From Causal Discovery to Dynamic Causal Inference in Neural Time Series

Dmitry Zaytsev; Valentina Kuskova; Michael Coppedge

cs.LGMar 21, 2026

From Causal Discovery to Dynamic Causal Inference in Neural Time Series

Authors: Dmitry Zaytsev, Valentina Kuskova, Michael Coppedge

Organizations: Lucy Family Institute for Data & Society, University of Notre Dame, Notre Dame, Indiana, USA · Political Science, University of Notre Dame, Notre Dame, Indiana, USA

Abstract

Time-varying causal models provide a powerful framework for studying dynamic scientific systems, yet most existing approaches assume that the underlying causal network is known a priori - an assumption rarely satisfied in real-world domains where causal structure is uncertain, evolving, or only indirectly observable. This limits the applicability of dynamic causal inference in many scientific settings. We propose Dynamic Causal Network Autoregression (DCNAR), a two-stage neural causal modeling framework that integrates data-driven causal discovery with time-varying causal inference. In the first stage, a neural autoregressive causal discovery model learns a sparse directed causal network from multivariate time series. In the second stage, this learned structure is used as a structural prior for a time-varying neural network autoregression, enabling dynamic estimation of causal influence without requiring pre-specified network structure. We evaluate the scientific validity of DCNAR using behavioral diagnostics that assess causal necessity, temporal stability, and sensitivity to structural change, rather than predictive accuracy alone. Experiments on multi-country panel time-series data demonstrate that learned causal networks yield more stable and behaviorally meaningful dynamic causal inferences than coefficient-based or structure-free alternatives, even when forecasting performance is comparable. These results position DCNAR as a general framework for using AI as a scientific instrument for dynamic causal reasoning under structural uncertainty.

Figures & tables

Figure 1. Comparison of DCNAR with Ridge VAR, TV-VAR, and LSTM (MC Dropout) across predictive (panel A, CRPS), distributional (panel B, local distributional accuracy - mean one-step-ahead CRPS), and causal diagnostics (panel C, nominal 90% prediction intervals across horizons) on the short-panel democracy dataset. Panel D shows representative counterfactual impulse response following a positive shock to freedom of expression (Albania). Four-panel comparison of DCNAR against Ridge VAR, TV-VAR, and LSTM baselines on the short-panel democracy dataset, showing CRPS across forecast horizons, mean one-step CRPS by model, empirical coverage of 90 percent prediction intervals, and a representative counterfactual impulse response for Albania following a positive shock to freedom of expression.

Figure 2. System-level counterfactual trajectories under DCNAR, summarized by the $L^{2}$ normalization across all democracy components, for Albania, the United States, and Mexico. Solid lines denote trajectories under a localized positive shock to freedom of expression at horizon $h=1$ ; dashed lines denote corresponding baseline trajectories without intervention. System-level counterfactual trajectories under DCNAR for Albania, the United States, and Mexico, showing the L2 norm of state across all democracy components over a 10-step horizon. Solid lines denote trajectories under a positive shock to freedom of expression at horizon 1; dashed lines denote baseline trajectories without intervention.

Condition	Edges	CRPS	Sign consistent / Monotonic
DCNAR (full pipeline)	166	0.01311	3/3 / 3/3
VAR-derived $\widehat{\mathbf{G}}$ (no neural discovery)	166	0.01502	3/3 / 3/3
Static $\boldsymbol{\Lambda}$ (no time-variation)	166	0.01306	3/3 / 3/3
Dense $\widehat{\mathbf{G}}$ (no sparsity)	240	0.01323	3/3 / 3/3

Table 1. Architectural ablation on the short panel. Replacing neural discovery with linear VAR–derived structure degrades CRPS by 14.5%; all variants preserve sign-consistent, monotonic impulse responses.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 3. Comparison of DCNAR on the 89-country, 75-year panel, with Ridge VAR, TV-VAR, and LSTM (MC Dropout) across predictive (panel A, CRPS), distributional (panel B, local distributional accuracy - mean one-step-ahead CRPS), and causal diagnostics (panel C, nominal 90% prediction intervals across horizons) on the extended-panel democracy dataset. Panel D shows representative counterfactual impulse response following a positive shock to freedom of expression (Albania).

