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

Empirical process of residuals for regression models with long memory errors