Comparison of unit root tests for time series with level shifts

Unit root tests are considered for time series which have a level shift at a known point in time. The shift can have a very general nonlinear form and additional deterministic mean and trend terms are allowed for. Prior to the tests the deterministic parts and other nuisance parameters of the data generation process are estimated in a first step. Then the series are adjusted for these terms and unit root tests of the Dickey-Fuller type are applied to the adjusted series. The properties of previously suggested tests of this sort are analyzed and a range of modifications is proposed which take into account estimation errors in the nuisance parameters. An important result is that estimation under the null hypothesis is preferable to estimation under local alternatives. This contrasts with results obtained by other authors for time series without level shifts.