Marginal likelihood for Markov-switching and change-point GARCH models

GARCH volatility models with fixed parameters are too restrictive for long time series due to breaks in the volatility process. Flexible alternatives are Markov-switching GARCH and change-point GARCH models. They require estimation by MCMC methods due to the path dependence problem. An unsolved issue is the computation of their marginal likelihood, which is essential for determining the number of regimes or change-points. We solve the problem by using particle MCMC, a technique proposed by  Andrieu et al. (2010). We examine the performance of this new method on simulated data, and we illustrate its use on several return series.