Efficiency comparisons of maximum-likelihood-based estimators in GARCH models

In this paper, we investigate the loss of asymptotic efficiency of semiparametric and quasi-maximum-likelihood estimators relative to maximum-likelihood estimators in models with generalized autoregressive conditional heteroscedasticity (GARCH). For a general time-varying location–scale model, the factors that contribute to differences in efficiency among the estimators can be divided in two categories. One pertains to the parametric specifications of the conditional mean and the conditional variance. The other corresponds to the shape characteristics of the conditional density of the standardized errors, summarized in the coefficients of skewness and kurtosis together with the Fisher information for location and scale. The quantification of these factors has practical implications since it can help to decide if the more complex semiparametric estimator provides sufficient efficiency gains with respect to the simplest quasi-maximum-likelihood estimator. We also prove that there is no probability density function, with the exception of the normal, for which the asymptotic efficiency of the three estimators is the same. Particular models are also considered, for which the efficiency comparisons are greatly simplified.