Volatility Modeling Using the Student's t Distribution

Over the last twenty years or so the Dynamic Volatility literature has produced a wealth of univariateand multivariate GARCH type models. While the univariate models have been relativelysuccessful in empirical studies, they suffer from a number ofweaknesses, such as unverifiable parameterrestrictions, existence of moment conditions and the retention of Normality. These problemsare naturally more acute in the multivariate GARCH type models, which in addition have theproblem 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 asalternative to the GARCH models. Its most important advantage is that it gives rise to internallyconsistent statistical models that do not require ad hoc parameter restrictions unlike the GARCHformulations. Chapters 1 and 2 provide an overview of my dissertation and recent developments in the volatilityliterature. In Chapter 3 we provide an empirical illustration of the PR approach for modelingunivariate volatility. Estimation results suggest that the Student's t AR model is a parsimoniousand 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 offreedom' parameter. A simulation study is used to examine:(i) the ability of the kurtosis coefficientto accurately capture the implied degrees of freedom, and (ii) the ability of Student's t GARCHmodel to estimate the true degree of freedom parameter accurately. Simulation results reveal thatthe kurtosis coefficient and the Student's t GARCH model (Bollerslev, 1987) provide biased andinconsistent estimators of the degree of freedom parameter. Chapter 5 develops the Students' t Dynamic Linear Regression (DLR) }model which allows usto 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 thatthe Student's t DLR model provides a promising way to model volatility. The main advantage ofthis model is that it is defined in terms of observable random variables and their lags, and notthe errors as is the case with the GARCH models. This makes the inclusion of relevant exogenous variablesa natural part of the model set up. In Chapter 6 we propose the Student's t VAR model which deals effectively with several keyissues raised in the multivariate volatility literature. In particular, it ensures positive definiteness ofthe variance-covariance matrix without requiring any unrealistic coefficient restrictions and providesa parsimonious description of the conditional variance-covariance matrix by jointly modeling theconditional mean and variance functions.