Nonparametric conditional variance and error density estimation in regression models with dependent errors and predictors

This paper considers nonparametric regression models with long memory errors and predictors. Unlike in weak dependence situations, we show that the estimation of the conditional mean has influence on the estimation of both, the conditional variance and the error density. In particular, the estimation of the conditional mean has a negative effect on the asymptotic behaviour of the conditional variance estimator. On the other hand, surprisingly, estimation of the conditional mean may reduce convergence rates of the residual-based Parzen-Rosenblatt density estimator, as compared to the errors-based one. Our asymptotic results reveal small/large bandwidth dichotomous behaviour. In particular, we present a method which guarantees that a chosen bandwidth implies standard weakly dependent-type asymptotics. Our results are confirmed by an extensive simulation study. Furthermore, our theoretical lemmas may be used in different problems related to nonparametric regression with long memory, like cross-validation properties, bootstrap, goodness-of-fit or quadratic forms.