Identifying and disentangling direct and spillover effects using multiple pre-treatment periods

Under interference, the treatment of some units affects the outcomes of others, so that the effect on a treated unit consists of a direct and a spillover component. A non-treated comparison group then generally identifies neither component, but the difference between the total effect on the treated and the spillover effect on the comparison units. We show that both components are identified separately when multiple pre-treatment periods are available, without an uncontaminated comparison group. Under no anticipation, pre-treatment outcomes are free of both components for every unit. Extrapolating the pre-treatment trend within a group, as in interrupted time series approaches, therefore identifies the counterfactual outcome in the absence of any treatment. Applied to the treated it gives the total effect on them, applied to the non-treated at a given level of exposure the spillover effect on them. Disentangling the two components in addition requires effect homogeneity across treated and non-treated units with the same exposure and covariates, as well as common support in exposure. We derive doubly robust and Neyman-orthogonal expressions for the effects of interest, which permits controlling for covariates by machine learning, and establish asymptotic normality. A simulation study confirms the root-$n$ behaviour of the estimators and the reliability of the standard errors. We apply the framework to the compulsory licensing of enemy-owned patents in the United States in 1919. Licensing raised patenting in the licensed subclasses and in technologically close ones alike, so that a comparison-group design understates the total effect.