Bent Nielsen
  (2000)
  'The Asymptotic Distribution of Unit Root Tests of Unstable
   Autoregressive Processes'
  To appear in Econometrica
   
  ABSTRACT: 
  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.