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This paper analyses the behaviour of a Wald-type test, i.e., the (E¢ cient) Fractional Dickey-Fuller (EFDF) test of I(1) against I(d); d < 1; relative to LM tests. Further, it extends the implementation of the EFDF test to the presence of deterministic trending components in the DGP. Tests of these hypotheses are important in many macroeconomic applications where it is crucial to distinguish between permanent and transitory shocks because shocks die out in I(d) processes with d < 1. We show how simple is the implementation of the EFDF in these situations and argue that, under xed alternatives, it is preferred to the LM test in Bahadur´ s sense. Finally, an empirical application is provided where the EFDF approach allowing for deterministic components is used to test for long-memory in the GDP p.c. of several OECD countries, an issue that has important consequences to discriminate between alternative growth theories.