Non-stationary log-periodogram regression

We study asymptotic properties of the log-periodogram semiparametric estimate of the memory parameter d for non-stationary (d⩾12) 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∈[12, 34) and still consistent for d∈[12, 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.