Multivariate Wavelet Whittle Estimation in Long‐range Dependence

Abstract

Multivariate processes with long-range dependent properties are found in a large number of applications including finance, geophysics and neuroscience. For real data applications, the correlation between time series is crucial. Usual estimations of correlation can be highly biased due to phase-shifts caused by the differences in the properties of autocorrelation in the processes. To address this issue, we introduce a semiparametric estimation of multivariate long-range dependent processes. The parameters of interest in the model are the vector of the long-range dependence parameters and the long-run covariance matrix, also called functional connectivity in neuroscience. This matrix characterizes coupling between time series. The proposed multivariate wavelet-based Whittle estimation is shown to be consistent for the estimation of both the long-range dependence and the covariance matrix and to encompass both stationary and nonstationary processes. A simulation study and a real data example are presented to illustrate the finite sample behaviour.

Wavelet-Based and Fourier-Based Multivariate Whittle Estimation: multiwave

“…Their good performances in comparison to Fourier have already been shown for example in univariate settings [6]. Let (φ(·), ψ(·)) be respectively a father and a mother wavelets, satisfying regularity conditions, as stated in [4]. At a given resolution j 0, for k ∈ Z, we define the dilated and translated functions φ j,k (·) = 2 −j/2 φ(2 −j · −k) and ψ j,k (·) = 2 −j/2 ψ(2 −j · −k).…”

“…The rate of convergence for the estimation of the long-range parameters d is similar to the MFW estimator and is minimax. We refer to [4] for the detailed study of the asymptotic behaviour of MWW estimation.…”

“…Even if it increases the computational time, such an initialization is important in high-dimensional settings. For example, in the MEG data set studied in [4], the optimization is done in R 274 . An initialization at the origin may not be able to reach the minimum even with a high number of iterations, due to the high dimension.…”

“…Default value for j 0 is set to 2 and for j 1 to the highest integer lower than log 2 (N ). The critical value to choose for estimation is j 0 , as it can be seen in the theoretical conditions for consistency [4] and in simulations studies. Similarly to the choice of the parameter m for MFW procedure, a compromise exists between choosing a small value of j 0 , which would introduce a bias due to the short-range properties of the time series, and a high value that will reduce the number of frequencies and thus increase variance.…”

“…Two semi-parametric estimation methods of the long-memory exponents and long-run covariance matrix of time series are implemented. The first one is the Fourier-based estimation proposed by [18] and the second one is a wavelet-based estimation described in [4]. The objective of this paper is to provide an overview of the R package multiwave with its practical application perspectives.…”