This paper proposes a theoretical explanation for the common empirical results in which different tests for cointegration give different answers. Using local to unity parametrization, this paper analytically computes the power of four tests for the null of no cointegration: The ADF test on the residuals of the cointegration regression, Johansen’s maximum eigenvalue test, the t-test on the Error Correction (EC) term, and Boswijk (1994) Wald test. The test statistics are shown to converge under a local alternative to random variables whose distributions are functions of Brownian Motions and Ornstein–Uhlenbeck processes and of a single nuisance parameter. The nuisance parameter is determined by the correlation at frequency zero of the errors in the cointegration relation with the shocks of the right-hand variables. I show that, when this correlation is high, system approaches, like the Johansen maximum eigenvalue or tests of the EC model, can exploit this correlation and significantly outperform single equation tests. Many of the varying results from applying different tests can be attributed to different values of this nuisance parameter.