A robust regression approach to synthetic control with interference

TL;DR

This work tackles the no-interference limitation of synthetic control by introducing a robust two-stage framework that combines latent-factor adjustment with sparse-outlier estimation to identify direct and interference effects. It develops two complementary asymptotic regimes: a fixed-$N$ setting leveraging a long post-intervention period and a large-$N$ setting leveraging cross-sectional strength with robust M-estimation and sparse interference. Theoretical results show selection consistency and asymptotic normality in the fixed-$N$ regime and consistency with asymptotic normality under sparsity in the large-$N$ regime, with conformal and bootstrap methods for inference. Empirically, the method detects meaningful interference in the US embassy relocation case and reveals spatial interference patterns in Beijing’s air-pollution policy, illustrating its practical ability to handle contaminated donor pools and unknown interference structures.

Abstract

Synthetic control methods are widely used for policy evaluation, but most existing approaches rule out interference among units, compromising validity when such effects are present. We develop a framework that accommodates contaminated donor pools and unknown interference patterns through two stages: factor-model adjustment for unobserved confounding, followed by robust regression in which direct and interference effects appear as a sparse outlier component. We study two asymptotic regimes. When the number of units is fixed and at least half are unaffected by interference, high-breakdown robust regression yields consistent identification of valid controls and asymptotically normal inference. When the number of units diverges, we allow for sparse large and dense weak interference, with robust M-estimation remaining valid even when the post-intervention period is short. Unlike existing approaches requiring prespecification of valid controls or parametric modeling of interference, our framework relies only on coarse sparsity information and enables formal inference on both direct and interference effects. We assess the proposed methods through simulations and two empirical applications. An analysis of the US embassy relocation to Jerusalem reveals significant interference effects on conflict outcomes in Jordan, and an analysis of Beijing's air pollution policy uncovers spatial interference patterns consistent with prevailing wind directions.

Figures (12)

• Figure 1: Finite-sample performance of estimators for $\bar{\beta}_1$ under fixed-$N$ setting
• Figure 2: Finite-sample performance of estimators for $\bar{\beta}_1$ under large-$N$ setting
• Figure 3: Estimated average direct and interference effects on weekly conflict numbers.
• Figure 4: Spatial pattern of estimated effects on PM$_{2.5}$ reduction in Beijing.
• Figure E1: Boxplots and MSE of different estimators for $\bar{\beta}_1$ under fixed-$N$ setting with perturbed loadings.

Theorems & Definitions (8)

• proof : Proof of Lemma S.\ref{['lem:ltslms']}
• proof : Proof of Lemma S.\ref{['lemma: factor loading']}
• proof : Proof of Lemma S.\ref{['lem:alpha']}
• proof : Proof of Theorem \ref{['thm:1']}
• proof : Proof of Proposition \ref{['prop:weight']}.
• proof : Proof of Lemma S.\ref{['lemma: maxrate']}
• proof : Proof of Theorem 2(i)
• proof : Proof of Theorem 2 (ii)