Summary: Local to unity limit theory is used in applications to construct confidence intervals (CIs) for autoregressive roots through inversion of a unit root test [J. H. Stock, “Confidence intervals for the largest autoregressive root in U.S. macroeconomic time series”, J. Monetary Econ. 28, No. 3, 435–459 (1991; doi:10.1016/0304-3932(91)90034-L)]. Such CIs are asymptotically valid when the true model has an autoregressive root that is local to unity \((\rho = 1 + \tfrac{c}{n})\), but are shown here to be invalid at the limits of the domain of definition of the localizing coefficient \(c\) because of a failure in tightness and the escape of probability mass. Failure at the boundary implies that these CIs have zero asymptotic coverage probability in the stationary case and vicinities of unity that are wider than \(O(n^{-1/3})\). The inversion methods of B. E. Hansen [“The grid bootstrap and the autoregressive model”, Rev. Econ. Stat. 81, No. 4, 594–607 (1999; doi:10.1162/003465399558463)] and A. Mikusheva [Econometrica 75, No. 5, 1411–1452 (2007; Zbl 1133.91046)] are asymptotically valid in such cases. Implications of these results for predictive regression tests are explored. When the predictive regressor is stationary, the popular J. Y. Campbell and M. Yogo [“”, J. Finance Econ. 81, No. 1, 27–60 (2006; doi:10.1016/j.jfineco.2005.05.008)] CIs for the regression coefficient have zero coverage probability asymptotically, and their predictive test statistic \(Q\) erroneously indicates predictability with probability approaching unity when the null of no predictability holds. These results have obvious cautionary implications for the use of the procedures in empirical practice.