Efficient Tests of Nonstationary Hypotheses

Abstract This article proposes tests for unit root and other forms of nonstationarity that are asymptotically locally most powerful against a certain class of alternatives and have asymptotic critical values given by the chi-squared distribution. Many existing unit root tests do not share these properties. The alternatives include fractionally and seasonally fractionally differenced processes. There is considerable flexibility in our choice of null hypothesis, which can entail one or more integer 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 nonstationarity against stationarity. “Overdifferencing” stationary null hypotheses can also be tested. The test statistic is derived via the score principle and is conveniently expressed in the frequency domain. The series tested are regression errors, which, when the hypothesized differencing is correct, are white noise or more generally have weak parametric autocorrelation. We establish the null and local limit distributions of the statistic under mild regularity conditions. We find that Bloomfield's exponential spectral model can provide an especially 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 several cases find that our tests reject the order of differencing that Box and Jenkins proposed. We also report Monte Carlo studies of finite-sample behavior of versions of our tests and comparisons with other tests.