On The Stationarity Of Markov-Switching Garch Processes

Generalized autoregressive conditional heteroskedasticity (GARCH) models with Markov-switching regimes are often used for volatility analysis of financial time series. Such models imply less persistence in the conditional variance than the standard GARCH model and potentially provide a significant improvement in volatility forecast. Nevertheless, conditions for asymptotic wide-sense stationarity have been derived only for some degenerated models. In this paper, we introduce a comprehensive approach for stationarity analysis of Markov-switching GARCH models, which manipulates a backward recursion of the model's second-order moment. A recursive formulation of the state-dependent conditional variances is developed, and the corresponding conditions for stationarity are obtained. In particular, we derive necessary and sufficient conditions for the asymptotic wide-sense stationarity of two different variants of Markov-switching GARCH processes and obtain expressions for their asymptotic variances in the general case of m-state Markov chains and (p,q)-order GARCH processes.The authors thank Professor Rami Atar for helpful discussions. The authors thank the co-editor Bruce Hansen and the three anonymous referees for their helpful comments and suggestions and in particular the referee who proposed a generalization of the proof in Appendix B. This research was supported by the Israel Science Foundation (grant 1085/05).