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 develop a new approach for estimating average treatment effects in the observational studies with unobserved group-level heterogeneity. A common approach in such settings is to use linear fixed effect specifications estimated by least squares regression. Such methods severely limit the extent of the heterogeneity between groups by making the restrictive assumption that linearly adjusting for differences between groups in average covariate values addresses all concerns with cross-group comparisons. We start by making two observations. First we note that the fixed effect method in effect adjusts only for differences between groups by adjusting for the average of covariate values and average treatment. Second, we note that weighting by the inverse of the propensity score would remove biases for comparisons between treated and control units under the fixed effect set up. We then develop three generalizations of the fixed effect approach based on these two observations. First, we suggest more general, nonlinear, adjustments for the average covariate values. Second, we suggest robustifying the estimators by using propensity score weighting. Third, we motivate and develop implementations for adjustments that also adjust for group characteristics beyond the average covariate values .
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cluster-level heterogeneity. The previous approach relied heavily on
group-level heterogeneity. A common approach in such settings is to use
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that
estimated by least squares regression. Such methods
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heterogeneity between clusters. These methods imply
extent of the heterogeneity between groups by making the restrictive assumption
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clusters
groups
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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
cross-group comparisons. We start by making two observations. First we note that the fixed effect method in effect adjusts only for differences between groups by adjusting for the average of covariate values and average treatment. Second, we note that weighting by the inverse of the propensity score would remove biases for comparisons between treated and control units under the fixed effect set up. We then develop three generalizations of the fixed effect approach based on these two observations. First, we suggest more general, nonlinear, adjustments for the
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Then we use modern causal inference methods to construct flexible and robust estimators
Second, we suggest robustifying the estimators by using propensity score weighting. Third, we motivate and develop implementations for adjustments that also adjust for group characteristics beyond the average covariate values