On the Properties of the Synthetic Control Estimator with Many Periods and Many Controls

TL;DR

This paper analyzes the asymptotic behavior of the Synthetic Control estimator with many periods and controls, showing conditions for unbiasedness and convergence under a linear factor model.

Contribution

It establishes conditions under which the SC estimator's factor loadings converge and remains unbiased with many controls and periods, even with correlated treatment assignment.

Findings

SC factor loadings converge to treated unit loadings

SC estimator remains asymptotically unbiased

Valid even with more controls than pre-treatment periods

Abstract

We consider the asymptotic properties of the Synthetic Control (SC) estimator when both the number of pre-treatment periods and control units are large. If potential outcomes follow a linear factor model, we provide conditions under which the factor loadings of the SC unit converge in probability to the factor loadings of the treated unit. This happens when there are weights diluted among an increasing number of control units such that a weighted average of the factor loadings of the control units asymptotically reconstructs the factor loadings of the treated unit. In this case, the SC estimator is asymptotically unbiased even when treatment assignment is correlated with time-varying unobservables. This result can be valid even when the number of control units is larger than the number of pre-treatment periods.