Fractal approach towards power-law coherency to measure cross-correlations between time series

TL;DR

This paper introduces three estimators for power-law coherency to analyze cross-correlations in time series, finding DMCA to be the most reliable, and discusses their finite sample properties and applicability.

Contribution

It presents novel estimators for power-law coherency based on established techniques, and evaluates their finite sample properties for empirical analysis.

Findings

DMCA-based estimator is the safest choice.

HXA is suitable for long series with over 10,000 observations.

DCCA-based method shows deteriorating properties with increasing series length.

Abstract

We focus on power-law coherency as an alternative approach towards studying power-law cross-correlations between simultaneously recorded time series. To be able to study empirical data, we introduce three estimators of the power-law coherency parameter $H_{\rho}$ based on popular techniques usually utilized for studying power-law cross-correlations -- detrended cross-correlation analysis (DCCA), detrending moving-average cross-correlation analysis (DMCA) and height cross-correlation analysis (HXA). In the finite sample properties study, we focus on the bias, variance and mean squared error of the estimators. We find that the DMCA-based method is the safest choice among the three. The HXA method is reasonable for long time series with at least $10^4$ observations, which can be easily attainable in some disciplines but problematic in others. The DCCA-based method does not provide favorable properties which even deteriorate with an increasing time series length. The paper opens a new venue towards studying cross-correlations between time series.