun model TARCH(1,1) modeli olarak belirlenmiştir. Bu model, BİST 100 endeks getirisi serisi üzerinde asimetrik etkilerin olduğunu göstermektedir. Çalışmada, altın getiri serisinin volatilitesinin ölçülmesi için en uygun model ise GARCH(1,1) modelidir.

Despite the criticisms on the validity of the CAPM, finance researchers continue to adopt the model in trying to describe the relationship between risk and return. The introduction of the GARCH(p,q)-M model provides an avenue for testing the model within the time-varying variance framework. This study employs the same model to address the issue within ten selected Asia Pacific countries. The result, though not comprehensive, shows that the CAPM still holds in explaining the risk-return relationship in China and Malaysia. The significant positive risk parameter coefficient suggests a positive linear relationship which indicates that investors are compensated for assuming high risk. Judging by the significant finding in China and Malaysia, this study provides evidence that the conditional CAPM is a useful tool for decision making in investments and corporate finance.

Abstract.  This paper analyses how outliers affect the identification of conditional heteroscedasticity and the estimation of generalized autoregressive conditionally heteroscedastic (GARCH) models. First, we derive the asymptotic biases of the sample autocorrelations of squared observations generated by stationary processes and show that the properties of some conditional homoscedasticity tests can be distorted. Second, we obtain the asymptotic and finite sample biases of the ordinary least squares (OLS) estimator of ARCH(p) models. The finite sample results are extended to generalized least squares (GLS), maximum likelihood (ML) and quasi-maximum likelihood (QML) estimators of ARCH(p) and GARCH(1,1) models. Finally, we show that the estimated asymptotic standard deviations are biased estimates of the sample standard deviations.

This study is undertaken to determine the relative impacts of the uncertainty of macroeconomic variables on investment and make policy recommendations that may help dampen their fluctuations. In the study, generalized autoregressive conditional heteroscedasticity (GARCH) model was applied in the estimation of uncertainty of the macroeconomic variables. In the analysis of the data, econometric results were obtained from cointegration test

Financial time series data cannot be adequately modelled by a normal distribution and empirical evidence on the non-normality assumption is very well documented in the financial literature; see [R.F. Engle, Autoregressive conditional heteroskedasticity with estimates of the variance of UK inflation, Econometrica 50 (1982) 987-1008] and [T. Bollerslev, Generalized autoregressive conditional heteroscedasticity, J. Econometrics 31 (1986) 307-327] for details. The kurtosis of various classes of RCA models has been the subject of a study by Appadoo et al. [

This paper uses a time-varying parameter model with generalized autoregressive conditional heteroscedasticity effects to examine the dynamic behavior of crude-oil prices for the period 1997-2008. Using data from four countries of the Gulf Cooperation Council, we find evidence of short-term predictability in oil-price changes over time, except for several short sub-periods. However, the hypothesis of convergence towards weak-form informational efficiency is rejected for all markets. In addition, we explore the possibility of structural breaks in the time-paths of the estimated predictability indices and detect only one breakpoint, for the oil markets in Qatar and United Arab Emirates. Our empirical results therefore call for new empirical research to further gauge the predictability characteristics and the determinants of oil-price changes.

The aim of this paper is to study the dynamic evolution of inflation rate. The model is constructed by extending the ARFIMA-GARCH to ARFIMA with a time varying GARCH model where the transition from one regime to another is evolving smoothly over time. We show by Monte Carlo experiments that the constancy parameter tests perform well. We apply then this new model on eight countries from Europe, Japan and Canada and find that this model is appropriate for six among these countries.

Generalized autoregressive conditional heteroscedasticity (GARCH) models have long been considered as one of the most successful families of approaches for volatility modeling in financial return series. In this paper, we propose an alternative approach based on methodologies widely used in the field of statistical machine learning. Specifically, we propose a novel nonparametric Bayesian mixture of Gaussian process regression models, each component of which models the noise variance process that contaminates the observed data as a separate latent Gaussian process driven by the observed data. This way, we essentially obtain a mixture Gaussian process conditional heteroscedasticity (MGPCH) model for volatility modeling in financial return series. We impose a nonparametric prior with power-law nature over the distribution of the model mixture components, namely the Pitman-Yor process prior, to allow for better capturing modeled data distributions with heavy tails and skewness. Finally, we provide a copulabased approach for obtaining a predictive posterior for the covariances over the asset returns modeled by means of a postulated MGPCH model. We evaluate the efficacy of our approach in a number of benchmark scenarios, and compare its performance to state-of-the-art methodologies.

