Many Experiments, Few Repetitions, Unpaired Data, and Sparse Effects: Is Causal Inference Possible?
This work addresses a critical challenge in causal inference for scenarios with sparse, unpaired data, offering a novel solution that could benefit fields like epidemiology or economics where such data structures are common.
The paper tackles the problem of estimating causal effects under hidden confounding with unpaired data, where covariates and outcomes are observed separately across many environments but with few observations per environment. The authors propose a GMM-type estimator with cross-fold sample splitting and prove its consistency as environments grow, achieving a consistent estimator where standard methods fail.
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.