Stationarity of a Markov-Switching GARCH Model

Abstract

This article investigates some structural properties of the Markov-switching GARCH process introduced by Haas, Mittnik, and Paolella. First, a sufficient and necessary condition for the existence of the weakly stationary solution of the process is presented. The solution is weakly stationary, and the causal expansion of the Markov-switching GARCH process is also established. Second, the general conditions for the existence of any integer-order moment of the square of the process are derived. The technique used in this article for the weak stationarity and the high-order moments of the process is different from that used by Haas, Mittnik, and Paolella and avoids the assumption that the process started in the infinite past with finite variance. Third, a sufficient and necessary condition for the strict stationarity of the Markov-switching GARCH process with possibly infinite variance is given. Finally, the strict stationarity of the so-called integrated Markov-switching GARCH process is also discussed.