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Title: "Design of Partial Population Experiments with an Application to Spillovers in Tax Compliance"

Abstract: "We develop a framework to design and analyze partial population experiments, a generalization of the cluster experimental design where clusters are assigned to different treatment intensities. Our framework allows for heterogeneity in outcome distributions across clusters and heterogeneous cluster sizes, which are pervasive in empirical settings but typically ignored in experimental design. We show that failing to account for this heterogeneity when designing an experiment can severely overestimate power and underestimate minimum detectable effects. We study the large-sample behavior of OLS estimators and their corresponding cluster-robust variance estimators with heterogeneous clusters, and use our results to derive simple formulas to calculate power, minimum detectable effects and optimal cluster assignment probabilities. We then set up a potential outcomes framework that justifies interpreting OLS estimands as causal effects. All our results apply to cluster experiments, which are a particular case of our framework. We apply our methods to design a large-scale experiment to estimate the spillover effects of a communication campaign on property tax compliance. We find an increase in tax compliance among individuals directly targeted with our mailing, as well as compliance spillovers on untreated individuals in street blocks with a high proportion of treated taxpayers."