Testing for a unit root in a time series with a level shift at unknown time

Unit root tests for time series with level shifts of general form are considered when the timing of the shift is unknown. It is proposed to estimate the nuisance parameters of the data generation process including the shift date in a first step and apply standard unit root tests to the residuals. The estimation of the nuisance parameters is done in such a way that the unit root tests on the residuals have the same limiting distributions as for the case of a known break date. Simulations are performed to investigate the small sample properties of the tests, and empirical examples are discussed to illustrate the procedure.