We provide a detailed summary of the large and vibrant emerging literature
that deals with the multivariate modeling of conditional volatility of
financial time series within the framework of stochastic volatility. The
developments and achievements in this area represent one of the great
success stories of financial econometrics. Three broad classes of
multivariate stochastic volatility models have emerged, one that is a direct
extension of the univariate class of stochastic volatility model, another
that is related to the factor models of multivariate analysis, and a third
that is based on the direct modeling of time-varying correlation matrices
via matrix exponential transformations, Wishart processes and other means.
We discuss each of the various model formulations, provide connections and
differences and show how the models are estimated. Given the interest in
this area, further significant developments can be expected, perhaps
fostered by the overview and details delineated in this paper, especially in
the fitting of high dimensional models.