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
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 L2 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 G (no neural discovery)
166
0.01502
3/3 / 3/3
Static Λ (no time-variation)
166
0.01306
3/3 / 3/3
Dense 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 L2 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
Sign changes
0.05
0.01616
Yes
Yes
0
0.10
0.01622
Yes
Yes
0
0.15
0.01624
Yes
Yes
0
0.20
0.01626
Yes
Yes
0
0.30
0.01629
Yes
Yes
0
0.40
0.01630
Yes
Yes
0
Appendix
Table 2. Bandwidth sensitivity. Mean CRPS varies by less than 2×10−4 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
Λ
CRPS
Sign / Mono.
DCNAR (full pipeline)
166
time-varying
0.01311
3/3 / 3/3
VAR-derived G
166
time-varying
0.01502
3/3 / 3/3
Static Λ
166
static
0.01306
3/3 / 3/3
Dense 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 λ(t) (left) versus TV-VAR self-influence coefficients Aii(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 Wijλj(t) . The learned structure remains stable across the observation window while node influence evolves smoothly.
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.
Dezhi Yang, Qiaoyu Tan, Carlotta Domeniconi +3
School of Software Shandong University Jinan, China · Department of Computer Science New York University Shanghai, China · Department of Computer Science George Mason University VA, USA +1
Leveraging deep learning for causal discovery in time series remains challenging because existing neural methods predominantly rely on component-wise architectures that fail to capture shared system dynamics or employ decoupled post-hoc graph extraction that risks overfitting to spurious correlations. We propose Mask2Cause, an end-to-end framework that recovers the underlying causal graph directly during the forecasting forward pass. Our approach introduces an Inverted Variable Embedding and an Adjacency-Constrained Masked Attention mechanism, trained with homoscedastic or heteroscedastic objectives to capture causal influences in both mean and variance. Empirical results on diverse benchmarks, from synthetic chaotic dynamics to realistic biological simulations, demonstrate state-of-the-art causal discovery with significantly reduced parameter complexity compared to standard baselines. We further show that inferred causal structures can be used to reduce parameter count of forecasting models by more than 70% on average while maintaining predictive accuracy.
Omar Muhammad, Pasupuleti Dhruv Shivkant, Deepak N. Subramani
Causal discovery with instantaneous effects in multivariate time series is challenging, as the instantaneous structure must be acyclic. Prior methods enforce this by either separating instantaneous and lagged estimation into multi-stage pipelines or imposing algebraic acyclicity constraints via complex augmented Lagrangian optimization, both of which incur high computational cost. In this work, we propose a different approach: we learn a differentiable permutation of variables using the Gumbel--Sinkhorn operator and triangularize the instantaneous coefficient matrix of a Structural Vector Autoregressive (SVAR) model in the learned order. This converts acyclicity from a hard constraint into a parameterization and keeps it valid throughout optimization. In doing so, our method enables unified, continuous optimization with gradient-based learning, leading to improved efficiency in time--series causal discovery. Across three real-world benchmarks, our method achieves the best overall performance compared with 12 baselines in both discovery accuracy and efficiency. On the large-scale benchmark, it further demonstrates strong scalability, achieving more than a 6x speedup over competing methods.