Testing I(1) against I(d) alternatives with Wald Tests in the presence of deterministic components

This paper analyses how to test I(1) against I(d), d<1, in the presence of deterministic components in the DGP, by extending a Wald-type test, i.e., the (Efficient) Fractional Dickey-Fuller (EFDF) test, to this case. Tests of these hypotheses are important in many economic applications where it is crucial to distinguish between permanent and transitory shocks because I(d) processes with d<1 are mean-reverting. On top of it, the inclusion of deterministic components becomes a necessary addition in order to analyze most macroeconomic variables. We show how simple is the implementation of the EFDF in these situations and argue that, in general, has better properties than LM tests. 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 growth theories, and on which there has been some controversy.