On the distribution of Dickey–Fuller unit root statistics when there is a break in the innovation variance

We show that Dickey & Fuller's [Distribution of the estimator for autoregressive time series with a unit root. J. Amer. Statist. Assoc. 74, 427–431; Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica 49, 1057–1072] normalized estimator and FF-statistics for the unit root null hypothesis will spuriously reject the null hypothesis if there is a break in the innovation variance relatively early in the sample.