The Role of the Propensity Score in Fixed Effect Models
We develop a new approach for estimating average treatment effects in the observational studies with unobserved cluster-level heterogeneity. The previous approach relied heavily on linear fixed effect specifications that severely limit the heterogeneity between clusters. These methods imply that linearly adjusting for differences between clusters in average covariate values addresses all concerns with cross-cluster comparisons. Instead, we consider an exponential family structure on the within-cluster distribution of covariates and treatments that implies that a low-dimensional sufficient statistic can summarize the empirical distribution, where this sufficient statistic may include functions of the data beyond average covariate values. Then we use modern causal inference methods to construct flexible and robust estimators.
We are grateful for comments by participants in the Harvard-MIT econometrics seminar, the SIEPR lunch at Stanford, the International Association of Applied Econometrics meeting in Montreal, Pat Kline, and Matias Cattaneo. The views expressed herein are those of the authors and do not necessarily reflect the views of the National Bureau of Economic Research.
I have consulted for Microsoft Corporation, Facebook, Amazon, and Lilly Corporation.