Behaviour of Dickey–Fuller tests when there is a break under the unit root null hypothesis

We show that Dickey and Fuller's [1979. Distribution of the estimator for autoregressive time series with a unit root. J. Amer. Statist. Assoc. 74, 427–431; 1981. Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica 49, 1057–1072.] normalized estimator, and F-statistics will spuriously reject the unit root null when the true data generating process is a unit root with a break.