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

Abstract

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.