Improving Maximum Quasi-Likelihood Estimators

Abstract

A quasi-likelihood model for a stochastic process is defined by parametric models for the conditional mean and variance processes given the past. The customary estimator for the parameter is the maximum quasi-likelihood estimator. We discuss some ways of improving this estimator. For simplicity we restrict attention to a Markov chain with conditional mean ϑx given the previous state x, and conditional variance a function of ϑ but not of x. Then the maximum quasi-likelihood estimator is the least squares estimator. It remains consistent if the conditional variance is misspecified. In this case, a better estimator is a weighted least squares estimator, with weights involving nonparametric predictors for the conditional variance. If the conditional variance is correctly specified, a better estimator is given by a convex combination of the least squares estimator and a function of an 'empirical' estimator for the variance.