VARX Granger Analysis: Modeling, Inference, and Applications

Vector Autoregressive models with exogenous input (VARX) provide a powerful framework for modeling complex dynamical systems like brains, markets, or societies. Their simplicity allows us to uncover linear effects between endogenous and exogenous variables. The Granger formalism is naturally suited for VARX models, but the connection between the two is not widely understood. We aim to bridge this gap by providing both the basic equations and easy-to-use code. We first explain how the Granger formalism can be combined with a VARX model using deviance as a test statistic. We also present a bias correction for the deviance in the case of L2 regularization, a technique used to reduce model complexity. To address the challenge of modeling long responses, we propose the use of basis functions, which further reduce parameter complexity. We demonstrate that p-values are correctly estimated, even for short signals where regularization influences the results. Additionally, we analyze the model's performance under various scenarios where model assumptions are violated, such as missing variables or indirect observations of the underlying dynamics. Finally, we showcase the practical utility of our approach by applying it to real-world data from neuroscience, physiology, and sociology. To facilitate its adoption, we make Matlab, Python, and R code available here: https://github.com/lcparra/varx