Testing for a unit root in the presence of a possible break in trend

We consider the issue of testing a time series for a unit root in the possible presence of a break in a linear deterministic trend at an unknown point in the series. We propose a new break fraction estimator which, where a break in trend occurs, is consistent for the true break fraction at rate Op(T). Unlike other available estimators, however, when there is no trend break our estimator converges to zero at rate Op(T). Used in conjunction with a quasi difference (QD) detrended unit root test that incorporates a trend break regressor, we show that these rates of convergence ensure that known break fraction null critical values are asymptotically valid. Unlike available procedures in the literature this holds even if there is no break in trend (the break fraction is zero). Here the trend break regressor is dropped from the deterministic component and standard QD detrended unit root test critical values then apply. We also propose a second procedure which makes use of a formal pre-test for a trend break in the series, including a trend break regressor only where the pre-test rejects the null of no break. Both procedures ensure that the correctly sized (near-) efficient unit root test that allows (does not allow) for a break in trend is applied in the limit when a trend break does (does not) occur.