Local Polynomial M-Estimation in Random Design Regression with Dependent Errors

Abstract

The asymptotic behaviour of the robust local polynomial M-estimator is investigated in the random design nonparametric regression model with short-range dependent and long-range- dependent errors. Asymptotic results are established by decomposing the estimator into two terms: a martingale term and a conditional expectation term. The local polynomial M-estimator is asymptotically normal when errors are short-range dependent. When the errors are long-range dependent, a more complex behaviour is observed that depends on the size of the bandwidth. If the bandwidth is small enough, the standard asymptotic normality persists. If the bandwidth is relatively large, the asymptotic result is more intricate and the long-range-dependent variables dominate. In both cases, the optimal bandwidth is investigated.