This paper investigates the asymptotic theory for a vector autoregressive moving average–generalized autoregressive conditional heteroskedasticity (ARMA-GARCH) model. The conditions for the strict stationarity, the ergodicity, and the higher order moments of the model are established. Consistency of the quasi-maximum-likelihood estimator (QMLE) is proved under only the second-order moment condition. This consistency result is new, even for the univariate autoregressive conditional heteroskedasticity (ARCH) and GARCH models. Moreover, the asymptotic normality of the QMLE for the vector ARCH model is obtained under only the second-order moment of the unconditional errors and the finite fourth-order moment of the conditional errors. Under additional moment conditions, the asymptotic normality of the QMLE is also obtained for the vector ARMA-ARCH and ARMA-GARCH models and also a consistent estimator of the asymptotic covariance.The authors thank the co-Editor, Bruce Hansen, and two referees for very helpful comments and suggestions and acknowledge the financial support of the Australian Research Council.

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.