Local Whittle estimation with (quasi-)analytic wavelets

TL;DR

This paper introduces a novel inference method using quasi-analytic wavelets to estimate long-memory and coupling parameters in multivariate long-memory time series, applicable beyond Gaussian or stationary assumptions.

Contribution

It develops a new estimation approach based on wavelet covariance and local Whittle approximation for complex, non-Gaussian, and non-stationary multivariate time series.

Findings

Estimates are consistent and effective in simulations.

Method performs well on real neuroscience data.

Provides insights into long-memory and connectivity in brain signals.

Abstract

In the general setting of long-memory multivariate time series, the long-memory characteristics are defined by two components. The long-memory parameters describe the autocorrelation of each time series. And the long-run covariance measures the coupling between time series, with general phase parameters. It is of interest to estimate the long-memory, long-run covariance and general phase parameters of time series generated by this wide class of models although they are not necessarily Gaussian nor stationary. This estimation is thus not directly possible using real wavelets decomposition or Fourier analysis. Our purpose is to define an inference approach based on a representation using quasi-analytic wavelets. We first show that the covariance of the wavelet coefficients provides an adequate estimator of the covariance structure including the phase term. Consistent estimators based on a local Whittle approximation are then proposed. Simulations highlight a satisfactory behavior of the estimation on finite samples on linear time series and on multivariate fractional Brownian motions. An application on a real neuroscience dataset is presented, where long-memory and brain connectivity are inferred.