  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.
