EFFICIENT TESTS OF NONSTATIONARY HYPOTHESES

This article proposes tests for unit root and other forms of nonstatio narity that are asymptotically locally most powerful against a certain class of alternatives and have asymptotic critical values given by th e chi-squared distribution. Many existing unit root tests do not share these properties. The alternatives include fractionally and seasonall y fractionally differenced processes. There is considerable flexibilit y in our choice of null hypothesis, which can entail one or more integ er or fractional roots of arbitrary order anywhere on the unit circle in the complex plane. For example, we can test for a fractional degree of integration of order 1/2; this can be interpreted as a test for no nstationarity against stationarity. ''Overdifferencing'' stationary nu ll hypotheses can also be tested. The test statistic is derived via th e score principle and is conveniently expressed in the frequency domai n. The series tested are regression errors, which, when the hypothesiz ed differencing is correct, are white noise or more generally have wea k parametric autocorrelation. We establish the null and local limit di stributions of the statistic under mild regularity conditions. We find that Bloomfield's exponential spectral model can provide an especiall y neat form for the test statistic. We apply the tests to a number of empirical time series originally analyzed by Box and Jenkins, and in s everal cases find that our tests reject the order of differencing that Box and Jenkins proposed. We also report Monte Carlo studies of finit e-sample behavior of versions of our tests and comparisons with other tests.