Volatility Modeling Using the Student’s t Distribution Maria S. Heracleous Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Economics Aris Spanos, Chair Richard Ashley Raman Kumar Anya McGuirk Dennis Yang August 29, 2003 Blacksburg, Virginia Keywords: Student’s t Distribution, Multivariate GARCH, VAR, Exchange Rates Copyright 2003, Maria S. Heracleous Volatility Modeling Using the Student’s t Distribution Maria S. Heracleous (ABSTRACT) Over the last twenty years or so the Dynamic Volatility literature has produced a wealth of uni- variateandmultivariateGARCHtypemodels.Whiletheunivariatemodelshavebeenrelatively successful in empirical studies, they suffer from a number of weaknesses, such as unverifiable param- eter restrictions, existence of moment conditions and the retention of Normality. These problems are naturally more acute in the multivariate GARCH type models, which in addition have the problem of overparameterization. This dissertation uses the Student’s t distribution and follows the Probabilistic Reduction (PR) methodology to modify and extend the univariate and multivariate volatility models viewed as alternative to the GARCH models. Its most important advantage is that it gives rise to internally consistent statistical models that do not require ad hoc parameter restrictions unlike the GARCH formulations. Chapters 1 and 2 provide an overview of my dissertation and recent developments in the volatil- ity literature. In Chapter 3 we provide an empirical illustration of the PR approach for modeling univariate volatility. Estimation results suggest that the Student’s t AR model is a parsimonious and statistically adequate representation of exchange rate returns and Dow Jones returns data. Econometric modeling based on the Student’s t distribution introduces an additional variable the degree of freedom parameter. In Chapter 4 we focus on two questions relating to the ‘degree of− freedom’ parameter. A simulation study is used to examine: (i) the ability of the kurtosis coefficient to accurately capture the implied degrees of freedom, and (ii) the ability of Student’s t GARCH model to estimate the true degree of freedom parameter accurately. Simulation results reveal that the kurtosis coefficient and the Student’s t GARCH model (Bollerslev, 1987) provide biased and inconsistent estimators of the degree of freedom parameter. Chapter 5 develops the Students’ t Dynamic Linear Regression (DLR) model which allows us to explain univariate volatility in terms of: (i) volatility in the past history of the series itself and (ii) volatility in other relevant exogenous variables. Empirical results of this chapter suggest that the Student’s t DLR model provides a promising way to model volatility. The main advantage of this model is that it is defined in terms of observable random variables and their lags, and not the errors as is the case with the GARCH models. This makes the inclusion of relevant exogenous variables a natural part of the model set up. In Chapter 6 we propose the Student’s t VAR model which deals effectively with several key issues raised in the multivariate volatility literature. In particular, it ensures positive definiteness of the variance-covariance matrix without requiring any unrealistic coefficient restrictions and provides a parsimonious description of the conditional variance-covariance matrix by jointly modeling the conditional mean and variance functions. Acknowledgments I am extremely grateful to my advisor, Aris Spanos, for encouraging me to pursue a Ph.D. and helping me navigate through the complex field of Econometrics. His inspiring teaching and invalu- able advice have been absolutely vital for the development of this research. My special thanks also go to Anya McGuirk for her genuine commitment, constructive suggestions and her enthusiastic support that enabled me to complete my dissertation. I would like to thank my other committee members, Richard Ashley, Raman Kumar and Dennis Yang for their valuable suggestions and interest in my work. I am also grateful to Abon Mozumdar and Andrew Feltenstein who were originally on my committee and gave me guidance when I first started working on this dissertation. My sincere thanks go to Sudipta Sarangi for his entertaining phone calls and constant encour- agement, and to Andreas Koutris for making sure I had fun while writing the dissertation. I am also thankful to the staff members of our department: Sherry Williams, Barbara Barker and Mike Cutlip for all their assistance. The department would not be the same without them. Special thanks go to my roommates: Ana Martin, for her endless stories during my first year of adjustment in Blacksburg and Yea Sun Chung for her patience during my last year of writing. For the three years in between I am particularly thankful to Stavros Tsiakkouris who helped me with all kinds of little things and made my life in Blacksburg hi-tech, easier and fun! Last but not least, I would like to thank my parents, Stelios and Panayiota, who have taught me the important things in life and have supported all my endeavors. My parents and my sister Elpida have given me great affection and support while patiently enduring my long absence from home. iii Contents 1 Introduction 1 1.1Background......................................... 