In this paper I present an alternative derivation of the asymptotic distribution of Kremers, Ericsson, and Dolado's (1992, Oxford Bulletin of Economics and Statistics 54, 325–348) conditional error correction model (ECM)–based t-test for cointegration with a single prespecified cointegrating vector. This alternative distribution, which is identical to the distribution of Hansen's (1995, Econometric Theory 11, 1148–1171) covariate augmented t-test for a unit root, is valid for weakly exogenous regressors and depends on a consistently estimable nuisance parameter that takes on values in the unit interval. I show analytically, using asymptotic power functions based on near-cointegrated alternatives, that the ECM t-test with a prespecified cointegrating vector can have much higher power than single equation tests for cointegration based on estimating the cointegrating vector. I also characterize situations in which the ECM t-test computed with a misspecified cointegrating vector will have high power.