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This paper presents a weighted optimization framework that unifies thebinary,multi-valued, continuous, as well as mixture of discrete and continuoustreatment, under the unconfounded treatment assignment. With a general lossfunction, the framework includes the average, quantile and asymmetric leastsquares causal effect of treatment as special cases. For this generalframework, we first derive the semiparametric efficiency bound for the causaleffect of treatment, extending the existing bound results to a wider class ofmodels. We then propose a generalized optimization estimation for the causaleffect with weights estimated by solving an expanding set of equations. Undersome sufficient conditions, we establish consistency and asymptotic normalityof the proposed estimator of the causal effect and show that the estimatorattains our semiparametric efficiency bound, thereby extending the existingliterature on efficient estimation of causal effect to a wider class ofapplications. Finally, we discuss etimation of some causal effect functionalssuch as the treatment effect curve and the average outcome. To evaluate thefinite sample performance of the proposed procedure, we conduct a small scalesimulation study and find that the proposed estimation has practical value. Toillustrate the applicability of the procedure, we revisit the literature oncampaign advertise and campaign contributions. Unlike the existing procedureswhich produce mixed results, we find no evidence of campaign advertise oncampaign contribution.

A Unified Framework for Efficient Estimation of General Treatment Models