Some results on random design regression with long memory errors and predictors

TL;DR

This paper analyzes the impact of long memory errors and predictors on nonparametric regression, deriving convergence rates, MISE calculations, and examining bandwidth selection methods.

Contribution

It provides general conditions for convergence rates in nonparametric regression with long memory data and compares bandwidth selection methods.

Findings

Long memory errors can affect MISE in regression.

Shape function estimators are less influenced by long memory.

Averaged Squared Error approximates MISE well, unlike cross-validation.

Abstract

This paper studies nonparametric regression with long memory (LRD) errors and predictors. First, we formulate general conditions which guarantee the standard rate of convergence for a nonparametric kernel estimator. Second, we calculate the Mean Integrated Squared Error (MISE). In particular, we show that LRD of errors may influence MISE. On the other hand, an estimator for a shape function is typically not influenced by LRD in errors. Finally, we investigate properties of a data-driven bandwidth choice. We show that Averaged Squared Error (ASE) is a good approximation of MISE, however, this is not the case for a cross-validation criterion.