Gaussian Semiparametric Estimation of Non‐stationary Time Series

Generalizing the definition of the memory parameter d in terms of the differentiated series, we showed in Velasco (Non‐stationary log‐periodogram regression, Forthcoming J. Economet., 1997) that it is possible to estimate consistently the memory of non‐stationary processes using methods designed for stationary long‐range‐dependent time series. In this paper we consider the Gaussian semiparametric estimate analysed by Robinson (Gaussian semiparametric estimation of long range dependence. Ann. Stat. 23 (1995), 1630–61) for stationary processes. Without a priori knowledge about the possible non‐stationarity of the observed process, we obtain that this estimate is consistent for d∈ (−½, 1) and asymptotically normal for d∈ (−½,¾) under a similar set of assumptions to those in Robinson's paper. Tapering the observations, we can estimate any degree of non‐stationarity, even in the presence of deterministic polynomial trends of time. The semiparametric efficiency of this estimate for stationary sequences also extends to the non‐stationary framework.