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We consider tests of null hypothesis of stationarity against a unit root alternative when the series is subject to structural change at an unknown point in time. Three extant test are reviewed which allow for an endogenously determined instantaneous structural break, and a related fourth procedure is introduced. We further propose tests which permit the structural change to be gradual rather than instantaneous, allowing the null hypothesis to be a stationarity about a smooth transition in linear trend. The size and power properties of the tests are investigated, and the test are applied to four economic time series.