Codifference as a practical tool to measure interdependence

TL;DR

This paper reviews the use of codifference as a practical measure of interdependence in stochastic processes, especially useful for heavy-tailed data where traditional tools like covariance fail.

Contribution

It introduces codifference as a versatile tool for analyzing interdependence, providing explicit formulas and demonstrating its application to real and simulated data.

Findings

Codifference equals covariance for Gaussian processes.

It remains relevant for processes with infinite variance.

Practical extraction from real data demonstrates its utility.

Abstract

Correlation and spectral analysis represent the standard tools to study interdependence in statistical data. However, for the stochastic processes with heavy-tailed distributions such that the variance diverges, these tools are inadequate. The heavy-tailed processes are ubiquitous in nature and finance. We here discuss codifference as a convenient measure to study statistical interdependence, and we aim to give a short introductory review of its properties. By taking different known stochastic processes as generic examples, we present explicit formulas for their codifferences. We show that for the Gaussian processes codifference is equivalent to covariance. For processes with finite variance these two measures behave similarly with time. For the processes with infinite variance the covariance does not exist, however, the codifference is relevant. We demonstrate the practical importance of the codifference by extracting this function from simulated as well as from real experimental data. We conclude that the codifference serves as a convenient practical tool to study interdependence for stochastic processes with both infinite and finite variances as well.