This paper considers the information available to invariant unit root
tests at and near the unit root. Because 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 upper bound upon the power of all invariant unit root tests is
shown to depend solely upon the information. This bound is illustrated via
a brief simulation study that also examines the impact that different
invariance requirements have on power.
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