Nonparametric Regression With Nearly Integrated Regressors Under Long Run Dependence

We study nonparametric estimation of regression function with nonstationary (integrated or nearly integrated) covariates and the error series of the regressor process following a fractional ARIMA model. A local linear estimation method is developed to estimate the unknown regression function. The asymptotic results of the resulting estimator at both interior points and boundaries are obtained. The asymptotic distribution is mixed normal, associated with the local time of an Ornstein-Uhlenbeck (O-U) fractional Brownian motion. Furthermore, we study the Nadaraya-Watson estimator and examine its asymptotic results. As a result, it shares exactly the same asymptotic results as those for the local linear estimator for the zero energy situation. But for the non-zero energy case, the local linear estimator is superior over the Nadaraya-Watson estimator in terms of optimal convergence rate. Moreover, a comparison of our results with the conventional results for stationary covariates is presented. Finally, a Monte Carlo simulation is conducted to illustrate the finite sample performance of the proposed estimator.