Many Experiments, Few Repetitions, Unpaired Data, and Sparse Effects: Is Causal Inference Possible?

TL;DR

This paper investigates the challenge of estimating causal effects with hidden confounding using unpaired data across multiple environments, proposing a GMM estimator that remains consistent with many environments but few observations per environment.

Contribution

It introduces a novel GMM-based estimator for causal inference in unpaired data settings with many environments and few observations, extending to sparse effects with regularization.

Findings

The proposed estimator is consistent as the number of environments increases.

The method effectively estimates sparse causal effects using regularization.

Standard two-sample IV estimators fail under many environments with limited data.

Abstract

We study the problem of estimating causal effects under hidden confounding in the following unpaired data setting: we observe some covariates $X$ and an outcome $Y$ under different experimental conditions (environments) but do not observe them jointly; we either observe $X$ or $Y$. Under appropriate regularity conditions, the problem can be cast as an instrumental variable (IV) regression with the environment acting as a (possibly high-dimensional) instrument. When there are many environments but only a few observations per environment, standard two-sample IV estimators fail to be consistent. We propose a GMM-type estimator based on cross-fold sample splitting of the instrument-covariate sample and prove that it is consistent as the number of environments grows but the sample size per environment remains constant. We further extend the method to sparse causal effects via $\ell_1$-regularized estimation and post-selection refitting.