Quasi-likelihood estimation of the single index conditional variance model

Abstract

This paper investigates estimation methods for the conditional variance function with a single index structure. We introduce two estimators of the single index parameter vector through maximizing local linear quasi-likelihood functions. The resulting parameter index estimators can achieve root-n consistency and the variance function estimator can maintain positivity. We show that the proposed methods can estimate the conditional variance with the same asymptotic efficiency as if the conditional mean function is given. Asymptotic distributions of the proposed estimators are also derived. Simulation studies and a real data application demonstrate our estimation approaches.