Locally Optimal Tests Against Seasonal Unit Roots

This paper builds on the existing literature on tests of the null hypothesis of deterministic seasonality in a univariate time-series process. Under the assumption of independent Gaussian errors, we derive the class of locally weighted mean most powerful invariant tests against unit roots at the zero and/or seasonal frequencies in a seasonally observed process. Representations for the limiting distributions of the proposed test statistics under sequences of local alternatives are derived, and the relationship with tests for corresponding moving average unit roots is explored. We also propose nonparametric modifications of these test statistics designed to have limit distributions which are free of nuisance parameters under weaker conditions on the errors. Our tests are shown to contain existing stationarity tests as special cases and to extend these tests in a number of useful directions.