1 1.2EconometricsofMultivariateVolatilityModels..................... 5 1.3ABriefOverview...................................... 8 2 Towards a Unifying Methodology for Volatility Modeling 14 2.1Introduction......................................... 14 2.2GARCHTypeModels:Univariate............................ 16 2.3GARCHTypeModels:Multivariate........................... 21 2.4ProbabilisticReductionApproach............................ 32 2.4.1 The VAR(1) from the Probabilistic Reduction Perspective . 36 2.5Conclusion......................................... 38 3 Univariate Volatility Models 40 3.1Introduction......................................... 40 3.2APicture’sWorthaThousandWords.......................... 41 3.3StatisticalModelComparisons.............................. 53 3.3.1 NormalAutoregressiveModel........................... 54 3.3.2 HeteroskedasticModels.............................. 55 iv 3.3.3 Student’s t Autoregressive Model with Dynamic Heteroskedasticity . 57 3.4EmpiricalResults...................................... 59 3.5Conclusion......................................... 69 4 Degrees of Freedom, Sample Kurtosis and the Student’s t GARCH Model 70 4.1SimulationSetUp..................................... 71 4.1.1 TheoreticalQuestions............................... 71 4.1.2 DataGeneration.................................. 72 4.2Results............................................ 74 4.2.1 Estimates of α4 andtheImpliedDegreesofFreedom.............. 75 4.2.2 Estimates of the Student’s t GARCHparameters................ 76 4.3Conclusion......................................... 82 5Student’st Dynamic Linear Regression 83 5.1 Student’s t DLRModel.................................. 84 5.1.1 Specification.................................... 84 5.1.2 MaximumLikelihoodEstimation......................... 88 5.2EmpiricalResults...................................... 91 5.2.1 DataandMotivation................................ 91 5.2.2 Empirical SpecificationandResults....................... 92 5.3Conclusion.........................................106 6 The Student’s t VA R M o d e l 1 0 7 6.1Introduction.........................................107 6.2StatisticalPreliminaries..................................108 6.2.1 Matrix Variate t distribution...........................109 6.2.2 SomeImportantResultsonToeplitzMatrices..................115 v 6.2.3 Matrix Calculus and Differentials.........................118 6.3 Student’s t VARModel..................................121 6.3.1 Specification....................................121 6.3.2 MaximumLikelihoodEstimation.........................130 6.4StatisticalModelComparisons..............................134 6.5Conclusion.........................................137 7Conclusion 139 AStudent’st VAR Derivatives 156 vi List of Figures 3-1 Standardized t-plot (DEM) .................................. 43 3-2 Standardized t-plot (FRF) .................................. 43 3-3 Standardized t-plot (CHF) .................................. 44 3-4 Standardized t-plot (GBP) .................................. 44 3-5 Standardized Normal P-P plot (FRF) ............................ 46 3-6 Standardized Student’s t P-P plot, ν =6, (FRF) ...................... 46 3-7 Standardized Student’s t P-P plot, ν =7, (FRF) ...................... 47 3-8 Standardized Student’s t P-P plot, ν =8, (FRF) ...................... 47 3-9 Standardized t-plot (DJ) ................................... 49 3-10 Standardized Normal P-P plot (DJ) ............................. 49 3-11 Standardized Student’s t P-P plot, ν =3, (DJ) ....................... 50 3-12 Standardized Student’s t P-P plot, ν =4, (DJ) ....................... 50 3-13 Standardized Student’s t P-P plot, ν =5, (DJ) ....................... 51 4-1 Kernel density for nu, ν =6, σ2 =1,n=50........................ 79 4-2 Kernel density for nu, ν =6, σ2 =1,n=100....................... 79 4-3 Kernel density for nu, ν =6, σ2 =1,n=500....................... 80 4-4 Kernel density for nu, ν =6, σ2 =1,n=1000....................... 80 4-5 Kernel density for nu, ν =6, σ2 =0.25,n=500..................... 81 vii 4-6 Kernel density for nu, ν =6, σ2 =4,n=500......................