Analysis of Financial Time Series, Third Edition

12.7 Stochastic Volatility Models

An important financial application of MCMC methods is the estimation of stochastic volatility models; see Jacquier, Polson, and Rossi (1994) and the references therein. We start with a univariate stochastic volatility model. The mean and volatility equations of an asset return rt are

12.20

12.21

where {xit|i = 1, … , p} are explanatory variables available at time t − 1, the βj are parameters, {ϵt} is a Gaussian white noise sequence with mean 0 and variance 1, {vt} is also a Gaussian white noise sequence with mean 0 and variance , and {ϵt} and {vt} are independent. The log transformation is used to ensure that ht is positive for all t. The explanatory variables xit may include lagged values of the return (e.g., xit = rt−i). In Eq. (12.21), we assume that |α1| < 1 so that the log volatility process ln ht is stationary. If necessary, a higher order AR(p) model can be used for ln ht.

Denote the coefficient vector of the mean equation by = (β0, β1, … , βp)′ and the parameter vector of the volatility equation by . Suppose that is the collection of observed ...