Causal Discovery in Structural VAR Models Under Equal Noise Variance

TL;DR

This paper introduces a new method for causal discovery in linear Gaussian structural VAR models with equal noise variance, addressing challenges posed by contemporaneous effects and coarse sampling in time series.

Contribution

It characterizes the observational equivalence class under equal noise variance and proposes ENVAR, a sparsity-based method to identify a representative causal structure.

Findings

ENVAR effectively recovers causal structures in synthetic data.

The method provides meaningful insights in an fMRI dataset.

Theoretical characterization of equivalence classes enhances causal inference.

Abstract

Causal discovery from multivariate time series is challenging when causal effects may occur both across time and within the same sampling interval. This issue is especially important in applications such as neuroscience, where the sampling rate may be coarse relative to the underlying dynamics and contemporaneous effects need not form an acyclic graph. We study causal discovery in linear Gaussian structural VAR models under an equal noise variance assumption, meaning that the structural noise terms have a common variance. Unlike the DAG-based cross-sectional equal noise variance setting, the time-series setting considered here does not generally yield point identification of a unique causal graph. Instead, multiple structural VAR parameterizations can induce the same stationary observed process law. We introduce a notion of observational equivalence tailored to this setting and show that the corresponding equivalence class is characterized by orthogonal transformations of the structural equations together with a global positive scale. This characterization leads to an equivalence-aware model discrepancy, the observational alignment discrepancy, which compares structural models modulo transformations that preserve the observed law. Building on this theory, we propose ENVAR, a sparsity-based procedure that searches over the induced observational equivalence class for a sparse normalized structural representative. We evaluate the proposed methodology on synthetic structural VAR data and on an fMRI dataset.