Testing for stationarity with a break

In this paper, we investigate a test for the null hypothesis of trend stationarity with a structural change against a unit root. We derive the limiting distribution of an Lagrange Multiplier (LM) test statistic and its characteristic function under a sequence of local alternatives. The local limiting power of the LM test depends on the persistence of the stationary component of the process, and the more persistent the process, the less powerful is the test statistic. We also propose a test statistic that does not depend on the fraction of the pre-break points to the sample size under the null hypothesis, which we call the PS test. Though it is convenient for the critical point not to depend on the break point, the PS test is found to be less powerful than the LM test under the alternative close to the null hypothesis. Finite sample simulations show that when the break point is known, the LM test tends to be oversized when the process is rather persistent, while the size distortion of the PS test is not so pronounced. On the other hand, the empirical sizes of both tests are close to the nominal one when the break point is estimated by the least-squares method, though the power decreases compared with the known break point case.

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