Modele hybrydowe MSV-MGARH z trzema procesami ukrytymi w badaniu zmienności cen na różnych rynkach

J. Osiewalski and A.Pajor (2007, 2009) and J. Osiewalski (2009) introduced hybrid multivariate stochastic variance — GARCH (MSV-MGARCH) models, where the conditional covariance matrix is the product of a univariate latent process and a matrix with a simple MGARCH structure (Engle's DCC or scalar BEKK). The aim was to parsimoniously describe volatility of a large group of assets. The proposed hybrid specifications, similarly as other models from the MSV class, require the Bayesian approach equipped with MCMC simulation tools. In order to jointly describe volatility on two different markets (or of two different groups of assets), J. Osiewalski and K.Osiewalski (2011) consider more complicated hybrid models with two latent processes. These new specifications seem very promising due to their good fit and moderate computational requirements. This paper is devoted to hybrid specifications with three latent processes, even more complicated and located on the edge of possibilities of conducting exact Bayesian analysis. We present full Bayesian inference for such models and propose efficient MCMC simulation strategy. Our approach is used to jointly model volatility of six daily time series representing three different groups: two stock indices, prices of gold and silver, prices of oil and natural gas. We formally compare joint modelling to individual bivariate volatility modelling for each of three groups.