Additional tests for a unit root allowing for a break in the trend function at an unknown time

We consider unit root tests that allow a shift in trend at an unknown time. We focus on the additive outlier approach, but also give results for the innovational outlier approach. Various methods of choosing the break date are considered. New limiting distributions are derived, including the case where a shift in trend occurs under the unit root null hypothesis. Limiting distributions are invariant to mean shifts but not to slope shifts. Simulations are used to assess finite sample size and power. We focus on the effects of a break under the null and the choice of break date.