Mirror image distributions and the Dickey-Fuller regression with a maintained trend

A puzzle of many empirical studies where Dickey-Fuller regressions with a maintained trend are run, is that the t-test of the AR-root and the trend t-statistic frequently appear to be of equal magnitude but with opposite sign. In this paper I demonstrate that when the true process contains a drift, then the asymptotic distributions of the two statistics will be the mirror image of each other and even in finite samples this property will be predominant as long as the standardized drift is sufficiently strong.