Identification and Semiparametric Estimation of Conditional Means from Aggregate Data

We introduce a new method for estimating the mean of an outcome variable within groups when researchers only observe the average of the outcome and group indicators across a set of aggregation units, such as geographical areas. Existing methods for this problem, also known as ecological inference, implicitly make strong assumptions about the aggregation process. We first formalize weaker conditions for identification which hold conditionally on covariates. To efficiently control for many covariates, we propose a debiased machine learning estimator that is based on nuisance functions restricted to a partially linear form. Our estimator admits a semiparametric sensitivity analysis which allows researchers to evaluate the impact of violations of the key identifying assumption. We also propose a nonparametric test for the identifying assumption itself. Finally, we derive asymptotically valid confidence intervals for local, unit-level estimates under additional assumptions. Simulations and validation on real-world data where ground truth is available demonstrate the advantages of our approach over existing methods. Open-source software is available which implements the proposed methods.