Diagnostic test for unstable autoregressive models

In this article, we investigate the limiting distribution of the Ljung and Box test statistic based on squared residuals in unstable autoregressive models [Chan, N.H. and Wei, C.Z., 1988, Limiting distribution of least squares estimates of unstable autoregressive processes. Annals of Statistics, 16, 367-401.]. We show that the limiting distribution is a chi-square distribution in which the degrees of freedom depends only on the number of lags of the residual autocorrelations. Further, we investigate the limiting distribution of Hong's [Hong, Y., 1996, Consistent testing for serial correlation of unknown form. Econometrica, 64, 837-864.] test statistic based on the original and squared residuals. It is shown that their limiting distributions are independent of the unit roots of the characteristic polynomial of the unstable model. The simulation results are listed and an application to quarterly US beer production data has been provided for illustration.