# A New Test of Multivariate Nonlinear Causality

## TL;DR

This paper introduces a revised multivariate nonlinear causality test that corrects previous statistical inference issues, providing consistent estimates and demonstrating good size and power through simulations.

## Contribution

It re-establishes the CLT for a new causality test statistic, correcting prior inaccuracies and improving inference reliability.

## Key findings

- The new test statistic's estimates are consistent.
- The test exhibits decent size and power in simulations.
- The paper corrects previous CLT assumptions for the test.

## Abstract

The multivariate nonlinear Granger causality developed by Bai et al. (2010) plays an important role in detecting the dynamic interrelationships between two groups of variables. Following the idea of Hiemstra-Jones (HJ) test proposed by Hiemstra and Jones (1994), they attempt to establish a central limit theorem (CLT) of their test statistic by applying the asymptotical property of multivariate $U$-statistic. However, Bai et al. (2016) revisit the HJ test and find that the test statistic given by HJ is NOT a function of $U$-statistics which implies that the CLT neither proposed by Hiemstra and Jones (1994) nor the one extended by Bai et al. (2010) is valid for statistical inference. In this paper, we re-estimate the probabilities and reestablish the CLT of the new test statistic. Numerical simulation shows that our new estimates are consistent and our new test performs decent size and power.