A regression model \[ y_{nt}= \beta_n' x_{nt}+ r_{nt}+ \varepsilon_{nt}, \quad t=1,2,\dots, n, \] is considered, where \(\beta_n\) are \(q_n\times 1\) unknown parameters, \(x_{nt}\) are observable \(q_n\times 1\) random vectors, and \(r_{nt}\) are random variables not necessarily observable, which may be viewed as “model bias”. The errors \(\{\varepsilon_{nt}\), \(t=1,\dots, t\}\) are i.i.d. random variables with common distribution \(H_n\). Given the observations \((x_{n1} y_{n1}),\dots, (x_{nn}, y_{nn})\), the residual epirical process is defined by \[ \widetilde{Y}_n(u)= n^{-1/2} \sum_{t=1}^n [I(H_n (\widetilde {\varepsilon}_{nt})\leq u)- u], \quad 0\leq u\leq 1, \] where \(\widetilde {\varepsilon}_{nt}= y_{nt}- \widetilde{\beta}_n' x_{nt}\) and \(\widetilde{\beta}_n\) is the least squares estimate of \(\beta_n\). The process can be imployed to form a Gaussian test statistic. M.V. Boldin [Theory Probab. Appl. 27, 866-871 (1982); translation from Teor. Veroyatn. Primen. 27, No. 4, 805-810 (1982; Zbl 0499.62083)] and D.A. Pierce [Biometrika 72, 293-297 (1985; Zbl 0571.62014)] applied it to stationary \(\text{AR}(q)\) models.
In the present paper an oscillation-like result is derived for the process \(\widetilde{Y}_n (u)\). The result is applied to autoregressive time series. For a stationary \(\text{AR}(\infty)\) process, the order of the fitted \(\text{AR} (q_n)\) process is determined and the limiting Gaussian process for \(\widetilde Y_n(u)\) is obtained. For an unstable \(A(q)\) process, the limiting process is no longer Gaussian if the characteristic polynomial has a root 1. For the explosive case, the Brownian bridge result given by H.L. Koul and Sh. Levental [Ann. Stat. 17, No. 4, 1784-1794 (1989; Zbl 0695.60042)] is reestablished by a short proof.