Markov Chain Monte Carlo versus Importance Sampling in Bayesian Inference of the GARCH Model

Abstract

Usually, the Bayesian inference of the GARCH model is preferably performed by the Markov Chain Monte Carlo (MCMC) method. In this study, we also take an alternative approach to the Bayesian inference by the importance sampling. Using a multivariate Student's t-distribution that approximates the posterior density of the Bayesian inference, we compare the perfor- mance of the MCMC and importance sampling methods. The overall performance can be measured in terms of statistical errors obtained for the same size of Monte Carlo data. The Bayesian inference of the GARCH model is performed by the MCMC method implemented by the Metropolis-Hastings algorithm and the importance sampling method for artificial return data and stock return data. We find that the statistical errors of the GARCH parameters from the importance sampling are smaller than or comparable to those obtained from the MCMC method. Therefore we conclude that the importance sampling method can also be applied effectively for the Bayesian inference of the GARCH model as an alternative method to the MCMC method.