Multivariate stochastic volatility using state space models

A Bayesian procedure is developed for multivariate stochastic volatility, using state space models. An autoregressive model for the log-returns is employed. We generalize the inverted Wishart distribution to allow for different correlation structure between the observation and state innovation vectors and we extend the convolution between the Wishart and the multivariate singular beta distribution. A multiplicative model based on the generalized inverted Wishart and multivariate singular beta distributions is proposed for the evolution of the volatility and a flexible sequential volatility updating is employed. The proposed algorithm for the volatility is fast and computationally cheap and it can be used for on-line forecasting. The methods are illustrated with an example consisting of foreign exchange rates data of 8 currencies. The empirical results suggest that timevarying correlations can be estimated efficiently, even in situations of high dimensional data. Some key words: volatility, multivariate, GARCH, time series, state space model, Bayesian forecasting, dynamic linear model, Kalman filter, generalized Wishart distribution.