The Asymptotic Distribution of Unit Root Tests of Unstable Autoregressive Processes.

Unit root testing has been developed through numerous papers since the work of Dickey and Fuller (1979). The idea is to test the hypothesis that the differences of an observed time series do not depend on its levels, or in other words, the levels of the time series has a unit root which can be removed by differencing. While it is in general possible to have multiple unit roots only the hypothesis of exactly one unit root is considered here. The available tests therefore hinge on two assumptions: (i) the levels of the time series has exactly one unit root which can be removed by differencing, and (ii) the remaining characteristic roots of the time series are stationary roots. In this paper it is proved that for the likelihood ratio test and a number of other likelihood based statistics the assumption (ii) is redundant whereas (i) is necessary. It is also shown that for some tests which are not likelihood based it is indeed necessary to assume that the differences have stationary roots