NORGES TEKNISK-NATURVITENSKAPELIGE UNIVERSITET Approximate Bayesian Inference for Multivariate Stochastic Volatility Models

In this report we apply Integrated Nested Laplace approximation (INLA) to a series of multivariate stochastic volatility models. These are a useful construct in financial time series analysis and can be formulated as latent Gaussian Markov Random Field (GMRF) models. This popular class of models is characterised by a GMRF as the second stage of the hierarchical structure and a vector of hyperparameters as the third stage. INLA is a new tool for fast, deterministic inference on latent GMRF models which provides very accurate approximations to the posterior marginals of the model. We compare the performance of INLA with that of some Markov Chain Monte Carlo (MCMC) algorithms run for a long time showing that the approximations, despite being computed in only a fraction of time with respect to MCMC estimations, are practically exact. The INLA approach uses numerical schemes to integrate out the uncertainty with respect to the hyperparameters. In this report we cope with problems deriving from an increasing dimension of the hyperparameter vector. Moreover, we propose different approximations for the posterior marginals of the hyperparameters of the model. We show also how Bayes factors can be efficiently approximated using the INLA tools thus providing a base for model comparison.