Nonparametric regression with nearly integrated regressors under long-run dependence

We study the nonparametric estimation of a regression function with nonstationary (integrated or nearly integrated) covariates and the error series of the regressor process following a fractional integrated autoregressive moving average 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 fractional Brownian motion. Furthermore, we study the Nadaraya–Watson estimator and we 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. However, for the non-zero energy case, the local linear estimator is superior to the Nadaraya–Watson estimator in terms of optimal convergence rate. We also present a comparison of our results with the conventional results for stationary covariates. Finally, we conduct a Monte Carlo simulation to illustrate the finite sample performance of the proposed estimator.