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The marginal treatment effect(MTE)model is an important tool for analyzing heterogeneous causal effects and an important complement to the local average treatment effect model.The MTE model postulates the existence of an instrumental variable(IV)that is continuously distributed.With the aid of the continuous IV,the MTE model can accommodate the treatment variable endogeneity and the treatment effect heterogeneity simultaneously.In addition,the MTE model has advantages such as rich in economic implications,able to identify the global rather than local average treatment effect(ATE),and so on.Based on the MTE model,identifying conditions for the global ATE are developed.The key condition is that the propensity score has a full support,or more generally,that the support of the propensity score contains the points of zero and one.Under the identifying conditions,the point identification of ATE is proved.Based on the identification result,a novel semiparametric trimmed weighted average estimator(TWAE)for ATE is suggested.Compared with the unweighted or untrimmed weighted average estimator,TWAE has finite variance and thus retains the possibility of convergence at a root-n rate,where n is the number of the sample size.The rate of convergence of TWAE is established in a general case.Moreover,two concrete examples are given to investigate the specific rates of convergence in different situations.Generally speaking,TWAE achieves root-n consistency when the distribution tail of the continuous IV is heavy relative to the distribution tail of the treatment error.A Monte Carlo simulation study demonstrates the theoretical results.Lastly,an empirical application to evaluating the return to compulsory education is provided to illustrate the usefulness of the proposed method.The construction of IV relies on the 1986 Compulsory Education Law(CEL)in China,which is a mandatory government intervention aimed at improving the completion rate of compulsory education.The existing literature usually defines the IV as the exposure intensity of CEL,which use only the variation across cohorts differentially exposed to the CEL due to differences in the timing of implementation across different provinces in China.To construct a continuous IV,the local enforcement strength of CEL is combined with the exposure intensity of CEL.Specifically,the continuous IV is defined as a weighted measure of the intensity of exposure to the CEL,where the weights are based on the geographic distance between the centroids of the home county and the corresponding provincial capital city.By this IV,the wage return to compulsory education is estimated to be0.561 2,which indicates that workers who completed the compulsory education get 56.12% higher wages on average than those who didn't,which is consistent with the existing results.In summary,the findings add theoretical supports to the applicability of the MTE model,and provide a theoretical basis for further investigating large sample properties of the MTE-based estimation of ATE.
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(1)文献中也把非混淆性称为selection on observables,而把处理变量的内生性相应地称为selection on unobservables,其中observables指的就是可观测的X,unobservables则指其他不可观测的因素。
(2)使得IV条件外生性成立的X与非混淆性假定中的X一般不相同,本文为了不引入过多的符号,把两种情形下的协变量不加区别地记为X。
(3)这是条件外生性的“严格”版本。实际上,这个条件独立假定可以放松为条件期望独立假定[17],即E[Yd|Z,X]=E[Yd|X],d=0,1。同理,上述非混淆性假定也可放松为E[Yd|D,X]=E[Yd|X],d=0,1。
(4)Heckman等认为,单调性假设的对象为全部个体而非单个个体,所以更恰当的称呼应该是一致性(uniformity)假设[11]。
(5)如果随机变量的分布对计数测度(counting measure)绝对连续,则称之为离散型随机变量;如果对勒贝格测度绝对连续,则称之为连续型随机变量。为简单起见,本文仅考虑连续型IV下平均处理效应的识别和估计问题,暂不考虑同时使用离散型和连续型IV时的处理效应模型。
(6)未对V的分布施加任何限制,所以g(Z)-V和g(Z)+V等价,此处使用减号仅为方便推导。
Basic Information:
DOI：10.20207/j.cnki.1007-3116.20260128.002
China Classification Code:O212.1
Citation Information:
[1]PAN Zhewen.Estimation of the Average Treatment Effect Based on a Continuous Instrumental Variable[J].Journal of Statistics and Information,2026,41(02):1-15.DOI:10.20207/j.cnki.1007-3116.20260128.002.
Fund Information:
国家自然科学基金面上项目“高维受限因变量模型的半参数估计：统计性质与最优化求解”（72173142）; 广东省基础与应用基础研究基金面上项目“高维归并回归模型的成对差分LASSO估计”（2022A1515010079）