[Paper Review] Granger Causality in Expectiles: an M-vine copula test

Roberto Fuentes-Martínez, Irene Crimaldi|arXiv (Cornell University)|Mar 24, 2026

Financial Risk and Volatility Modeling0 citations

TL;DR

Proposes a model-free measure of Granger causality in expectiles and develops a multivariate M-vine copula test to detect it, including consistency proofs, simulations, and stock market applications.

ABSTRACT

A model-free measure of Granger causality in expectiles is proposed, generalizing the traditional mean-based measure to arbitrary positions of the conditional distribution. Expectiles are the only law-invariant risk measures that are both coherent and elicitable, making them particularly well-suited for studying distributional Granger causality where risk quantification and forecast evaluation are both relevant. Based on this measure, a test is developed using M-vine copula models that accounts for multivariate Granger causality with $d+1$ series under non-linear and non-Gaussian dependence, without imposing parametric assumptions on the joint distribution. Strong consistency of the test statistic is established under some regularity conditions. In finite samples, simulations show accurate size control and power increasing with sample size. A key advantage is the joint testing capability: causal relationships invisible to pairwise tests can be detected, as demonstrated both theoretically and empirically. Two applications to international stock market indices at the global and Asian regional level illustrate the practical relevance of the proposed framework.

Motivation & Objective

• Motivate distributional Granger causality beyond the mean by using expectiles as a coherent, elicitable risk measure.
• Define a model-free GC measure at a given expectile level.
• Develop a multivariate M-vine copula test to assess GC with non-linear, non-Gaussian dependence.
• Prove consistency of the test and evaluate finite-sample performance through simulations and applications.

Proposed method

• Define GC in the tau-th expectile using asymmetric quadratic loss and expectile minimizers.
• Construct a model-free GC measure GC_tau(Z -> X) as a log-ratio of expectile losses.
• Estimate expectations via simulation from an M-vine copula model fitted to data.
• Compute the test statistic as a log-difference of two losses with and without Z, using the fitted M-vine to generate predictions.
• Obtain p-values under the null by simulating under the null via sequential extraction of copulas from the M-vine structure and generating B replicates.
• Assess size and power through Monte Carlo simulations under various DGPs, including non-linear and tail-dependent settings.
• Compare with pairwise tests and kernel-based quantile causality tests, and contrast with linear Granger causality in the mean.

Experimental results

Research questions

• RQ1 Can Granger causality in expectiles be measured in a model-free way using expectile losses?
• RQ2 Does a multivariate M-vine copula framework reliably detect joint Granger causality at different expectile levels, including tails?
• RQ3 How does the proposed test perform relative to pairwise tests and linear mean-based GC tests under nonlinear and heavy-tailed data?
• RQ4 Do empirical applications (global and Asian stock indices) reveal joint causality detectable only by the multivariate test?

Key findings

• The GC_tau measure is non-negative and equals zero iff there is no Granger causality in the tau-th expectile (and reduces to the mean-based measure at tau = 1/2).
• The M-vine copula test provides a strongly consistent estimator and maintains good finite-sample size control, with power increasing with sample size.
• Joint testing via the M-vine framework detects causal relationships invisible to pairwise tests, especially in tail regions.
• In simulations, the test outperforms kernel-based quantile causality tests, particularly in finite samples and tail dependence.
• Compared to the classical linear Granger F-test, the proposed method captures nonlinear dependencies and attains higher power in non-linear DGPs, while maintaining comparable size in linear settings.
• Empirical applications on international stock indices demonstrate joint causality patterns not evident from pairwise analyses.