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).