Quasi -maximum Likelihood Estimation of Dynamic Models with Time Varying Covariances

This paper studies the properties of the quasi -maximum likelihood estimator (QMLE) and related test statistics in dynamic models that jointly parameterize conditional means and conditional covariances when a normal log likelihood is maximized but the assumption of normality is violated. Because the score of the normal log likelihood has the martingale difference property under fairly general regularity conditions provided only that the first two conditional moments are correctly specified, the QMLE is generally consistent and has a limiting normal distribution. Easily computable formulas for asymptotic standard errors that are valid under nonnormality are also available. Further, we show how robust LM tests for the adequacy of the jointly parameterized mean and variance can be computed from simple auxiliary regressions. An appealing feature of these robust inference procedures is that only first derivatives of the conditional mean and variance functions are called for. In addition, the robust tests lose nothing in terms of asymptotic local power if the normality assumption is true. A Monte Carlo study indicates that the asymptotic results carry over to finite samples. Estimation of several AR and AR-GARCH time series models reveals that in most situations the robust form of the test statistics compare favorably to the two standard (nonrobust) formulations of the Wald and LM tests. Also, for the GARCH models and the sample sizes analyzed here, the bias in the exact MLE or the QMLE appear to be relatively small, and typically there is only minor loss in efficiency for the parameters in the conditional mean from not modelling the heteroskedasticity.