Stationarity for a Markov-switching Box–Cox transformed threshold GARCH process

In order to capture three important dynamic characteristics of time series, the asymmetry, regimes, and conditional heteroskedasticity, based on Hwang and Basawa's [2004. Stationarity and moment structure for Box–Cox transformed threshold GARCH(1,1) processes. Statist. Probab. Lett. 68, 209–220] and Haas et al. [2004. A new approach to Markov-switching GARCH models. J. Financial Econometrics 2, 493–530] models, this paper proposes a Markov-switching Box–Cox transformed threshold GARCH model. Some structural properties of this new GARCH process are considered. First, a sufficient and necessary condition for the existence of the weakly and strictly stationary solution of the process is presented, respectively. Second, the general conditions for the existence of high-order moments of the process are derived. The technique used in this paper for the weak stationarity and the high-order moments of the process is different from that used in Haas et al. [2004. A new approach to Markov-switching GARCH models. J. Financial Econometrics 2, 493–530], and avoids the assumption that the process started in the infinite past with finite variance.