Tests for the stationarity null due to Kwiatkowski et al. (1992) continue to be an indispensable part of tool kits for empirical researchers when investigating time series property of aggregate variables. As well-documented in the literature (see for instance, Caner and Kilian, 2001), the tests display considerable size distortions, if the data generated under the null is highly persistent. The paper oers an asymptotic explanation in a local-to-unity framework. Our analytical derivations unveil that the tests fail to converge without a re-normalization. The surprising nding suggests that the size bias deteriorates as sample size increases, but declines as bandwidth number increases, consistent with simulation evidence. The derivations however give little clue to how to mitigate the size bias, because of an inability to consistently estimate the local-to-unity parameter. While it is natural to appeal to the bootstrapping, it proves infeasible to construct a sensible re-sampling scheme, based on the unobserved compo- nent model from which the observed series is generated. We resolve the diculty by drawing bootstrap samples from a parametric ARIMA model, second-order equivalent in moments to the unobserved component model. Even in the presence of highly per- sistent processes, our bootstrap tests are found to yield very satisfactory control over the rejection probability at little cost of power loss.