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Biometrika 1987 74(3):535-547; doi:10.1093/biomet/74.3.535
© 1987 by Biometrika Trust
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Towards a unified asymptotic theory for autoregression

P. C. B. PHILLIPS

Cowles Foundation for Research in Economics, Yale University Yale Station, New Haven, Connecticut 06520-2125, U.S.A.

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 possibilities 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.

Key Words: Autoregression • Brownian motion • Diffusion • Near-integrated process • Noncentrality parameter • Unit root


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