Empirical process of residuals for regression models with long memory errors

TL;DR

This paper investigates the asymptotic behavior of residual empirical processes in regression models with long memory errors, highlighting differences from error-based processes and enabling goodness-of-fit testing.

Contribution

It provides new theoretical results on the convergence rates of residual empirical processes with long memory errors and their use in statistical tests.

Findings

Residual empirical process rates differ from error-based processes.

Asymptotic distributions enable goodness-of-fit tests.

Simulation studies support theoretical findings.

Abstract

We consider the residual empirical process in random design regression with long memory errors. We establish its limiting behaviour, showing that its rates of convergence are different from the rates of convergence for to the empirical process based on (unobserved) errors. Also, we study a residual empirical process with estimated parameters. Its asymptotic distribution can be used to construct Kolmogorov-Smirnov, Cram\'{e}r-Smirnov-von Mises, or other goodness-of-fit tests. Theoretical results are justified by simulation studies.