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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 (M^GLS) 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 (P_T^GLS) suggested by ERS. Also, we compute the non-centrality parameter used for the local GLS detrending that permits the test P_T^GLS to have 50% asymptotic power at that value. The asymptotic critical values of …