VARX Granger Analysis: Modeling, Inference, and Applications

TL;DR

This paper introduces a VARX model for analyzing delayed interactions in complex systems, bridging the gap between VARX and Granger causality, with practical code and demonstrations across multiple disciplines.

Contribution

It provides fundamental equations, implementation code, and validation for VARX models to better understand causality in complex systems, addressing a gap in existing methods.

Findings

VARX effectively captures delayed external-internal interactions.

The model reduces spurious correlations by factoring external influences.

Regularization and basis functions improve model performance with limited data.

Abstract

Complex systems, such as brains, markets, and societies, exhibit internal dynamics influenced by external factors. Disentangling delayed external effects from internal dynamics within these systems is often challenging. We propose using a Vector Autoregressive model with eXogenous input (VARX) to capture delayed interactions between internal and external variables. While this model aligns with Granger's statistical formalism for testing "causal relations", the connection between the two is not widely understood. Here, we bridge this gap by providing fundamental equations, user-friendly code, and demonstrations using simulated and real-world data from neuroscience, physiology, sociology, and economics. Our examples illustrate how the model avoids spurious correlation by factoring out external influences from internal dynamics, leading to more parsimonious explanations of the systems. We also provide methods for enhancing model efficiency, such as L2 regularization for limited data and basis functions to cope with extended delays. Additionally, we analyze model performance under various scenarios where model assumptions are violated. MATLAB, Python, and R code are provided for easy adoption: https://github.com/lcparra/varx