Double and Single Descent in Causal Inference with an Application to High-Dimensional Synthetic Control

TL;DR

This paper explores the double descent phenomenon in high-dimensional causal inference models, demonstrating that more complex models with many parameters can outperform simpler ones, especially with many control units, through a unified theoretical framework.

Contribution

It provides a theoretical perspective on high-dimensional causal inference models, showing how complex models can be viewed as model-averaging estimators, improving performance.

Findings

High-dimensional models can outperform simpler models in causal inference.

Adding control units improves imputation even beyond perfect pre-treatment fit.

Complex models can be interpreted as model-averaging over simpler models.

Abstract

Motivated by a recent literature on the double-descent phenomenon in machine learning, we consider highly over-parameterized models in causal inference, including synthetic control with many control units. In such models, there may be so many free parameters that the model fits the training data perfectly. We first investigate high-dimensional linear regression for imputing wage data and estimating average treatment effects, where we find that models with many more covariates than sample size can outperform simple ones. We then document the performance of high-dimensional synthetic control estimators with many control units. We find that adding control units can help improve imputation performance even beyond the point where the pre-treatment fit is perfect. We provide a unified theoretical perspective on the performance of these high-dimensional models. Specifically, we show that more complex models can be interpreted as model-averaging estimators over simpler ones, which we link to an improvement in average performance. This perspective yields concrete insights into the use of synthetic control when control units are many relative to the number of pre-treatment periods.