Bayesian Inference on GARCH Models using the Gibbs Sampler

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Abstract
This paper explains how the Gibbs sampler can be used to perform Bayesian inference on GARCH models. Although the Gibbs sampler is usually based on the analytical knowledge of the full conditional posterior densities, such knowledge is not available in regression models with GARCH errors. We show that the Gibbs sampler can be combined with a unidimensional deterministic integration rule applied to each coordinate of the posterior density. The full conditional densities are evaluated and inverted numerically to obtain random draws of the joint posterior. The method is shown to be feasible and competitive compared to importance sampling and the Metropolis-Hastings algorithm. It is applied to estimate an asymmetric GARCH model for the return on a stock exchange index, and to compute predictive densities of option prices on the index.
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