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Causal Inference with Groupwise Matching

Ratzanyel Rincón, Kyungchul Song

Abstract

This paper examines methods of causal inference based on groupwise matching when we observe multiple large groups of individuals over several periods. We formulate causal inference validity through a generalized matching condition, generalizing the parallel trend assumption in difference-in-differences designs. We show that difference-in-differences, synthetic control, and synthetic difference-in-differences designs are distinguished by the specific matching conditions that they invoke. Through regret analysis, we demonstrate that difference-in-differences and synthetic control with differencing are complementary; the former dominates the latter if and only if the latter's extrapolation error exceeds the former's matching error up to a term vanishing at the parametric rate. The analysis also reveals that synthetic control with differencing is equivalent to difference-in-differences when the parallel trend assumption holds for both the pre-treatment and post-treatment periods. We develop a statistical inference procedure based on synthetic control with differencing and present an empirical application demonstrating its usefulness.

Causal Inference with Groupwise Matching

Abstract

This paper examines methods of causal inference based on groupwise matching when we observe multiple large groups of individuals over several periods. We formulate causal inference validity through a generalized matching condition, generalizing the parallel trend assumption in difference-in-differences designs. We show that difference-in-differences, synthetic control, and synthetic difference-in-differences designs are distinguished by the specific matching conditions that they invoke. Through regret analysis, we demonstrate that difference-in-differences and synthetic control with differencing are complementary; the former dominates the latter if and only if the latter's extrapolation error exceeds the former's matching error up to a term vanishing at the parametric rate. The analysis also reveals that synthetic control with differencing is equivalent to difference-in-differences when the parallel trend assumption holds for both the pre-treatment and post-treatment periods. We develop a statistical inference procedure based on synthetic control with differencing and present an empirical application demonstrating its usefulness.

Paper Structure

This paper contains 36 sections, 23 theorems, 204 equations, 4 figures, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Causal Inference with Groupwise Matching
  3. The Set-Up
  4. Generalized Matching
  5. Generalized Matching Conditions under a Linear Factor Model
  6. Causal Inference Methods using Generalized Matching
  7. Matching with Weights Based on Group Sizes
  8. Randomized Controlled Trials
  9. Unconfoundedness Condition with Discrete Covariates
  10. Difference-in-Differences
  11. Difference-in-Differences with Discrete Covariates
  12. Difference-in-Differences with Staggered Adoption
  13. Matching with Weights Based on Pre-Treatment Fit
  14. Synthetic Control
  15. Synthetic Control with Differencing
  16. ...and 21 more sections

Key Result

Proposition 2.1

Suppose that Assumption assump: factor holds and let $\lambda \in \Delta_{|\mathcal{T}_0|-1}$ and $w \in \Delta_{K-1}$. (i) GMC holds at $(\lambda,w)$. (ii) GMC holds at $(\tilde{\lambda},w)$ for all $\tilde{\lambda} \in \Delta_{|\mathcal{T}_0|-1}$. (iii) Then, (iii) $\Rightarrow$ (ii) $\Rightarrow$ (i). If, furthermore, $\mathbf{F}(\lambda)$ is full row rank for some $\lambda \in \Delta_{|\mathc

Figures (4)

  • Figure 1: Computation Time.
  • Figure 2: Length of Post-Treatment Confidence Intervals.
  • Figure 3: Estimated Effects on Arizona's Share of Non-citizen Hispanic.
  • Figure 4: Robustness Checks for ATT.

Theorems & Definitions (26)

  • Example 2.1: Difference-in-Differences with Discrete Covariates
  • Example 2.2: Difference-in-Differences with Staggered Adoption
  • Definition 2.1
  • Proposition 2.1
  • Proposition 3.1
  • Proposition 4.1
  • Theorem 4.1
  • Proposition 4.2
  • Theorem 5.1
  • Theorem 5.2
  • ...and 16 more