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A cointegration test statistic based upon estimation of an error correction model can be approximately normally distributed when no cointegration is present. By contrast, the equivalent Dickey-Fuller statistic applied to residuals from a static relationship has a non-standard asymptotic distribution. When cointegration exists, the error-correction test generally is more powerful than the Dickey-Fuller test. These differences arise because the latter imposes a possibly invalid common factor restriction. The issue is general and has ramifications for system-based cointegration tests. Monte Carlo analysis and an empirical study of U.K. money demand demonstrate the differences in power.