The Available Information for Invariant Tests of a Unit Root

This paper considers the information available to invariant unit root tests at and near the unit root. Since all invariant tests will be functions of the maximal invariant, the Fisher information in this statistic will be the available information. The main finding of the paper is that the available information for all tests invariant to a linear trend is zero at the unit root. This result applies for any sample size, over a variety of distributions and correlation structures and is robust to the inclusion of any other deterministic component. In addition, an explicit bound upon the power of all invariant unit root tests is shown to depend solely upon the information. This bound is illustrated via comparison with the local-to-unity power envelope and a brief simulation study illustrates the impact that the requirements of invariance have on power.