Stationarity and the existence of moments of a family of GARCH processes

This paper investigates some structural properties of a family of GARCH processes. A simple sufficient condition for the existence of the αδ-order stationary solution of the processes is derived, where α∈(0,1] and δ>0. The solution is strictly stationary and ergodic, and the causal expansion of the family of GARCH processes is also established. Furthermore, the necessary and sufficient condition for the existence of the moments is obtained. The technique used in this paper for the moment conditions is different from that used in He and Terasvirta (J. Econom. 92 (1999a) 173), and avoids the assumption that the process started at some finite value infinitely many periods ago. Moreover, the conditions for the strict stationarity of the model and the existence of its moments are simple to check and should prove useful in practice.