This paper develops an asymptotic theory for a ﬁrst order autoregression with a root near unity. Deviations from the unit root theory are measured through a noncentrality parameter c . When c < 0 we have a local alternative that is stationary and when c > 0 the local alternative is explosive. As c approaches the limits of its domain of deﬁnition (±∞) it is shown that the asymptotic distributions known to apply under ﬁxed stationary and explosive alternatives are obtained as special cases. Moreover, when c = 0 we have the standard unit root theory. Thus, the asymptotic theory that we present goes a long way towards unifying earlier theory for these individual special cases. The general theory is expressed in terms of functionals of the Wiener process.
Phillips, Peter C.B., "Towards a Unified Asymptotic Theory for Autoregression" (1986). Cowles Foundation Discussion Papers. 1024.