Weak convergence of the sequential empirical processes of residuals in nonstationary autoregressive models

This paper establishes the weak convergence of the sequential empirical process K̂_n of the estimated residuals in nonstationary autoregressive models. Under some regular conditions, it is shown that K̂_n converges weakly to a Kiefer process when the characteristic polynomial does not include the unit root 1; otherwise K̂_n converges weakly to a Kiefer process plus a functional of stochastic integrals in terms of the standard Brownian motion. The latter differs not only from that given by Koul and Levental for an explosive AR(1) model but also from that given by Bai for a stationary ARMA model.