Simple Standard Errors for Complicated (Quasi-)Experimental Designs

We develop standard errors for estimates of causal effects in settings where the distribution of treatment exposures is known but potentially exhibits complicated dependence patterns across units. Such dependence arises, for example, in settings with spillovers/interference across units or when the treatment assignment is spatially correlated. Our variance estimator generalizes several properties of the Neyman variance estimator for completely randomized experiments to general designs: it is unbiased for an upper bound on the variance from the design-based perspective; the conservativeness of this upper bound depends on the heterogeneity of treatment effects; and it is also valid under certain forms of super-population sampling of the potential outcomes. Our estimator reduces to standard approaches in simple designs, such as the Neyman variance estimator in completely randomized experiments and the cluster-robust variance estimator in cluster-randomized experiments. We show that the proposed estimator is admissible among the class of quadratic-form variance estimators, and derive asymptotic conditions under which it is consistent for an upper bound on the asymptotic variance of the treatment effect estimator. In simulations and applications based on two empirical studies, one with peer effects and another with a spatially correlated treatment, we find that the proposed variance estimator performs well, and in some cases yields standard errors that are 2-3 times shorter than those of existing alternatives.