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Granger Causality in Expectiles: an M-vine copula test

Roberto Fuentes-Martínez, Irene Crimaldi

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.

Granger Causality in Expectiles: an M-vine copula test

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 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.
Paper Structure (13 sections, 3 theorems, 60 equations, 7 tables)

This paper contains 13 sections, 3 theorems, 60 equations, 7 tables.

Table of Contents

  1. Introduction
  2. Granger causality in expectiles
  3. M-vine test for Granger causality in expectiles
  4. Simulation Study
  5. Granger causality in the mean (i.e. $\tau=1/2$) for $(Y,Z)\to X$: comparison with the classic Granger causality F-test
  6. Empirical Applications
  7. Causality among Global Stock Markets
  8. Causality among Asian Stock Markets
  9. Conclusion
  10. Appendix
  11. Theoretical results
  12. Power-assessment model P3: non-pairwise but joint Granger causality in the mean
  13. Power-assessment model P3: pairwise and joint Granger causality in $\tau$-expectiles, with $\tau\neq 1/2$

Key Result

Theorem 1

Let $X$ be a real random variable with ${\mathbb E}[X^2]<\infty$ and let $X_1,X_2,\dots$ be i.i.d. real random variables with the same distribution of $X$ and fix $\tau\in (0,1)$. Denote by $\mu_{\tau}$ the $\tau$-expectiles of $X$, i.e. the unique minimizer (over $m$) of Denote by $\widehat{\mu}_{\tau,N}$ the empirical $\tau$-expectile, i.e. the random variable $\widehat{\mu}_{\tau,N}(X_1,\dots,

Theorems & Definitions (10)

  • Definition 1
  • Definition 2
  • Remark 1
  • Remark 2
  • Theorem 1: Strong consistency of the empirical $\tau$-expectile
  • proof
  • Corollary 1
  • proof
  • Corollary 2
  • proof