Large-Sample Properties of the Synthetic Control Method under Selection on Unobservables

TL;DR

This paper investigates the asymptotic properties of the synthetic control method in panel data with many units, showing it can produce normal estimators under certain conditions related to unobserved heterogeneity and pre-treatment data.

Contribution

It provides theoretical analysis of the synthetic control method's large-sample behavior under selection on unobservables, extending its applicability as an alternative to Difference-in-Differences.

Findings

SC method yields asymptotically normal estimators with enough pre-treatment periods.

Input features' ability to approximate unobserved heterogeneity is crucial.

Applicable to a broad class of linear panel data models.

Abstract

We analyze the synthetic control (SC) method in panel data settings with many units. We assume the treatment assignment is based on unobserved heterogeneity and pre-treatment information, allowing for both strictly and sequentially exogenous assignment processes. We show that the critical property that determines the behavior of the SC method is the ability of input features to approximate the unobserved heterogeneity. Our results imply that the SC method delivers asymptotically normal estimators for a large class of linear panel data models as long as the number of pre-treatment periods is sufficiently large, making it a natural alternative to the Difference-in-Differences.