Necessary and sufficient moment conditions for the GARCH(r,s) and asymmetric power GARCH(r,s) models

Although econometricians have been using Bollerslev's (1986, Journal of Econometrics 31, 307-327) GARCH(r,s) model for over a decade, the higher order moment structure of the model remains unresolved. The sufficient condition for the existence of the higher order moments of the GARCH(r,s) model was given by Ling (1999a, Journal of Applied Probability 36, 688-705). This paper shows that Ling's condition is also necessary. As an extension, the necessary and sufficient moment conditions are established for Ding, Granger, and Engle's (1993, Journal of Empirical Finance, 1, 83-106) asymmetric power GARCH(r,s) model.