Virtually all empirical studies that assume a time-varying conditional variance use a quasi-maximum likelihood estimator (QMLE). If the density from which the likelihood is constructed is assumed to be Gaussian, the QMLE is known to be consistent under correct specification of both the conditional mean and conditional variance. We show that if both the assumed density and the true density are symmetric a QMLE remains consistent. If, however, either the assumed density or the true density is asymmetric, a QMLE is generally not consistent. To ensure that a QMLE is consistent under asymmetric densities, we include the conditional standard deviation as a regressor. We calculate the efficiency loss associated with the added regressor if the densities are symmetric and show that for a QMLE of the conditional variance parameters of a GARCH process there is no efficiency loss. Finally, we develop a test of consistency of a QMLE from the significance of the additional regressor.

This note proves the consistency and asymptotic normality of the Quasi-Maximum Likelihood Estimator (QMLE) of the parameters of a GARCH model with martingale difference centered squared innovations. The results are obtained under mild conditions and generalize and improve those in Lee and Hansen (1994) for the local QMLE in semi-strong GARCH(1,1) models. In particular, no restrictions on the conditional mean are imposed. Our proofs closely follow those in Francq and Zakoian (2004) for independent and identically distributed innovations.