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A Practical Approach to Causal Inference over Time
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Maps stable VARs to linear (possibly cyclic, confounded) SCMs at equilibrium via a long-run normalized-mean transform, and defines additive and forcing interventions for causal inference over time on observational time series.
Problem
Estimating causal effects of interventions over time on dynamical systems from observational time series, where SCMs lack temporal dynamics and autoregressive models lack causal semantics (Granger causality only).
Approach
Defines time-indexed causal interventions on discrete-time stochastic processes, shows the equilibrium of a VAR (after a long-run normalized-mean transform Z_t) equals a linear SCM with A = sum of A_i, and gives additive and forcing intervention implementations on VARs so causal effects can be forecast from observational data.
Key results
- On German and Pendulum, VAR achieves observational 1-step MAE .008 on German (size 100) vs DLinear .009, TiDE .011, TSMixer .014, close to Oracle .004.
- On Census 5-step horizon, TiDE (.014) slightly outperforms VAR (.017), while VAR matches or beats the other baselines on most settings.
- Interventional forecasting on German at 10-step horizon yields additive MAE .043e-2 and forcing MAE .364e-2 with 100 samples.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use VAR instead of DLinear for observational forecasting | Forecasting models · German and Pendulum, sizes 100 and 500, horizons 1 and 10 | MAE: Pendulum, size 100, horizon 10: .440 -> .420; Pendulum, size 500, horizon 10: .405 -> .401 | Mixed |
| Use VAR instead of TiDE or TSMixer for observational forecasting | Forecasting models · German and Pendulum | MAE | Helped |
| Use TiDE instead of VAR for observational forecasting | Forecasting models · Census, horizon 5 | MAE: .017 -> .014 | Helped |
| Use additive rather than forcing intervention | Causal VAR · German, size 100, horizon 10 | MAE: .043_{.028} vs .364_{.297} | Helped |
| Use additive rather than forcing intervention | Causal VAR · German, size 500, horizon 10 | MAE: .018_{.014} vs .115_{.081} | Helped |
| Use additive rather than forcing intervention | Causal VAR · German, horizons 1, sizes 100 and 500 | MAE: .000_{.000} vs .000_{.000} | No clear effect |
Surprising findings
- Forcing interventions can break VAR stability even when the observational system is stable (shown with a toy VAR(1) example at F=1).
- The equilibrium SCM of a stable VAR is linear and may be cyclic and confounded, yet interventions on the VAR and the SCM commute.
Limitations
- Linear VARs may be less effective for strongly nonlinear dynamics.
- Requires prior knowledge of the causal graph; causal discovery is challenging otherwise.
- Forcing interventions can make the intervened system unstable; nonlinear SCM mapping is not yet completed.
Experimental setup
Datasets: German (synthetic) Pendulum (synthetic) Census (US Census Bureau, 50 countries, 1992-2023)
Benchmarks: German Pendulum Census
Metrics: MAE
Baselines: DLinear TiDE TSMixer Oracle forecaster
Builds on: Structural VAR Granger causality cyclic SCMs Wold decomposition
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