Before revision:  I show that the commonly-used linear-in-means  regression identifies a weighted sum of spillover effects with some negative weights, and characterize the estimand that is recovered by a simple difference in means in the presence of spillovers. My results reveal the potential pitfalls of failing to flexibly account for spillover effects in policy evaluation. I then propose nonparametric estimators  that overcome these issues and are consistent and asymptotically normal under a precise relationship between the number of parameters of interest, the total sample size and the treatment assignment mechanism.
After revision:  I show that the commonly-used linear-in-means regression identifies a weighted sum of spillover effects with some negative weights, and characterize the estimand that is recovered by a simple difference in means between treated and controls identifies a combination of direct and spillover effects entering with different signs. I propose nonparametric estimators that overcome these issues and are consistent and asymptotically normal under a precise relationship between the number of parameters of interest, the total sample size and the treatment assignment mechanism.