A Generalization of a Gaussian Semiparametric Estimator on Multivariate Long-Range Dependent Processes

In this paper we propose and study a general class of Gaussian Semiparametric Estimators (GSE) of the fractional differencing parameter in the context of long-range dependent multivariate time series. We establish large sample properties of the estimator without assuming Gaussianity. The class of models considered here satisfies simple conditions on the spectral density function, restricted to a small neighborhood of the zero frequency and includes important class of VARFIMA processes. We also present a simulation study to assess the finite sample properties of the proposed estimator based on a smoothed version of the GSE which supports its competitiveness.

💡 Research Summary

This paper introduces a broad class of Gaussian semiparametric estimators (GSE) for the fractional differencing parameter d in multivariate long‑range dependent (LRD) time series. The authors start by noting that most existing multivariate LRD estimators either rely on strict Gaussian assumptions or impose heavy restrictions on the spectral density. To overcome these limitations, they formulate a minimal set of conditions on the spectral density matrix f(λ) in a neighborhood of the zero frequency:
(f(\lambda)=G,\lambda^{-2d}+R(\lambda)) with a positive‑definite matrix G, a vector of differencing orders d, and a remainder term R(λ)=o(\lambda^{-2d}) as λ→0. This formulation includes VARFIMA(p,d,q) processes and many other multivariate ARFIMA‑type models.

The estimator is built from the periodogram evaluated at the lowest mₙ Fourier frequencies. A weight function w(λ) (typically a symmetric kernel) is applied, and the estimator (\hat d_n) minimizes the criterion
(Q_n(d)=\sum_{j=1}^{m_n} w(\lambda_j)\bigl