The purpose of this paper is to characterize three commonly used double unit root tests in terms of their asymptotic local power. To this end, we study a class of nearly doubly integrated processes which in the limit will behave as a weighted integral of a double indexed Ornstein-Uhlenbeck process. Based on a numerical examination of the analytical distributions, a comparison of the tests is made via their asymptotic local power functions.

We examine some of the consequences on commonly used unit root tests when the underlying series is integrated of order two rather than of order one. It turns out that standard augmented Dickey-Fuller type of tests for a single unit root have excessive density in the explosive region of the distribution. The lower (stationary) tail, however, will be virtually unaffected in the presence of double unit roots. On the other hand, the Phillips-Perron class of semi-parametric tests is shown to diverge to plus infinity asymptotically and thus favoring the explosive alternative. Numerical simulations are used to demonstrate the analytical results and some of the implications in finite samples.