Function-Valued Causal Influence in Nonlinear Time Series
Abstract
Causal discovery in time series is increasingly performed using nonlinear machine-learning models, yet the resulting causal relationships are almost always summarized by scalar edge scores. We argue that this practice obscures the true object learned by nonlinear autoregressive models: a state-dependent function whose effect varies across regimes, magnitudes, and contexts. We formalize function-valued causal influence for additive, contribution-decomposable architectures and show that scalar causal scores constitute a severe information bottleneck, conflating between-state variation with within-state residual noise. Using Neural Additive Vector Autoregression as a representative architecture, we introduce a practical framework based on Individual Conditional Expectation for estimating causal response functions directly from trained models. Through controlled synthetic experiments, we demonstrate that edges with indistinguishable scalar scores can exhibit qualitatively different functional behaviors, including monotonic, thresholded, saturating, and sign-changing effects. An applied case study on democratic development further shows that function-valued analysis reveals regime-specific and asymmetric causal structure systematically missed by score-centric approaches.
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BibTeX
@misc{kuskova2026functionvalued,
title = {{Function-Valued Causal Influence in Nonlinear Time Series}},
author = {Valentina V. Kuskova and Dmitry Zaytsev and Michael Coppedge},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.26408},
eprint = {2605.26408},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.26408},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Function-Valued Causal Influence in Nonlinear Time Series
AU - Valentina V. Kuskova
AU - Dmitry Zaytsev
AU - Michael Coppedge
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2605.26408
UR - https://arxiv.org/abs/2605.26408
AB - Causal discovery in time series is increasingly performed using nonlinear machine-learning models, yet the resulting causal relationships are almost always summarized by scalar edge scores. We argue that this practice obscures the true object learned by nonlinear autoregressive models: a state-dependent function whose effect varies across regimes, magnitudes, and contexts. We formalize function-valued causal influence for additive, contribution-decomposable architectures and show that scalar causal scores constitute a severe information bottleneck, conflating between-state variation with within-state residual noise. Using Neural Additive Vector Autoregression as a representative architecture, we introduce a practical framework based on Individual Conditional Expectation for estimating causal response functions directly from trained models. Through controlled synthetic experiments, we demonstrate that edges with indistinguishable scalar scores can exhibit qualitatively different functional behaviors, including monotonic, thresholded, saturating, and sign-changing effects. An applied case study on democratic development further shows that function-valued analysis reveals regime-specific and asymmetric causal structure systematically missed by score-centric approaches.
ER -