Local $M$-estimation for conditional variance function with dependent data

Abstract

In this paper, a local $M$-estimation for the conditional variance function in heteroscedastic regression models under stationary $\alpha $-mixing dependent samples is developed. The local $M$-estimator is based on the local linear smoothing technique and the $M$-estimation technique, and it is shown to be not only asymptotically equivalent to the local linear estimator but also robust. The weak consistency as well as the asymptotic normality of the local $M$-estimator for the conditional variance function are obtained under mild conditions.