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Time Series Treatment Effects Analysis with Always-Missing Controls

Juan Shu, Qiyu Han, George Chen, Xihao Cao, Kangming Luo, Dan Pallotta, Shivam Agrawal, Yuping Lu, Xiaoyu Zhang, Jawad Mansoor, Jyoti Anand

TL;DR

This paper tackles causal inference in time series when the control group is unobservable during treatment, such as holiday periods. The authors introduce a predictive-model–based synthetic-control framework with a temporal adaptive loss to recover counterfactual outcomes $\hat{Y}_{i,t}(0)$ from non-treatment periods and compute treatment effects $\hat{\delta}_t$. Under a baseline AR(1) process, they establish asymptotic normality: $\sqrt{N}(\hat{\delta}-\delta)\xrightarrow{d} \mathcal{N}(\mathbf{0},\Sigma)$ with diagonal $\Sigma$ and $\Sigma_{t,t}=\sigma^2/(1-\phi^2)$. Empirical validation on Walmart M5 data shows accurate holiday-effect estimation and robust consistency across years. The framework generalizes to other time-series causal-inference settings lacking observable controls.

Abstract

Estimating treatment effects in time series data presents a significant challenge, especially when the control group is always unobservable. For example, in analyzing the effects of Christmas on retail sales, we lack direct observation of what would have occurred in late December without the Christmas impact. To address this, we try to recover the control group in the event period while accounting for confounders and temporal dependencies. Experimental results on the M5 Walmart retail sales data demonstrate robust estimation of the potential outcome of the control group as well as accurate predicted holiday effect. Furthermore, we provided theoretical guarantees for the estimated treatment effect, proving its consistency and asymptotic normality. The proposed methodology is applicable not only to this always-missing control scenario but also in other conventional time series causal inference settings.

Time Series Treatment Effects Analysis with Always-Missing Controls

TL;DR

This paper tackles causal inference in time series when the control group is unobservable during treatment, such as holiday periods. The authors introduce a predictive-model–based synthetic-control framework with a temporal adaptive loss to recover counterfactual outcomes from non-treatment periods and compute treatment effects . Under a baseline AR(1) process, they establish asymptotic normality: with diagonal and . Empirical validation on Walmart M5 data shows accurate holiday-effect estimation and robust consistency across years. The framework generalizes to other time-series causal-inference settings lacking observable controls.

Abstract

Estimating treatment effects in time series data presents a significant challenge, especially when the control group is always unobservable. For example, in analyzing the effects of Christmas on retail sales, we lack direct observation of what would have occurred in late December without the Christmas impact. To address this, we try to recover the control group in the event period while accounting for confounders and temporal dependencies. Experimental results on the M5 Walmart retail sales data demonstrate robust estimation of the potential outcome of the control group as well as accurate predicted holiday effect. Furthermore, we provided theoretical guarantees for the estimated treatment effect, proving its consistency and asymptotic normality. The proposed methodology is applicable not only to this always-missing control scenario but also in other conventional time series causal inference settings.

Paper Structure

This paper contains 15 sections, 2 theorems, 31 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Methodology
  3. Preliminaries
  4. Estimating synthetic control
  5. Temporal Adaptive loss
  6. Theoretical Properties
  7. Model Specification for Treatment Effect
  8. Forecasting Counterfactual Outcomes and Estimating the Treatment Effects
  9. Theoretical Guarantees
  10. Experiment
  11. Setup
  12. Consistency of Estimated Synthetic control
  13. Holiday impact Prediction
  14. Conclusion
  15. Disclosure of Interests.

Key Result

theorem thmcountertheorem

As $N \to \infty$, the estimator $\hat{\delta} = [\hat{\delta}_{T_0 + 1}, \dots, \hat{\delta}_{T_0 + d}]^\top$ satisfies where $\delta = [\delta_{T_0 + 1}, \dots, \delta_{T_0 + d}]^\top$ is the true treatment effect vector, and $\Sigma$ is a $d \times d$ diagonal covariance matrix with elements and $\delta_{i,j} = I\{i=j\}$ for $I(\cdot)$ as an indicator function.

Figures (1)

  • Figure 1: (A). Thanksgiving. (B).Christmas. Each row corresponds to one department.

Theorems & Definitions (4)

  • theorem thmcountertheorem
  • lemma thmcounterlemma
  • proof
  • proof