Power of tests for unit roots in the presence of a linear trend

Dickey and Fuller [Econometrica (1981) Vol. 49, pp. 1057-1072] suggested unit-root tests for an autoregressive model with a linear trend conditional on an initial observation. The testing problem has nuisance parameters so power studies often work with a slightly different model with a random initial value in which nuisance parameters can easily be eliminated by an invariant reduction of the model. We show that invariance arguments can also be used when comparing power within a conditional model. In the context of the conditional model, the Dickey-Fuller test is shown to be more stringent than a number of unit-root tests motivated by models with random initial value. The power of the Dickey-Fuller test can be improved by making assumptions to the initial value. The practitioner therefore has to trade-off robustness and power, as assumptions about initial values are hard to test, but can give more power.