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We develop a class of invariant tests for the null hypothesis that a time series is a martingale dierence sequence against the alternative that it belongs to the class of perturbed fractionally integrated (long memory) processes. The class of tests we develop are indexed by a user-chosen long memory parameter, d > 0, and are locally most powerful (under Gaussianity) where the true long memory parameter coincides with this value. We show that this class of tests contains a number of widely used tests as special cases. A complete taxonomy of limiting null distribution theory for the class of statistics (indexed by d) is provided. These distributions depend on d, but can be straightforwardly simulated in practice. We also compare the asymptotic local power properties of the tests under appropriate Pitman drift sequences.
