Near observational equivalence and theoretical size problems with unit root tests

Said and Dickey (1984, Biometrika 71, 599-608) and Phillips and Perron (1988, Biometrika 75, 335-346) have derived unit root tests that have asymptotic distributions free of nuisance parameters under very general maintained models. Under models as general as those assumed by these authors, the size of the unit root test procedures will converge to one, not the size under the asymptotic distribution. Solving this problem requires restricting attention to a model that is small, in a topological sense, relative to the original. Sufficient condition's for solving the asymptotic size problem yield some suggestions for improving finite-sample Size performance of standard tests.