Semiparametric Doubly Robust Estimation For Continuous Treatment With Spillover Effects

Many experimental and observational studies are affected by interference units.Interference is said to be present when a treatment,exposure or intervention,on one unit has an effect on the response of another unit.As previous studies have mostly focused on the case of binary treatment,the spillover effects of continuous treatment is rarely considered and the treatment is mostly based on an extension of the propensity score method.Therefore,this paper proposes semiparametric doubly robust estimation of continuous treatment with spillover effects.Under the framework of potential outcomes,this paper analyze spillover effects and establish the Stable Unit Treatment on Neighborhood Value Assumption which based on the structure of the social network.A semiparametric doubly robust estimator of spillover effects is constructed by regressing the pseudo-outcome(?)of Single-index model.At first,the efficient influence function is constructed based on the doubly robust mapping,and the pseudo-outcome is found in the efficient influence function,which uses Single-index model to estimate the parameters.Secondly,aiming at the problem of dual endogenous variables of individual treatment and neighbors’ treatment caused by spillover effects,the regression of pseudo-outcome is constructed on individual treatment Ti and neighbors’ treatment Si,which uses one-index model to estimate the link function φ(q).Finally,the consistency theorem(theorem 3.2)and normality theorem(theorem 3.3)for the semiparametric doubly robust estimator of spillover effects are given.At the same time,the proof of the corresponding theorems is given.It is found that the proposed estimator has good properties,i.e.,after finding a doubly robust mapping,the conditional expectation of the mapping is equal to the dose-response curve function of interest in the presence of both individual and neighbors’ treatments,as long as one of the two nuisance functions is correctly specified.At the same time,the single-index model is used to introduce the doubly robust construction into the spillover effects,which can reflect the interaction between individual treatment and neighbors’ treatment.The semiparametric doubly robust estimation of continuous causal spillover effects proposed in this paper is a useful addition to previous studies that investigate both the case of continuous treatment,and consider the double robustness of the model.The method proposed in this paper can be naturally combined with general machine learning methods for estimation.