Asymptotic inference for unstable auto- regressive time series with drifts

Abstract

An autoregressive time series is said to be unstable if all of its characteristics roots lie on or outside the unit circle. This paper aims at developing the asymptotic inference for the least squares estimates of unstable autoregressive time series with drifts. Our framework allows for both constant and periodic drifts. The presence of a nonzero drift in the series affects the asymptotic behavior in a surprising and interesting way. For models with constant nonzero drifts, conventional asymptotic normal theory is attained only for unit root time series but not in general. For models with periodic drifts, degenerate nonnormal asymptotics are resulted for unit root series. However for models with roots lying on the unit circle which are not equal to unity, nonnormal asymptotics are always attained for both constant and periodic drifts unless there are confounding effects between the periodicities of the drifts and the locations of the roots. Our approach is based on a componentwise argument used in Chan and Wei (1988). The theories developed here extend much earlier work and help to clarify the intrinsic critical phenomena in the field of unstable time series.