Alternative estimators of the covariance matrix in GARCH models

With most of the available software packages, estimates of the parameter covariance matrix in a GARCH model are usually obtained from the outer products of the first derivatives of the log-likelihoods (BHHH estimator). However, other estimators could be defined and used, analogous to the covariance matrix estimators in maximum likelihood studies described in the literature for other types of models (linear regression model, linear and nonlinear simultaneous equations, Probit and Tobit models). These alternative estimators can be derived from: (1) the Hessian (observed information), (2) the estimated information (expected Hessian), (3) a mixture of Hessian and outer products matrix (White's QML covarjance matrix). Signifacant differences among these estimates can be interpreted as an indication of misspecification, or can be due to systematic inequalities between alternative estimators in small samples. Unlike other types of models, from our Monte Carlo study we do not encounter very large differences, presumably because GARCH estimation is usually applied when the sample size is rather large. However, analogously to otber types of models we find in this Monte Carlo study that, even in absence of misspecification, the sign of the differences between some estimators is almost systematic. This suggests that, as for other types of models, the choice of the covariance estimator is not neutral, but the results of hypotheses testing are not strongly affected by such a choice.