Estimating a Continuous Treatment Model with Spillovers: A Control Function Approach

In this paper, we consider the estimation of a continuous treatment effect model in the presence of treatment spillovers through social networks. We assume that one's outcome is affected not only by his/her own treatment but also by the average of his/her neighbors' treatments, both of which are treated as endogenous variables. Using a control function approach with appropriate instrumental variables, in conjunction with some functional form restrictions, we show that the conditional mean potential outcome can be nonparametrically identified. We also consider a more empirically tractable semiparametric model and develop a three-step estimation procedure for this model. The consistency and asymptotic normality of the proposed estimator are established under certain regularity conditions. As an empirical illustration, we investigate the causal effect of the regional unemployment rate on the crime rate using Japanese city data.