On the behaviour of fixed-b trend break tests under fractional integration

Testing for the presence of a broken linear trend when the nature of the persistence in the data is unknown is not a trivial problem, since the test needs to be both asymptotically correctly sized and consistent, regardless of the order of integration of the data. In a recent paper, Sayginsoy and Vogelsang (2011) [SV] show that tests based on fixed-b asymptotics provide a useful solution to this problem in the case where the shocks may be either weakly dependent or display strong dependence within the near-unit root class. In this paper we analyse the performance of these tests when the shocks may be fractionally integrated, an alternative model paradigm which allows for either weak or strong dependence in the shocks. We demonstrate that the fixed-b trend break statistics converge to well-defined limit distributions under both the null and local alternatives in this case (and retain consistency against fixed alternatives), but that these distributions depend on the fractional integration parameter δ. As a result, it is only when δ is either zero or one that the SV critical values yield sized tests. Consequently, we propose a procedure which employs _-adaptive critical values to remove the size distortions in the SV test. In addition, use of δ-adaptive critical values also allows us to consider a simplification of the SV test which is (asymptotically) correctly sized across δ but can also provide a significant increase in power over the standard SV test when δ=1.