Towards a Unified Asymptotic Theory for Autoregression

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.