In order to describe the comovements in both conditional mean and conditional variance of high dimensional nonstationary time series by dimension reduction, we introduce the conditional heteroscedasticity with factor structure to the error correction model. The new model is called the error correction-volatility factor model. Some specification and estimation approaches are developed. In particular, the determination of the number of factors is discussed. Our setting is general in the sense that we impose neither i.i.d assumption on idiosyncratic components in the factor structure nor independence between factors and idiosyncratic errors. We illustrate the proposed approach with a Monte Carlo simulation and a real data example.

This paper reviews the literature on GARCH-type models proposed to represent the dynamic evolution of conditional variances. Effects of level outliers on the diagnostic and estimation of GARCH models are also studied. Both outliers and conditional heteroscedasticity can generate time series with excess kurtosis and autocorrelated squared observations. Consequently, both phenomena can be confused. However, since outliers are generated by unexpected events and the conditional variances are predictable, it is important to identify which one is producing the observed features in the data. We compare two alternative procedures for dealing with the simultaneous presence of outliers and conditional heteroscedasticity in time series. The first one is to clean the series of outliers before fitting a GARCH model. The second is to estimate first the GARCH model and then to clean of outliers by using the residuals adjusted by its conditional variance. It is shown that both approaches may result in different estimated conditional variances.

In this article we propose a statistical model to adjust, interpolate and forecast the term structure of interest rates. This model is based on extensions for the term structure model of interest rates proposed by [Diebold & Li, 2006], through a Bayesian estimation using Markov Chain Monte Carlo. The proposed extensions involve the use of a more flexible parametric form for the yield curve, making all parameters time-varying using a structure of latent factors, and adding a stochastic volatility structure to control the presence of conditional heteroscedasticity observed in the interest rates. The Bayesian estimation enables the exact distribution of estimators in finite samples, and as a sub product, the estimation enables obtaining the distribution of forecasts for the term structure of interest rates. The methodology developed does not need a pre-interpolation of the yield curve as it happens in some econometric models of term structure. We do an empirical exercise of this methodology in which we adjust daily data of the term structure of interest rates implicit in Swap DI-PRÉ contracts traded in the Mercantile and Futures Exchange (BM&F)

Securities Transaction Taxes have received much attention over the last few years with countries and global organizations trying to control the level of speculations, especially since the Global Financial Crisis. This study examines the impact of an increase in the level of securities transaction tax on traded quantity of shares and time series behaviour of stock returns using data from two prominent national stock exchanges of India. We find that when the tax on equity transactions increases from 0.1% to 0.125%, the quantity of traded shares (volume) decreases by more than twenty five percent. Since the volatility of returns on stocks is not constant through time, conditional heteroscedasticity models are used to estimate the volatility of stock returns.

The paper aimed at modelling the density of inflation based on time-varying conditional variance, skewness and kurtosis model developed by Leon, Rubio, and Serna (2005) who model higher-order moments as GARCH-type processes by applying a Gram-Charlier series expansion of the normal density function. Additionally, it extended their work by allowing both conditional skewness and kurtosis to have an asymmetry term.

We attempt empirical detection and characterization of power laws in financial time series. Fractional Brownian motion is defined. After testing for multifractality we calculate the multifractal spectrum of the series. The multifractal nature of stock prices leads to volatility clustering (conditional heteroscedasticity) and long memory (slowly decaying autocorrelation). Wavelet Transform Modulus Maxima approach to multifractal spectrum estimation proved certain advantages to the structure function approach as shown by . We apply wavelet based methodology as wavelets are an appropriate tool for non-stationary time series.

