[Paper Review] Cross-codifference for bidimensional VAR(1) models with infinite variance

Aleksandra Grzesiek, Marek Teuerle+1 more|arXiv (Cornell University)|Feb 6, 2019

Financial Risk and Volatility ModelingEconomics, Econometrics and Finance22 references3 citations

TL;DR

This paper introduces the cross-codifference as a dependence measure for bidimensional VAR(1) processes with infinite variance, extending the codifference concept from univariate to multivariate time series. It derives analytical expressions for cross-codifference under sub-Gaussian and Gaussian innovations, showing it reduces to cross-covariance in the Gaussian case and enables consistent parameter estimation via Monte Carlo validation.

ABSTRACT

In this paper we consider the problem of a measure that allows us to describe the spatial and temporal dependence structure of multivariate time series with innovations having infinite variance. By using recent results obtained in the problem of temporal dependence structure of univariate stochastic processes, where the auto-codifference was used, we extend its idea and propose a cross-codifference measure for a general vector autoregressive model of order 1 (VAR(1)). Next, we derive an analytical results for VAR(1) model with Gaussian and sub-Gaussian innovations, that are characterized by finite and infinite variance, respectively. We emphasize that obtained expressions perfectly agree with the empirical counterparts. Moreover, we show that for the considered processes the cross-codifference simplifies to the well-established cross-covariance measure in case of Gaussian white noise. Last part of the work is devoted to the statistical estimation of VAR(1) parameters based on the empirical cross-codifference. Again, we demonstrate via Monte Carlo simulations that proposed methodology works correctly.

Motivation & Objective

• To address the lack of a robust dependence measure for multivariate time series with infinite variance, particularly when classical cross-covariance fails.
• To extend the univariate codifference concept to bivariate vector autoregressive models of order 1 (VAR(1)).
• To derive analytical expressions for cross-codifference under sub-Gaussian and Gaussian innovations.
• To demonstrate that the cross-codifference reduces to classical cross-covariance in the Gaussian case.
• To develop and validate a statistical estimation method for VAR(1) parameters using empirical cross-codifference.

Proposed method

• Proposes a new dependence measure, the cross-codifference, defined as the negative logarithm of the characteristic function of the difference between lagged components of a VAR(1) process.
• Derives analytical expressions for cross-codifference using characteristic function factorization and infinite series decomposition of VAR(1) innovations.
• Applies the method to both sub-Gaussian (infinite variance) and Gaussian (finite variance) innovations, showing consistency across both cases.
• Uses characteristic function properties of stable and symmetric distributions to simplify the codifference expressions.
• Derives an empirical estimator of the cross-codifference based on sample characteristic functions.
• Validates the estimator via Monte Carlo simulations, demonstrating accuracy and consistency in parameter recovery.

Experimental results

Research questions

• RQ1 Can a codifference-based measure be generalized to multivariate time series with infinite variance, specifically for bidimensional VAR(1) models?
• RQ2 How does the proposed cross-codifference behave under sub-Gaussian (infinite variance) and Gaussian (finite variance) innovations?
• RQ3 Does the cross-codifference reduce to the classical cross-covariance in the Gaussian case, confirming consistency with established measures?
• RQ4 Can the empirical cross-codifference be used effectively to estimate VAR(1) model parameters in finite samples?
• RQ5 How robust is the cross-codifference estimator under varying parameter configurations and sample sizes?

Key findings

• The cross-codifference is analytically derived for both sub-Gaussian and Gaussian innovations in a bidimensional VAR(1) model, providing a unified dependence measure.
• For Gaussian innovations, the cross-codifference simplifies exactly to the classical cross-covariance, validating its consistency with established theory.
• The derived analytical expressions for cross-codifference perfectly match their empirical counterparts in simulations, confirming theoretical accuracy.
• The empirical cross-codifference estimator is consistent and effective in recovering VAR(1) model parameters, as demonstrated by Monte Carlo simulations.
• The method enables reliable dependence structure analysis and parameter estimation in multivariate time series with infinite variance, where standard cross-covariance fails.