Locally stationary long memory estimation

Spectral analysis of strongly dependent time series data has a long history in applications in a variety of fields, such as, e.g., telecommunication, meteorology, hydrology or, more recently, financial and economical data analysis. There exists a wide literature on parametrically or semi-parametrically modelling such processes using a long-memory parameter d, including more recent work on wavelet estimation of d. As a generalization of these latter approaches, in this work we allow the long-memory parameter d to be varying over time. Hence, we give up the somewhat restrictive assumption of second-order stationarity of the observed process (or its increments, respectively, after differencing a finite number of times). We embed our approach into the framework of locally stationary processes which, over the past decade, has been developed for weakly dependent time series with a time-varying spectral structure. In this paper we adopt a semi-parametric approach for estimating the time-varying parameter d in order to avoid fitting a parametric model,such as ARFIMA, to the observed data. We show weak consistency and a central limit theorem for our log-regression wavelet estimator of the time-dependent d in a Gaussian context. Both simulations and a real data example complete our work on providing a fairly general approach