Anna Pajor

Multivariate models of asset returns are very important in financial applications. Asset allocation, risk assessment and construction of an optimal portfolio require estimates of the covariance matrix between the returns of assets (see e.g. Aguilar and West (2000), Pajor (2005a, 2005b)). Similarly, hedges require a covariance matrix of all the assets in the hedge. There are two main types of volatility models for asset returns: the Generalised Autoregressive Conditional Heteroscedasticity (GARCH) and the Stochastic Volatility (SV) families. The GARCH models define the timevarying covariance matrix as a deterministic function of past squared innovations and lagged conditional variances and covariances, whereas the conditional covariance matrix in the SV models is treated as an unobserved component that follows some separate multivariate stochastic process. The first multivariate SV model proposed in the literature by Harvey, Ruiz and Shephard (1994) allowed the variances of multivariate returns to vary over time, but constrained the correlations to be constant. Pitt and Shephard (1999) proposed a factor SV model, which allows a parsimonious representation of the time series evolution of covariances when the number of series being modelled is very large. Simple multivariate factor models for SV processes have been suggested, but not applied, by Jacquier, Polson and Rossi (1995, 1999). Tsay (2002) proposed the SV process based on the Cholesky decomposition of the conditional covariance matrix. A practical drawback of stochastic volatility