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.
Discrete Fourier transform, fractional Brownian motion, fractional integration, log periodogram regression, long memory parameter, nonstationarity, semiparametric estimation and testing, unit root