Abstract.The residual autocorrelations in nonstationary autoregressive processes with autoregressive characteristic roots on the unit circle are considered. Limiting distributions of the residual autocovariances and the residual autocorrelations are shown to be the same as the limiting distributions when parameters are estimated with all roots on the unit circle known. The portmanteau statistic is shown to have a x2limiting distribution. The Canadian lynx data set is analysed to illustrate our theory. The portmanteau test seems also useful when the characteristic roots are close to the unit circle.