Testing for unit roots in time series with level shifts

Tests for unit roots in univariate time series with level shifts are proposed and investigated. The level shift is assumed to occur at a known time. It may be a simple one-time shift which can be captured by a dummy variable or it may have a more general form which can be modeled by some general nonlinear transition function. There may also be more than one shift point and there may be other deterministic terms such as a linear trend term or seasonal components. It is proposed to estimate the deterministic parts of the series in a first step by a generalized least squares procedure subtract the estimated deterministic terms from the series and apply standard unit root tests to the residuals. It is shown that the tests have known asymptotic distributions under the null hypothesis of a unit root and nearly optimal asymptotic power under local alternatives. The procedure is applied to German macroeconomic time series which have a level shift in 1990 where the reunification took place.