Linear Estimation of Global Average Treatment Effects

We study the problem of estimating the average causal effect of treating every member of a population, as opposed to none, using an experiment that treats only some. We consider settings where spillovers have global support and decay slowly with (a generalized notion of) distance. We derive the minimax rate over both estimators and designs, and show that it increases with the spatial rate of spillover decay. Estimators based on OLS regressions like those used to analyze recent large-scale experiments are consistent (though only after de-weighting), achieve the minimax rate when the DGP is linear, and converge faster than IPW-based alternatives when treatment clusters are small, providing one justification for OLS's ubiquity. When the DGP is nonlinear they remain consistent but converge slowly. We further address inference and bandwidth selection. Applied to the cash transfer experiment studied by Egger et al (2022) these methods yield a 20% larger estimated effect on consumption.