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Primary Submission Category: Synthetic Control Method

On the Asymptotics of Synthetic Control Methods

Authors: Claudia Shi, Achille Nazaret, David Blei,

Presenting Author: Claudia Shi*

Synthetic control is a method for estimating the causal effects of large-scale interventions, such as the statewide effects of policy changes.

The idea of SC is to approximate the treated unit as a weighted combination of the control units.

The SC estimators use the pre-intervention outcomes to learn the weights and use those weights to approximate the counterfactual outcomes of the treated unit.

The existing asymptotic framework suggests that as the number of time points goes to infinity, we can get an unbiased estimate of the SC.

However, this asymptotic framework may be unrealistic in practice, because we often only have data from a few time points.

In this paper, we build on the fine-grained model in Shi et al. [Shi+22] and introduce a novel asymptotic framework for synthetic control.

In the fine-grained model, the units of analysis are “small units” (e.g. individuals in states), rather than “large units” (e.g. states).

With the formulation, we show that the variance in the SC estimate could be explained by each large unit containing a finite number of small units (i.e., there are a finite number of individuals in each state).

We derive a variance quantification method and a variance-minimizing estimator. The significance of these methods is that they reduce the variance in the SC estimate by leveraging external data. We first study the properties of these methods using synthetic data, we then apply them to a real-world case study.