Under the canonical setting of observational studies for causal inference, we show that the average treatment effect under the overlap weight, the weight that is proportional to the conditional variance of the treatment given the covariates, is bounded between the average treatment effects on the treated and control, under a monotonic relationship between the propensity score and the conditional average treatment effect. We further extend the result to weighted local average treatment effects, under the canonical setting with a binary instrumental variable and a binary treatment. We also extend the results to other weights. Based on the theory, we recommend the ``CP-plot'' of the estimated conditional average treatment effect against the estimated propensity score.

翻译：在因果推断的标准观察性研究设定下，我们证明：在倾向得分与条件平均处理效应满足单调关系时，基于重叠权重的平均处理效应（该权重与给定协变量下处理变量的条件方差成正比）介于处理组平均处理效应与控制组平均处理效应之间。我们进一步将该结果推广至二元工具变量与二元处理变量的标准设定下的加权局部平均处理效应，并扩展至其他权重形式。基于该理论，我们推荐绘制估计的条件平均处理效应相对于估计的倾向得分的“CP图”。