# On the Misspecification of Linear Assumptions in Synthetic Control

## TL;DR

This paper investigates the limitations of the linearity assumption in synthetic control methods, providing bounds on misspecification error and developing new estimators that incorporate additional data to improve causal inference accuracy.

## Contribution

It introduces theoretical bounds on misspecification error and proposes new synthetic control estimators using supplementary data to reduce bias from linearity assumption violations.

## Key findings

- New estimators outperform standard synthetic controls on synthetic data.
- Bounds on misspecification error are small for minor violations.
- Application to California tobacco data suggests true effects outside bounds.

## Abstract

The synthetic control (SC) method is a popular approach for estimating treatment effects from observational panel data. It rests on a crucial assumption that we can write the treated unit as a linear combination of the untreated units. This linearity assumption, however, can be unlikely to hold in practice and, when violated, the resulting SC estimates are incorrect. In this paper we examine two questions: (1) How large can the misspecification error be? (2) How can we limit it? First, we provide theoretical bounds to quantify the misspecification error. The bounds are comforting: small misspecifications induce small errors. With these bounds in hand, we then develop new SC estimators that are specially designed to minimize misspecification error. The estimators are based on additional data about each unit, which is used to produce the SC weights. (For example, if the units are countries then the additional data might be demographic information about each.) We study our estimators on synthetic data; we find they produce more accurate causal estimates than standard synthetic controls. We then re-analyze the California tobacco-program data of the original SC paper, now including additional data from the US census about per-state demographics. Our estimators show that the observations in the pre-treatment period lie within the bounds of misspecification error, and that the observations post-treatment lie outside of those bounds. This is evidence that our SC methods have uncovered a true effect.