Spurious rejections by Dickey–Fuller tests in the presence of a break under the null

It is well known that if a series is generated by a process that is stationary around a broken trend, conventional Dickey–Fuller tests can have very low power. In this paper, the converse phenomenon is studied and illustrated. Suppose that the true generating process is integrated of order one, but with a break. Then it is shown that, if the break occurs early in the series, routine application of standard Dickey–Fuller tests can lead to a very serious problem of spurious rejection of the unit root null hypothesis.