Hybrid MSV-MGARCH models, in particular the MSF-SBEKKspecification, proved useful in multivariate modelling of returns on financialand commodity markets. The initial MSF-MGARCH structure, called LN-MSF-MGARCH here, is obtained by multiplying the MGARCH conditionalcovariance matrixHtby a scalar random variablegtsuch that{lngt, t∈Z} is aGaussian AR(1) latent process with auto-regression parameterφ. Here we alsoconsider an IG-MSF-MGARCH specification, which is a hybrid generalisationof conditionally StudenttMGARCH models, since the latent process{gt}is nolonger marginally log-normal (LN), but forφ= 0it leads to an inverted gamma(IG) distribution forgtand to thet-MGARCH case. If φ6= 0, the latentvariablesgtare dependent, so (in comparison to thet-MGARCH specification)we get an additional source of dependence and one more parameter. Dueto the existence of latent processes, the Bayesian approach, equipped withMCMC simulation techniques, is a natural and feasible statistical tool to dealwith MSF-MGARCH models. In this paper we show how the distributionalassumptions for the latent process together with the specification of theprior density for its parameters affect posterior results, in particular theones related to adequacy of thet-MGARCH model. Our empirical findingsdemonstrate sensitivity of inference on the latent process and its parameters,but, fortunately, neither on volatility of the returns nor on their conditionalcorrelation. The new IG-MSF-MGARCH specification is based on a morevolatile latent process than the older LN-MSF-MGARCH structure, so thenew one may lead to lower values of φ– even so low that they can justify thepopulart-MGARCH model.