On the asymptotic distribution of the DF-GLS test statistic

In a very influential paper Elliott et al. (Efficient Tests for an Autoregressive Unit Root, Econometrica 64, 813?836, 1996) show that no uniformly most powerful test for the unit root testing problem exits, derive the relevant power envelope and characterize a family of point-optimal tests. As a by-product, they also propose a ?GLS detrended? version of the conventional Dickey?Fuller test, denoted DF?GLS, that has since then become very popular among practitioners, much more so than the point-optimal tests. In view of this, it is quite strange to find that, while conjectured in Elliott et al. (1996), so far there seems to be no formal proof of the asymptotic distribution of the DF?GLS test statistic. By providing three separate proofs the current paper not only substantiates the required result, but also provides insight regarding the pros and cons of different methods of proof.