GLS detrending, efficient unit root tests and structural change

We extend the class of M-tests for a unit root analyzed by Perron and Ng (Rev. Econ. Studies 63 (1996) 435) and Ng and Perron (Econometrica 69 (2001) 1519) to the case where a change in the trend function is allowed to occur at an unknown time. These tests (MGLS) adopt the GLS detrending approach developed by Elliott et al. (Econometrica 64 (1996) 813) (ERS) following the results of Dufour and King (J. Econometrics 47 (1991) 115). Following Perron (Econometrica 57 (1989) 1361), we consider two models: one allowing for a change in slope and the other for both a change in intercept and slope. We derive the asymptotic distributions of the tests as well as that of the feasible point optimal test (PTGLS) suggested by ERS. Also, we compute the non-centrality parameter used for the local GLS detrending that permits the test PTGLS to have 50% asymptotic power at that value. The asymptotic critical values of the tests are tabulated. We show that the MGLS and PTGLS tests have an asymptotic power function close to the power envelope. A simulation study analyzes the size and power in finite samples under various methods to select the truncation lag for the autoregressive spectral density estimator. An empirical application is also provided.