Non-stationary log-periodogram regression

We study asymptotic properties of the log-periodogram semiparametric estimate of the memory parameter d for non-stationary(d greater than or equal to 1/2) time series with Gaussian increments, extending the results of Robinson (1995) for stationary and invertible Gaussian processes. We generalize the definition of the memory parameter d for non-stationary processes in terms of the (successively) differentiated series. We obtain that the log-periodogram estimate is asymptotically normal for d is an element of [1/2, 3/4) and still consistent for d is an element of [1/2, 1). We show that with adequate data tapers, a modified estimate is consistent and asymptotically normal distributed for any d, including both non-stationary and non-invertible processes. The estimates are invariant to the presence of certain deterministic trends, without any need of estimation. (C) 1999 Elsevier Science S.A. All rights reserved.