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Time-Aware Synthetic Control

Saeyoung Rho, Cyrus Illick, Samhitha Narasipura, Alberto Abadie, Daniel Hsu, Vishal Misra

TL;DR

Time-Aware Synthetic Control (TASC) extends synthetic control by embedding panel data into a linear Gaussian state-space model with a persistent trend, enabling explicit use of time ordering via $x_t = A x_{t-1} + q_{t-1}$ and $y_t = H x_t + r_t$. It uses an EM algorithm to learn parameters and Kalman filtering with Rauch–Tung–Striebel smoothing to perform counterfactual inference after treatment, treating post-intervention target data as missing. Empirical results on simulated data and real-world cases (e.g., Proposition 99 in California, cricket, and NBA score trajectories) show TASC delivers improved predictive accuracy in high-noise, trend-rich settings and produces narrower confidence intervals than baselines like CIM. The work highlights the value of incorporating temporal dynamics alongside low-rank structure in causal panel data, while noting limitations (single time series, linear trend) and pointing to future extensions toward multivariate, nonlinear state-space models and more scalable learning methods.

Abstract

The synthetic control (SC) framework is widely used for observational causal inference with time-series panel data. SC has been successful in diverse applications, but existing methods typically treat the ordering of pre-intervention time indices interchangeable. This invariance means they may not fully take advantage of temporal structure when strong trends are present. We propose Time-Aware Synthetic Control (TASC), which employs a state-space model with a constant trend while preserving a low-rank structure of the signal. TASC uses the Kalman filter and Rauch-Tung-Striebel smoother: it first fits a generative time-series model with expectation-maximization and then performs counterfactual inference. We evaluate TASC on both simulated and real-world datasets, including policy evaluation and sports prediction. Our results suggest that TASC offers advantages in settings with strong temporal trends and high levels of observation noise.

Time-Aware Synthetic Control

TL;DR

Time-Aware Synthetic Control (TASC) extends synthetic control by embedding panel data into a linear Gaussian state-space model with a persistent trend, enabling explicit use of time ordering via and . It uses an EM algorithm to learn parameters and Kalman filtering with Rauch–Tung–Striebel smoothing to perform counterfactual inference after treatment, treating post-intervention target data as missing. Empirical results on simulated data and real-world cases (e.g., Proposition 99 in California, cricket, and NBA score trajectories) show TASC delivers improved predictive accuracy in high-noise, trend-rich settings and produces narrower confidence intervals than baselines like CIM. The work highlights the value of incorporating temporal dynamics alongside low-rank structure in causal panel data, while noting limitations (single time series, linear trend) and pointing to future extensions toward multivariate, nonlinear state-space models and more scalable learning methods.

Abstract

The synthetic control (SC) framework is widely used for observational causal inference with time-series panel data. SC has been successful in diverse applications, but existing methods typically treat the ordering of pre-intervention time indices interchangeable. This invariance means they may not fully take advantage of temporal structure when strong trends are present. We propose Time-Aware Synthetic Control (TASC), which employs a state-space model with a constant trend while preserving a low-rank structure of the signal. TASC uses the Kalman filter and Rauch-Tung-Striebel smoother: it first fits a generative time-series model with expectation-maximization and then performs counterfactual inference. We evaluate TASC on both simulated and real-world datasets, including policy evaluation and sports prediction. Our results suggest that TASC offers advantages in settings with strong temporal trends and high levels of observation noise.
Paper Structure (27 sections, 1 theorem, 7 equations, 18 figures, 9 algorithms)

This paper contains 27 sections, 1 theorem, 7 equations, 18 figures, 9 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Synthetic Control Methods
  4. Latent Variable Models for Synthetic Control
  5. State-Space Model for TASC
  6. More Advanced Models for TASC
  7. Allowing time-invariant portion of latent states
  8. Allowing seasonality
  9. Allowing multiple time series
  10. Comparison to Other Models
  11. When TASC Model Is Advantageous
  12. TASC Algorithms
  13. Learning from Pre-Intervention Data
  14. Counterfactual Inference with Post-Intervention Data
  15. Empirical Evaluation on Simulated Data
  16. ...and 12 more sections

Key Result

Proposition A.1

Consider the panel data following Equations eq.model.latent and eq.model.observation. Then, the following holds.

Figures (18)

  • Figure 1: General data structure for synthetic control
  • Figure 2: Post-intervention RMSE of TASC when the time indices are kept in their original order (left) and when the indices are permuted (right)
  • Figure 3: Post-intervention RMSE of TASC and benchmark methods on datasets generated with low observation noise (small $R$): small $Q$ (left) and large $Q$ (right)
  • Figure 4: Post-intervention RMSE of TASC and benchmark methods on datasets generated with high observation noise (large $R$): small $Q$ (left) and large $Q$ (right)
  • Figure 5: Post-intervention RMSE under low observation noise and small $Q$, evaluated across five future prediction horizons (10 time periods each)
  • ...and 13 more figures

Theorems & Definitions (3)

  • Proposition A.1
  • proof : Proof Sketch
  • Remark A.2