Local linear regression estimation for time series with long-range dependence

Abstract

Consider the nonparametric estimation of a multivariate regression function and its derivatives for a regression model with long-range dependent errors. We adopt local linear fitting approach and establish the joint asymptotic distributions for the estimators of the regression function and its derivatives. The nature of asymptotic distributions depends on the amount of smoothing resulting in possibly non-Gaussian distributions for large bandwidth and Gaussian distributions for small bandwidth. It turns out that the condition determining this dichotomy is different for the estimates of the regression function than for its derivatives; this leads to a double bandwidth dichotomy whereas the asymptotic distribution for the regression function estimate can be non-Gaussian whereas those of the derivatives estimates are Gaussian. Asymptotic distributions of estimates of derivatives in the case of large bandwidth are the scaled version of that for estimates of the regression function, resembling the situation of estimation of cumulative distribution function and densities under long-range dependence. The borderline case between small and large bandwidths is also examined.