Most of discrete stochastic processes used in financial econometrics belong to one of the two leading classes: the GARCH (Generalised AutoRegressive Conditionally Heteroskedastic) class or the SV (Stochastic Volatility or Stochastic Variance) class. In both, the parametric distributional assumptions (like conditional normality or the conditional Student t distribution) are usually made. However, despite the large number of observations that are often available, classical statistical inference is not easy for any of these classes, because either the (asymptotic) theory is not well established or the computations are very time-consuming, especially when the latent variables are present in the model (as it is in the SV case). The Bayesian approach to statistical inference, equipped with the numerical tools based on the Markov Chain Monte Carlo (MCMC) methods, is an interesting alternative. It provides the researcher with fully probabilistic, non-asymptotic and very intuitive inference procedures.

The paper reviews the Bayesian statistical approach to parameter inference and forecasting as well as to model comparison and pooling (averaging). The aim is to show how Bayesian techniques are used to compare different non-nested multivariate specifications proposed in financial econometrics. Since some of the models are very unparsimonious, the actual comparison is restricted to bivariate cases. The review of some recent empirical work is presented, which shows that the SV class (based on latent processes describing volatility), can fit the data much better. But the most flexible SV models use at least as many latent processes as there is financial time series to be modelled, which makes them of little practical interest in really multivariate settings. Also, the best GARCH models involve too many parameters to be of practical interest in such cases. Thus, a hybrid GARCH-SV specification based on just one latent process and a relatively parsimonious GARCH-type model is proposed.

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1. Bayesian Value-at-Risk and Expected Shortfall for a portfolio (multi- and univariate approaches)