Figure 4. System-level counterfactual trajectories under DCNAR on the 89-country, 75-year panel, summarized by the $L^{2}$ normalization across all democracy components, for Albania, the United States, and Mexico. Solid lines denote trajectories under a localized positive shock to freedom of expression at horizon $h=1$ ; dashed lines denote corresponding baseline trajectories without intervention.

Bandwidth $h$	Mean CRPS	Sign consistent	Monotonic
0.05	0.01616	Yes	Yes
0.10	0.01622	Yes	Yes
0.15	0.01624	Yes	Yes
0.20	0.01626	Yes	Yes
0.30	0.01629	Yes	Yes
0.40	0.01630	Yes	Yes

Appendix

Table 2. Bandwidth sensitivity. Mean CRPS varies by less than

2\times 10^{-4}

across the full range, and impulse responses remain smooth, monotonic, and sign-consistent (zero sign changes) for every country and bandwidth.

Figure 5. DCNAR predictive accuracy (mean CRPS) across bandwidth values. The bandwidth used in the main experiments ( $h=0.20$ ) is highlighted. Predictive accuracy is essentially flat across the full range.

Condition	Edges	$\boldsymbol{\Lambda}$	CRPS	Sign / Mono.
DCNAR (full pipeline)	166	time-varying	0.01311	3/3 / 3/3
VAR-derived $\widehat{\mathbf{G}}$	166	time-varying	0.01502	3/3 / 3/3
Static $\boldsymbol{\Lambda}$	166	static	0.01306	3/3 / 3/3
Dense $\widehat{\mathbf{G}}$	240	time-varying	0.01323	3/3 / 3/3

Appendix

Table 3. Architectural ablation on the short panel. Replacing neural discovery with a linear VAR–derived structure degrades CRPS by 14.5%. All variants preserve sign-consistent, monotonic impulse responses across the three countries examined.

Figure 6. Impulse-response behavior under each ablation condition (DCNAR full pipeline solid; ablated conditions dashed) for shocks to freedom of expression in Albania, the United States, and Mexico. The VAR-derived structure systematically overestimates shock persistence (slower decay), while the other variants closely track the full pipeline.

Figure 7. Dynamic coherence: DCNAR node-influence paths $\lambda(t)$ (left) versus TV-VAR self-influence coefficients $A_{ii}(t)$ (right) for Albania. DCNAR uses the learned structural prior; TV-VAR uses dense, unstructured coefficients. DCNAR paths stabilize after the early-1990s transition, whereas TV-VAR coefficients remain comparatively erratic.

Figure 8. DCNAR effective network snapshots for Albania at three time points (1996, 2002, 2008), showing the effective edge weights $W_{ij}\,\lambda_{j}(t)$ . The learned structure remains stable across the observation window while node influence evolves smoothly.

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Learning the dynamic causal structure of time series is a challenging problem. Most existing approaches rely on distributional or structural invariance to uncover underlying causal dynamics, assuming stationary or partially stationary causality. However, these assumptions often conflict with the complex, time-varying causal relationships observed in real-world systems. This motivates the need for methods that address fully dynamic causality, where both instantaneous and lagged dependencies evolve over time. Such a setting poses significant challenges for the efficiency and stability of causal discovery. To address these challenges, we introduce DyCausal, a dynamic causal structure learning framework. DyCausal leverages convolutional networks to capture causal patterns within coarse-grained time windows, and then applies linear interpolation to refine causal structures at each time step, thereby recovering fine-grained and time-varying causal graphs. In addition, we propose an acyclic constraint based on matrix norm scaling, which improves efficiency while effectively constraining loops in evolving causal structures. Comprehensive evaluations on both synthetic and real-world datasets demonstrate that DyCausal achieves superior performance compared to existing methods, offering a stable and efficient approach for identifying fully dynamic causal structures from coarse to fine.

Abstract

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