Asymptotic Theory For a Vector Arma-Garch Model

摘要

This paper investigates the asymptotic theory for a fector autoregressive moving average-generalized autoregressive conditional heteroskedasticity (ARMA-GARCH) model. The conditions for the strict stationary, the ergyodicity, and the higher order moments of the model are established. Consistency of the quasi-maximum-likelihood estimator (QMLE) is proved uner only the second-order moment condition. This consistency result is new, even for the univariate auto-regressive conditional heteroskedasiticity (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.