Stationarity And High-order Moments For Markov-Switching GARCH Models

Asset-pricing Theory shows that forecast returns, time-varying risk premium, stochastic bubbles, learing mechanism, etc., presents nonlinear structure. A flexible method to capture it is to introduce discrete mixture of distributions, which has time-varying conditional mean and variance and lep-tokurtic unconditional distribution, see Timmermann (2000)~[1]. And Markov-switching model proposed by Hamilton (1989)~[2] is popularly adopted. The use of Markov-switching models to capture the volatility dynamics of financial time series has grown considerably during past years, in part because they give rise to a plausible interpretation of nonlinearities. Nevertheless, GARCH-type models remain ubiquitous in order to allow for nonlinearities associated with time-varying volatility. Existing methods of combining the two approaches are unsatisfactory, as they either suffer from severe estimation difficulties or else their dynamic properties are not well understood. A generalization to Markov-switching GARCH models was developed by Gray (1996)~[3] and subsequently modified by Klaassen (2002)~[4]. Whereas these models can not present satisfying interpretation with the viewpoint of GARCH-typr models(see Haas, Mittnik & Paolella (2004)~[5]. Therefore, Haas, Mittnik & Paolella (2004)~[5] presented a new Markov-switching GARCH model. The disaggregation of the variance process offered by the new model is more plausible than in the existing variants. The approach is illustrated with several exchange rate return series. The results suggest that a promising volatility model is an independent switching GARCH process with a possibly skewed conditional mixture density.This paper investigates some structural properties of the Markov-switchingGARCH processes introduced by Haas, Mittnik & Paolella(2004)[5].A sufficient and necessary condition for the existence of the second-order stationary solution is derived. And the explicit expansion of the stationary solution Markov-switching GARCH process is also established. The technique used in this paper for the stationarity of is different from that used in Haas, Mittnik & Paolella(2004)[5l, and avoids the assumption that the process started in the infinite past with finite variance. Furthermore, using the expansion of the stationary solution, existence for the high-order moments of the process is also presented.