The autocorrelation structure of the Markov-switching asymmetric power GARCH process

In a recent paper, Liu [Liu, J.-C., 2007. Stationarity for a Markov-switching Box-Cox transformed threshold GARCH process. Statistics and Probability Letters 77, 1428-1438] proposed a generalization of the Markov-switching GARCH model of Haas et al. [Haas, M., Mittnik, S., Paolella, M.S., 2004. A new approach to Markov-switching GARCH models. Journal of Financial Econometrics 2, 493-530] to allow for a nonlinear relation between past shocks and future volatility as well as for the leverage effect, which refers to the observation that stock market volatility reacts differently to positive and negative shocks. For the new model, Liu [Liu, J.-C., 2007. Stationarity for a Markov-switching Box-Cox transformed threshold GARCH process. Statistics and Probability Letters 77, 1428-1438] derived conditions for stationarity and the existence of moments. This article complements Liu's [Liu, J.-C., 2007. Stationarity for a Markov-switching Box-Cox transformed threshold GARCH process. Statistics and Probability Letters 77, 1428-1438] results in two directions. First, a simple method for calculating the moments and the autocorrelation structure of the power-transformed absolute process is devised, which is of vital interest in applied GARCH modeling. Second, in an application to stock returns, the relevance of the extended Markov-switching GARCH process proposed by Liu [Liu, J.-C., 2007. Stationarity for a Markov-switching Box-Cox transformed threshold GARCH process. Statistics and Probability Letters 77, 1428-1438], as compared to simpler versions, is illustrated.