Linear estimation of average global 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. This is the policy-relevant estimand when deciding whether to scale up an intervention based on the results of an RCT, for example, but differs from the usual average treatment effect in the presence of spillovers. We consider both estimation and experimental design given a bound (parametrized by $\eta > 0$) on the rate at which spillovers decay with the ``distance'' between units, defined in a generalized way to encompass spatial and quasi-spatial settings, e.g. where the economically relevant concept of distance is a gravity equation. Over all estimators linear in the outcomes and all cluster-randomized designs the optimal geometric rate of convergence is $n^{-\frac{1}{2+\frac{1}{\eta}}}$, and this rate can be achieved using a generalized ``Scaling Clusters'' design that we provide. We then introduce the additional assumption, implicit in the OLS estimators used in recent applied studies, that potential outcomes are linear in population treatment assignments. These estimators are inconsistent for our estimand, but a refined OLS estimator is consistent and rate optimal, and performs better than IPW estimators when clusters must be small. Its finite-sample performance can be improved by incorporating prior information about the structure of spillovers. As a robust alternative to the linear approach we also provide a method to select estimator-design pairs that minimize a notion of worst-case risk when the data generating process is unknown. Finally, we provide asymptotically valid inference methods.