Likelihood inference for Markov switching GARCH(1,1) models using sequential Monte Carlo

Abstract

Markov switching (MS-)GARCH(1,1) models allow for structural changes in volatility dynamics between a finite number of regimes. Since the regimes are not observed, computation of the likelihood requires integrating over an exponentially increasing number of regime paths, which is intractable. An existing smooth likelihood estimation procedure for sequential Monte Carlo (SMC), that is currently limited to hidden Markov models with a one-dimensional state variable, is modified to enable likelihood estimation and maximisation for MS-GARCH(1,1) models, a model which requires two dimensions, volatility and regime, to evolve its hidden state process. Furthermore, the modified SMC procedure is shown to be easily adapted to fitting MS-GARCH(1,1) models even when there are missing observations. The proposed methodology is validated with simulated data and is also illustrated with analysis of two financial time series, the daily returns on the S&P 500 index and on the Henry Hub natural gas spot price, with the latter series containing a gap caused by shutdown in response to hurricane Rita in 2005.