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Estimation and Inference for Synthetic Control Methods with Spillover Effects

Jianfei Cao, Connor Dowd

TL;DR

This paper extends synthetic control methods to settings with spillover effects by adopting a Rubin-style framework where the full effect vector is linear: $\alpha = A\gamma$. It develops an asymptotically unbiased estimator that leverages per-unit synthetic controls to identify both direct and spillover effects under known spillover structures, and it provides an inference procedure based on Andrews end-of-sample tests that remains valid with spillovers and under factor models that are stationary or cointegrated. The authors introduce a misspecification diagnostic, the $\kappa_A$ statistic, and compare against pure-donor approaches, showing typically smaller misspecification bias for the proposed method. The methodology is applied to California's Proposition 99, uncovering spillovers to neighboring states and demonstrating improved robustness to spillover misspecification. The work advances inference in SCM with spillovers, extends to multiple treated units and post-treatment periods, and offers extensive Monte Carlo validation and practical guidance for empirical researchers.

Abstract

Estimation and inference procedures for synthetic control methods often do not allow for the existence of spillover effects, which are plausible in many applications. In this paper, we consider estimation and inference for synthetic control methods, allowing for spillover effects. We propose estimators for both direct treatment effects and spillover effects and show that they are asymptotically unbiased. In addition, we propose an inferential procedure and show that it is asymptotically unbiased. Our estimation and inference procedure applies to cases with multiple treated units and/or multiple post-treatment periods, and to ones where the underlying factor model is either stationary or cointegrated. We discuss the bias from misspecified spillover structures and propose a test for correct specification. We apply our method to a classic empirical example that investigates the effect of California's tobacco control program as in Abadie et al. (2010) and find evidence of spillovers. We contrast our method with the pure-donor approach through a sensitivity analysis.

Estimation and Inference for Synthetic Control Methods with Spillover Effects

TL;DR

This paper extends synthetic control methods to settings with spillover effects by adopting a Rubin-style framework where the full effect vector is linear: . It develops an asymptotically unbiased estimator that leverages per-unit synthetic controls to identify both direct and spillover effects under known spillover structures, and it provides an inference procedure based on Andrews end-of-sample tests that remains valid with spillovers and under factor models that are stationary or cointegrated. The authors introduce a misspecification diagnostic, the statistic, and compare against pure-donor approaches, showing typically smaller misspecification bias for the proposed method. The methodology is applied to California's Proposition 99, uncovering spillovers to neighboring states and demonstrating improved robustness to spillover misspecification. The work advances inference in SCM with spillovers, extends to multiple treated units and post-treatment periods, and offers extensive Monte Carlo validation and practical guidance for empirical researchers.

Abstract

Estimation and inference procedures for synthetic control methods often do not allow for the existence of spillover effects, which are plausible in many applications. In this paper, we consider estimation and inference for synthetic control methods, allowing for spillover effects. We propose estimators for both direct treatment effects and spillover effects and show that they are asymptotically unbiased. In addition, we propose an inferential procedure and show that it is asymptotically unbiased. Our estimation and inference procedure applies to cases with multiple treated units and/or multiple post-treatment periods, and to ones where the underlying factor model is either stationary or cointegrated. We discuss the bias from misspecified spillover structures and propose a test for correct specification. We apply our method to a classic empirical example that investigates the effect of California's tobacco control program as in Abadie et al. (2010) and find evidence of spillovers. We contrast our method with the pure-donor approach through a sensitivity analysis.

Paper Structure

This paper contains 41 sections, 10 theorems, 91 equations, 8 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Model Specification
  3. A Rubin model with spillover effects
  4. Known spillover structures
  5. An Asymptotically Unbiased Estimator
  6. SCM without spillover effects
  7. The proposed estimator under spillovers
  8. The factor model as an example
  9. Invertibility of $A'MA$
  10. Intuition and a toy example
  11. Results on rank of $I-B$
  12. Statistical Inference
  13. Cases without spillover effects
  14. Cases with spillover effects
  15. Discussion
  16. ...and 26 more sections

Key Result

Theorem 1

Suppose Assumption unbiased assumption holds. Then, $\widehat{\alpha}-(\alpha+Gu_{T+1})\rightarrow_p0$ as $T\rightarrow \infty$, where $G=A(A'MA)^{-1}A'(I-B)'$. Moreover, $E[Gu_{T+1}]=0$.

Figures (8)

  • Figure 2: Evaluation of Treatment and Spillover Effects on Cigarette Sales. Panel (a) illustrates the trends in per-capita cigarette sales in California against two synthetic controls: one derived from the standard synthetic control method (SCM) and the other from our spillover-adjusted method (SP). The intervention onset is marked by the vertical dashed line at the passage of Proposition 99. Panel (b) displays the estimated treatment effect on per-capita cigarette sales, showing the difference between actual sales in California and those predicted by both SCM and SP models. The shaded regions highlight the time periods where our test rejects the null hypothesis of no spillover effect at 5% significant level. Error bars represent 95% confidence intervals for the SP estimates post-treatment. Panel (c) presents the estimated spillover effects on cigarette sales in the neighboring states of Arizona (AZ), Nevada (NV), and Oregon (OR). For visual clarity, 95% confidence intervals are included only for Nevada.
  • Figure 3: Sensitivity Analysis of Spillover Structure Misspecification. This figure illustrates the relationship between the magnitude of missed spillover effects ($\bar{\alpha}$) and the potential bias in estimators. The parameter $p$ denotes the number of units where spillovers have been missed. The biases $\delta_{1,SP}$ and $\delta_{1,PD}$ represent the (asymptotic) biases of the proposed method and the usual synthetic control method using only pure donors, respectively. The bounds of the identified bias sets for both methods are shown. Benchmarks within the figure include the largest and smallest treatment effect estimates, $\alpha_{max}^{CA}$ and $\alpha_{min}^{CA}$, alongside the largest estimated spillover effect, $\alpha_{max}^{NV}$. Notably, the intersection point of $\alpha_{max}^{CA}$ with the upper bound of the $\delta_{1,SP}$ bias set is highlighted. Panels (a) and (b) correspond to misspecification scenarios with $p=1$ and $p=2$ missed spillover units, respectively.
  • Figure 4: $\widehat{b}$ Estimate over Time with 95% Confidence Interval.
  • Figure 5: Illustration of Placebo test with and without Spillover Effects. Area with lines is 95% probability region of the error of the treated unit. Filled area is 95% probability region of null distribution formed in placebo test. A test is rejected when the error of the treated units falls outside of the filled area.
  • Figure 6: Illustration of Andrews' Test with and without Spillover Effects. Area with lines is 95% probability region of the error of the treated unit. Filled area is 95% probability region of null distribution formed in Andrews' test. A test is rejected when the error of the treated units falls outside of the filled area.
  • ...and 3 more figures

Theorems & Definitions (13)

  • Example 1
  • Theorem 1
  • Lemma 1
  • Example 2
  • Proposition 1
  • Theorem 2
  • Lemma 2
  • Theorem 3
  • Lemma 3
  • Proposition 2
  • ...and 3 more