Motivated by Gaussian tests for a time series, we are led to
investigate the asymptotic behavior of the residual empirical processes of
stochastic regression models. These models cover the fixed design regression
models as well as general AR$(q)$ models. Since the number of the regression
coeffi-cients is allowed to grow as the sample size increases, the obtained
results are also applicable to nonlinear regression and stationary AR$(\infty)$
models. In this paper, we first derive an oscillation-like result for the
residual em-pirical process. Then, we apply this result to autoregressive time
series. In particular, for a stationary AR$(\infty)$ process, we are able to
determine the order of the number of coefficients of a fitted AR$(q_n)$ model
and obtain the limiting Gaussian processes. For an unstable AR$(q)$ process, we
show that if the characteristic polynomial has a unit root 1, then the limiting
process is no longer Gaussian. For the explosive case, one of our side results
also provides a short proof for the Brownian bridge results given by Koul and
Levental.