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Incentive-Aware Synthetic Control: Accurate Counterfactual Estimation via Incentivized Exploration

Daniel Ngo, Keegan Harris, Anish Agarwal, Vasilis Syrgkanis, Zhiwei Steven Wu

Abstract

We consider the setting of synthetic control methods (SCMs), a canonical approach used to estimate the treatment effect on the treated in a panel data setting. We shed light on a frequently overlooked but ubiquitous assumption made in SCMs of "overlap": a treated unit can be written as some combination -- typically, convex or linear combination -- of the units that remain under control. We show that if units select their own interventions, and there is sufficiently large heterogeneity between units that prefer different interventions, overlap will not hold. We address this issue by proposing a framework which incentivizes units with different preferences to take interventions they would not normally consider. Specifically, leveraging tools from information design and online learning, we propose a SCM that incentivizes exploration in panel data settings by providing incentive-compatible intervention recommendations to units. We establish this estimator obtains valid counterfactual estimates without the need for an a priori overlap assumption. We extend our results to the setting of synthetic interventions, where the goal is to produce counterfactual outcomes under all interventions, not just control. Finally, we provide two hypothesis tests for determining whether unit overlap holds for a given panel dataset.

Incentive-Aware Synthetic Control: Accurate Counterfactual Estimation via Incentivized Exploration

Abstract

We consider the setting of synthetic control methods (SCMs), a canonical approach used to estimate the treatment effect on the treated in a panel data setting. We shed light on a frequently overlooked but ubiquitous assumption made in SCMs of "overlap": a treated unit can be written as some combination -- typically, convex or linear combination -- of the units that remain under control. We show that if units select their own interventions, and there is sufficiently large heterogeneity between units that prefer different interventions, overlap will not hold. We address this issue by proposing a framework which incentivizes units with different preferences to take interventions they would not normally consider. Specifically, leveraging tools from information design and online learning, we propose a SCM that incentivizes exploration in panel data settings by providing incentive-compatible intervention recommendations to units. We establish this estimator obtains valid counterfactual estimates without the need for an a priori overlap assumption. We extend our results to the setting of synthetic interventions, where the goal is to produce counterfactual outcomes under all interventions, not just control. Finally, we provide two hypothesis tests for determining whether unit overlap holds for a given panel dataset.
Paper Structure (35 sections, 12 theorems, 67 equations, 3 figures, 2 algorithms)

This paper contains 35 sections, 12 theorems, 67 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Overview of Paper.
  3. Related Work
  4. Causal Inference and Synthetic Control Methods.
  5. Bayesian Persuasion and Incentivized Exploration.
  6. Model
  7. Notation.
  8. Our Panel Data Setting.
  9. Recommendations and Beliefs
  10. Preliminaries
  11. On Estimating Unit-Specific Counterfactual Outcomes
  12. Vertical vs. Horizontal Regression.
  13. Principal Component Regression.
  14. Incentivized Exploration for Synthetic Control
  15. Extension to Synthetic Interventions
  16. ...and 20 more sections

Key Result

Theorem 3.1

Fix a time horizon $T$, pre-intervention time period $T_0$, and number of donor units $n^{(0)}$. For any algorithm used to estimate the average post-intervention outcome under control for a test unit, there exists a problem instance such that the produced estimate has constant error whenever the ass

Figures (3)

  • Figure 1: Summary of our setting.
  • Figure 2: Counterfactual estimation error for units of type $1$ under control using \ref{['alg:ie_noise']} (blue) and synthetic control without incentives (orange) with increasing number of units and different lengths of the latent factors. Results are averaged over $50$ runs, with the shaded regions representing one standard deviation.
  • Figure 3: Counterfactual estimation error for units of type $1$ under control using \ref{['alg:ie_noise']} (blue) and synthetic control without incentives (orange) with different gaps between the prior mean reward of control and treatment. Results are averaged over $50$ runs, with the shaded regions representing one standard deviation.

Theorems & Definitions (28)

  • Definition 2.2
  • Definition 2.3: Interaction History
  • Definition 2.4: Recommendation Policy
  • Definition 2.5: Unit Belief
  • Definition 2.6: Unit Type
  • Definition 2.7: Bayesian Incentive-Compatibility
  • Theorem 3.1
  • proof
  • Proposition 3.3
  • Theorem 4.2: Informal; detailed version in \ref{['lem:bic-two-types']}
  • ...and 18 more