This study employs firm-specific announcements as a proxy for information flows and investigates the information-volatility relation using high-frequency data from the Australian Stock Exchange. Our analysis reveals a positive and significant impact of the arrival rate of the selected news variable on the conditional variance of stock returns, even after controlling for the potential effects of trading volume and high opening volatility. Furthermore, the inclusion of the news variable in the conditional variance equation of the generalized autoregressive conditional heteroscedastic model also reduces volatility persistence, especially with intraday data. Combined with the evidence that news arrivals display a very strong pattern of autocorrelation, our results are consistent with the Mixture of Distribution Hypothesis, which attributes conditional heteroscedasticity of stock returns to time-dependence in the news arrival process.

This study is undertaken to determine the relative impacts of the uncertainty of macroeconomic variables on investment and make policy recommendations that may help dampen their fluctuations. In the study, generalized autoregressive conditional heteroscedasticity (GARCH) model was applied in the estimation of uncertainty of the macroeconomic variables. In the analysis of the data, econometric results were obtained from cointegration test and ordinary least squares. Thus, the study revealed the following: (1) existence of long run relationship between some of the macroeconomic variables and investment, (2) possibility of convergence of the variables from the short run to the long run with slow speed of adjustment and (3) the uncertainty of most of the macroeconomic variables impact negatively on investment in Nigeria. The study recommends the use of appropriate policy instruments that will bring about macroeconomic stability which provide conducive and enabling environment for private investment to thrive.

Abstract.  This paper analyses how outliers affect the identification of conditional heteroscedasticity and the estimation of generalized autoregressive conditionally heteroscedastic (GARCH) models. First, we derive the asymptotic biases of the sample autocorrelations of squared observations generated by stationary processes and show that the properties of some conditional homoscedasticity tests can be distorted. Second, we obtain the asymptotic and finite sample biases of the ordinary least squares (OLS) estimator of ARCH(p) models. The finite sample results are extended to generalized least squares (GLS), maximum likelihood (ML) and quasi-maximum likelihood (QML) estimators of ARCH(p) and GARCH(1,1) models. Finally, we show that the estimated asymptotic standard deviations are biased estimates of the sample standard deviations.

Since volatility is perceived as an explicit measure of risk, financial economists have long been concerned with accurate measures and forecasts of future volatility and, undoubtedly, the Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model has been widely used for doing so. It appears, however, from some empirical studies that the GARCH model tends to provide poor volatility forecasts in the presence of additive outliers. To overcome the forecasting limitation, this paper proposes a robust GARCH model (RGARCH) using least absolute deviation estimation and introduces a valuable estimation method from a practical point of view. Extensive Monte Carlo experiments substantiate our conjectures. As the magnitude of the outliers increases, the one-step-ahead forecasting performance of the RGARCH model has a more significant improvement in two forecast evaluation criteria over both the standard GARCH and random walk models. Strong evidence in favour of the RGARCH model over other competitive models is based on empirical application. By using a sample of two daily exchange rate series, we find that the out-of-sample volatility forecasts of the RGARCH model are apparently superior to those of other competitive models.

In this paper we propose an approach to modelling non-linear conditionally heteroscedastic time series characterised by asymmetries in both the conditional mean and variance. This is achieved by combining a TAR model for the conditional mean with a Changing Parameters Volatility (CPV) model for the conditional variance. Empirical results are given for the daily returns of the S&P 500, NASDAQ composite and FTSE 100 stock market indexes.

Previous studies reach no consensus on the relationship between risk and return using data from one market. We argue that the world market factor should not be ignored in assessing the risk-return relationship in a partially integrated market. Applying a bivariate generalized autoregressive conditional heteroscedasticity in mean (GARCH-M) model to the weekly stock index returns from the UK and the world market, we document a significant positive relationship between stock returns and the variance of returns in the UK stock market after controlling for the covariance of the UK and the world market return. In contrast, conventional univariate GARCH-M models typically fail to detect this relationship. Nonnested hypothesis tests supplemented with other commonly used model selection criteria unambiguously demonstrate that our bivariate GARCH-M model is more likely to be the true model for UK stock market returns than univariate GARCH-M models. Our results have implications for empirical assessments of the risk-return relationship, expected return estimation, and international diversification.

