Tests for Unit Roots and the Initial Observation

The paper analyzes the impact of the initial observation on the problem of testing for unit roots. To this end, we derive a family of optimal tests that maximize a weighted average power criterion with respect to the initial observation. We then investigate the relationship of this optimal family to unit root tests in an asymptotic framework. We find that many popular unit root tests are closely related to specific members of the optimal family, but the corresponding members employ very different weightings for the initial observation. The popular Dickey-Fuller tests, for instance, are closely related to optimal tests which put a large weight on extreme deviations of the initial observation from the deterministic component, whereas other popular tests put more weight on moderate deviations. At the same time, the power of the various unit root tests varies dramatically with the initial observation. This paper therefore helps to explain the results of comparative power studies of unit root tests, and allows a much deeper understanding of the merits of particular tests in specific circumstances.