Bayesian analysis of stochastic volatility models : modelling and applicatons

It has long been recognised that the return volatility of financial assets tends to vary over time with persistence. One way of modelling this feature is allowing the conditional variance to be a function of previous observations and past variances, which leads to the ARCH—type models developed by Engle (1982). An alternative to the ARCH framework is a model which specifies the variance to follow a latent stochastic process, known as the stochastic volatility (SV) model. A fundamental theme of this thesis is the extension of current SV model specifications in a Bayesian framework: firstly to increase model flexibility in capturing prominent empirical characteristics of financial time series data and secondly widen the range of applications of SV models. Firstly, we consider a heavy-tailed SV model with leverage effect, modelled by a (negative) correlation between the innovations of returns and latent volatilities. Choy et al. (2008) investigated this model by assigning a bivariate Student—t distribution as a scale mixture of bivariate normal distributions. We propose a novel alternative formulation by first deriving a conditional Student—t distribution for the returns and a marginal Student-t distribution for the volatilities. This is followed by an employment of a scale mixture of normals representation for these two Student—t distributions. Our approach separates the sources of outliers by distinguishing between outliers generated by the return process or by the volatility process, and hence providing an improvement over the approach of Choy et al. (2008). With motivation from the finding that the SV model with a conditional normal distribution does not adequately account for the leptokurtic distribution of the returns, we study the use of different heavy—tailed distributions in SV models and investigate the effects of choosing different conditional distributions for the returns on both the estimates of model parameters and latent volatilities. More specifically, we propose the use of a generalised—t (GT) distribution, which incorporates commonly used distributions in the SV literature, including the Student—t and exponential power distributions, into a unified distributional framework, expressed as a scale mixture of uniform distributions. The SV model with GT distribution is further extended to accommodate the asymmetric behaviour between the returns and volatilities, adopting an alternative specification to the bivariate SV model in which the leverage effect is modelled through a constant correlation coefficient of the bivariate distribution. In this alternative specification, both the sign and the magnitude of the previous return is included in the volatility equation (Asai and McAleer (2005a); So et al. (2002)). Asai and McAleer (2005a) showed that a specific case of their proposed asymmetric SV model with normal error distribution leads to time varying correlation and we generalise this result to the case of the GT distribution. This time varying feature provides a more flexible way to model the dynamic of asymmetric effect compared to the constant correlation between the return and volatility processes. In view of another prominent characteristic, we extend the SV model to take into account the skewness observed in financial asset returns by replacing the normality assumption of the returns with the asymmetric Laplace (AL) distribution. In addition, past observations and exogenous variables are incorporated into the model to examine possible effects on the returns. When the AL distribution is used as the error distribution for the returns, the model can be interpreted from a different perspective: a quantile regression model where heteroscedasticity is controlled by a SV model for the volatility. This study is a stochastic extension of Chen et a1. (2009) who assumed a GARCH specification to account for heteroscedasticity in a quantile regression problem. Furthermore, we consider a multivariate extension of the SV model to explain the cross correlation between multivariate time series. In response to the lack of literature in the modelling of electricity price and price volatility of the Australian wholesale spot electricity market using SV models, we explore the applicability of multivariate SV models in this field. Different specifications regarding both the correlation structure and distributional assumption of the multivariate SV models are adopted and compared using empirical data. For statistical inference, these extended models are implemented using Bayesian Markov chain Monte Carlo method with Gibbs sampler algorithm. A series of simulation experiments and empirical studies are conducted to demonstrate the properties and applicability of the proposed models. Extended models are shown to provide satisfactory performance in both simulation studies and real data analyses, and they have made substantial contributions to enrich the SV model literature