Broadband Semiparametric Estimation of the Memory Parameter of a Long‐Memory Time Series Using Fractional Exponential Models

Abstract

We consider a fractional exponential, or FEXP estimator of the memory parameter of a stationary Gaussian long‐memory time series. The estimator is constructed by fitting a FEXP model of slowly increasing dimension to the log periodogram at all Fourier frequencies by ordinary least squares, and retaining the corresponding estimated memory parameter. We do not assume that the data were necessarily generated by a FEXP model, or by any other finite‐parameter model. We do, however, impose a global differentiability assumption on the spectral density except at the origin. Because of this, and its use of all Fourier frequencies, we refer to the FEXP estimator as a broadband semiparametric estimator. We demonstrate the consistency of the FEXP estimator, and obtain expressions for its asymptotic bias and variance. If the true spectral density is sufficiently smooth, the FEXP estimator can strongly outperform existing semiparametric estimators, such as the Geweke–Porter‐Hudak (GPH) and Gaussian semiparametric estimators (GSE), attaining an asymptotic mean squared error proportional to (log n)/n, where n is the sample size. In a simulation study, we demonstrate the merits of using a finite‐sample correction to the asymptotic variance, and we also explore the possibility of automatically selecting the dimension of the exponential model using Mallows’CL criterion.