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. When this parameter is negative we have a local alternative that is stationary; when it is positive, the local alternative is explosive; and when it is zero we have the standard unit root theory. Our asymptotic theory accommodates these alternatives and helps to unify earlier theory in which the unit root case appears as a singularity of the asymptotics. The general theory is expressed in terms of functionals of a simple diffusion process. The theory has applications to continuous time estimation and to the analysis of the asymptotic power of tests for a unit root under a sequence of local alternatives.