Markov chain Monte Carlo methods for stochastic volatility models

This paper is concerned with simulation-based inference in generalized models of stochastic volatility defined by heavy-tailed Student-t distributions (with unknown degrees of freedom) and exogenous variables in the observation and volatility equations and a jump component in the observation equation. By building on the work of Kim, Shephard and Chib (Rev. Econom. Stud. 65 (1998) 361), we develop efficient Markov chain Monte Carlo algorithms for estimating these models. The paper also discusses how the likelihood function of these models can be computed by appropriate particle filter methods. Computation of the marginal likelihood by the method of Chib (J. Amer. Statist. Assoc. 90 (1995) 1313) is also considered. The methodology is extensively tested and validated on simulated data and then applied in detail to daily returns data on the S&P 500 index where several stochastic volatility models are formally compared under different priors on the parameters.