This paper analyses cyclical behaviour of Orange stock price listed in French stock exchange over 01/03/2000 to 02/02/2017 by testing the nonlinearities through a class of conditional heteroscedastic nonparametric models. The linearity and Gaussianity assumptions are rejected for Orange Stock returns and informational shocks have transitory effects on returns and volatility. The forecasting results show that Orange stock prices are short-term predictable and nonparametric NAR-ARCH model has better performance over parametric MA-APARCH model for short horizons. Plus, the estimates of this model are also better comparing to the predictions of the random walk model. This finding provides evidence for weak form of inefficiency in Paris stock market with limited rationality, thus it emerges arbitrage opportunities.

Volatility forecasting is a major area in the pricing of derivative securities, such as stock and index options. In this paper, we compare three methods of forecasting volatility. These are the naive method based on historical sample variance, the exponentially weighted moving average (EWMA) method, and the generalised autoregressive conditional heteroscedasticity (GARCH) model. Out-ofsample forecasts of monthly return variances generated by these three methods are compared. The results strongly favour the EWMA method.

This article investigates the existence of contagion between countries on the basis of an analysis of returns for stock indices over the period 1994 to 2003. The econometrics methodology used is that of multivariate Generalized Autoregressive Conditional Heteroscedasticity (GARCH) family volatility models, particularly the Dynamic Conditional Correlation (DCC) models in the form proposed by Engle and Sheppard (2001). The returns were duly corrected for a series of country-specific fundamentals. The relevance of this procedure is highlighted in the literature by the work of Pesaran and Pick (2003). The results obtained in this article provide evidence favourable for the hypothesis of regional contagion in both Latin America and Asia. As a rule, contagion spread from the Asian crisis to Latin America, but not in the opposite direction.

Novel periodic extensions of dynamic long memory regression models with autoregressive conditional heteroskedastic errors are considered for the analysis of daily electricity spot prices. The parameters of the model with mean and variance specifications are estimated simultaneously by the method of approximate maximum likelihood. The methods are implemented for time series of 1, 200 to 4, 400 daily price observations. Apart from persistence, heteroskedasticity and extreme observations in prices, a novel empirical finding is the importance of day-of-the-week periodicity in the autocovariance function of electricity spot prices. In particular, daily log prices from the Nord Pool power exchange of Norway are modeled effectively by our framework, which is also extended with explanatory variables. For the daily log prices of three European emerging electricity markets (EEX in Germany, Powernext in France, APX in The Netherlands), which are less persistent, periodicity is also highly significant.

The volatility clustering frequently observed in financial/economic time series is often ascribed to GARCH and/or stochastic volatility models. This paper demonstrates the usefulness of reconceptualizing the usual definition of conditional heteroscedasticity as the (h = 1) special case of h-step-ahead conditional heteroscedasticity, where the conditional volatility in period t depends on observable variables up through period t-h. Here it is shown that, for h > 1, h-stepahead conditional heteroscedasticity arises-necessarily and endogenously-from nonlinear serial dependence in a time series; whereas one-step-ahead conditional heteroscedasticity (i.e., h = 1) requires multiple and heterogeneously-skedastic innovation terms. Consequently, the best response to observed volatility clustering may often be to model the nonlinear serial dependence which is likely causing it, rather than 'tacking on' an ad hoc volatility model. Even where such nonlinear modeling is infeasible-or where volatility is quantified using, say, a model-free implied volatility measure rather than squared returns-these results suggest a reconsideration of the usefulness of lag-one terms in volatility models. An application to observed daily stock returns is given.

The purpose of this study is to examine the relationship between firm size and time-varying betas of UK stocks. We extend the Schwert and Seguin (1990)(Journal of Finance 45, 1120–1155) methodology by explicitly modeling conditional heteroscedasticity in the market model residual returns. Our results show that the time-varying coefficient is not statistically significant for both small and large firm stock indexes. We also find that accounting for GARCH effects in the Schwert-Seguin market model yields beta estimates that are markedly differently from those when conditional heteroscedasticity is ignored. Event studies that ignore conditional heteroscedasticity may bias the abnormal returns of small and large firms, thereby leading to a different conclusion regarding the significance of an information event.