Asymptotic theory for a vector ARMA-GARCH model. Econometric Theory 19

This paper investigates the asymptotic theory for a vector ARMA-GARCH model. The conditions for the strict stationarity, ergodicity, and the higher-order moments of the model are established. Consistency of the quasi- maxi-mum likelihood estimator (QMLE) is proved under only the second-order mo-ment condition. This consistency result is new, even for the univariate 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, as well as a consistent estimator of the asymptotic covariance.