On residual empirical processes of stochastic regression models with applications to time series

Motivated by Gaussian tests for a time series, we are led to investigate th e asymptotic behavior of the residual empirical processes of stochastic reg ression models. These models cover the fixed design regression models as we ll as general AR(q) models. Since the number of the regression coefficients is allowed to grow as the sample size increases, the obtained results are also applicable to nonlinear regression and stationary AR(infinity) models. In this paper, we first derive an oscillation-like result for the residual empirical process. Then, we apply this result to autoregressive time serie s. In particular, for a stationary AR(infinity) process, we are able to det ermine the order of the number of coefficients of a fitted AR(q(n)) model a nd obtain the limiting Gaussian processes. For an unstable AR(q) process, w e show that if the characteristic polynomial has a unit root 1, then the li miting process is no longer Gaussian. For the explosive case, one of our si de results also provides a short proof for the Brownian bridge results give n by Koul and Levental.