Log periodogram (LP) regression is shown to be consistent and to have a mixed normal limit distribution when the memory parameter d = 1. Gaussian errors are not required. Tests of d = 1 based on LP regression are consistent against d < 1 alternatives but inconsistent against d > 1 alternatives. A test based on a modiﬁed LP regression that is consistent in both directions is provided.
Phillips, Peter C.B., "Unit Root Log Periodogram Regression" (1999). Cowles Foundation Discussion Papers. 1494.