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.