Behaviour of Dickey-Fuller unit-root tests under trend misspecification

We analyse the case where a unit-root test is based on a Dickey-Fuller regression the only deterministic term of which is a fixed intercept. Suppose, however, as could well be the case, that the actual data-generating process includes a broken linear trend. It is shown theoretically, and verified empirically, that under the I(1) null and I(0) alternative hypotheses the Dickey-Fuller test can display a wide range of different characteristics depending on the nature and location of the break.