The L²-structures of standard and switching-regime GARCH models

This paper analyzes the probabilistic structure of Markov-switching GARCH(p,q) models, in which the volatility process is driven by a finite state-space Markov chain. We give necessary and sufficient conditions for the existence of moments of any order. We find that the squares and higher order powers of the process have the L² structures of ARMA processes, and hence admit ARMA representations. These results are applicable to standard GARCH models and have statistical implications in terms of order identification and parameter estimation.