Abstract
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.
Highlights
Random design regression with long memory errorsConsider the random design regression modelYi = m(Xi) + σ(Xi)εi, i = 1, . . . , n. (1.1)We shall assume that the predictors Xi, i ≥ 1, are random variables with unit variance and density f = fX, independent of εi, i ≥ 1
The error sequence is assumed to be centered with unit variance and density fε
In the aforementioned paper the author shows that in case of independent errors and predictors, estimation of the conditional mean does not influence the rates of convergence for an estimator of fε
Summary
In the aforementioned paper the author shows that in case of independent errors and predictors, estimation of the conditional mean does not influence the rates of convergence for an estimator of fε. The goal of this paper is to present the full asymptotic theory for the conditional variance and the error density estimation in the model (1.1), when errors and/or predictors have long memory. Such situations are very often encountered in, especially, financial time series. There, the authors deal with the parametric mean case, m(x) = β0 + β1x, but predictors are allowed to form different long memory sequences, including linear processes and stochastic volatility models. The latter are important especially in modeling financial time series, which are typically uncorrelated, but have long memory in squares
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