  This paper studies the residual empirical process of long- and short-memory
time series regression models and establishes its uniform expansion under a
general framework. The results are applied to the stochastic regression models
and unstable autoregressive models. For the long-memory noise, it is shown that
the limit distribution of the Kolmogorov-Smirnov test statistic studied in Ho
and Hsing [Ann. Statist. 24 (1996) 992-1024] does not hold when the stochastic
regression model includes an unknown intercept or when the characteristic
polynomial of the unstable autoregressive model has a unit root. To this end,
two new statistics are proposed to test for the distribution of the long-memory
noises of stochastic regression models and unstable autoregressive models.
(With Correction.)
