Via the leading unit-root case, the problem of testing on a lagged dependent variable is characterized by a nuisance parameter which is present only under the alternative [see Andrews and Ploberger, Econometrica (1994) Vol. 62, pp. 1318–1414]. This has proven to be a barrier to the construction of optimal tests. Moreover, in their absence it is impossible to objectively assess the absolute power properties of existing tests. Indeed, feasible tests based upon the optimality criteria used here are found to have numerically superior power properties to both the original Dickey and Fuller [Econometrica (1981) Vol. 49, pp. 1057–1072] statistics and the efficient detrended versions suggested by Elliott et al. [Econometrica (1996) Vol. 64, pp. 813–836] and analysed in Burridge and Taylor [Oxford Bulletin of Economics and Statistics (2000) Vol. 62, pp. 633–645].