On the Stationarity of Markov-Switching GARCH Processes

GARCH models with Markov-switching regimes are often used for volatility analysis of …nancial time series. Such models imply less persistence in the conditional variance than the standard GARCH model, and potentially provide a signi…cant 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 su¢ cient conditions for the asymptotic wide-sense stationarity of two di¤erent 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. 1 Introduction Volatility analysis of …nancial time series is of major importance in many …nancial applications. The generalized autoregressive conditional heteroskedasticity (GARCH) model [1] has been applied quite extensively in the …eld of econometrics, both by practitioners and by researchers, and shown to be useful for the analysis and forecasting the volatility of time-varying processes such as those pertaining to …nancial markets. Incorporating GARCH models with a hidden Markov chain, where each state of the chain (regime) allows a di¤erent GARCH behavior and thus a di¤erent volatility structure, extends the dynamic formulation of the model and potentially enables improved forecasts of the volatility [2–7]. Unfortunately, the volatility of a GARCH process with switching-regimes depends on the entire history of the process, including the regime path, which makes the derivation of a volatility estimator impractical. Cai [8] and Hamilton and Susmel [9] applied the idea of regime-switching parameters into ARCH speci…cation. The conditional variance of an ARCH model depends only on past observations, and accordingly the restriction to ARCH models avoids problems of in…nite path dependency. Gray [2], Klaassen [3] and