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It is widely known that when there are errors with a moving‐average root close to −1, a high order augmented autoregression is necessary for unit root tests to have good size, but that information criteria such as the AIC and the BIC tend to select a truncation lag (k) that is very small. We consider a class of Modified Information Criteria (MIC) with a penalty factor that is sample dependent. It takes into account the fact that the bias in the sum of the autoregressive coefficients is highly dependent on k and adapts to the type of deterministic components present. We use a local asymptotic framework in which the moving‐average root is local to −1 to document how the MIC performs better in selecting appropriate values of k. In Monte‐Carlo experiments, the MIC is found to yield huge size improvements to the DF^GLS and the feasible point optimal P_T test developed in Elliott, Rothenberg, and Stock (1996). We also extend the …