We evaluate the impact of heavy‐tailed innovations on some popular unit root tests. In the context of a near‐integrated series driven by linear process shocks, we demonstrate that their limiting distributions are altered under infinite variance vis‐à‐vis finite variance. Reassuringly, however, simulation results suggest that the impact of heavy‐tailed innovations on these tests is relatively small. We use the framework of Amsler and Schmidt () whereby the innovations have local‐to‐finite variances being generated as a linear combination of draws from a thin‐tailed distribution (in the domain of attraction of the Gaussian distribution) and a heavy‐tailed distribution (in the normal domain of attraction of a stable law). We also explore the properties of augmented Dickey–Fuller tests that employ Eicker–White standard errors, demonstrating that these can yield significant power improvements over conventional tests.