This paper presents methods to study the causal effect of a binary treatment on a functional outcome with observational data. We define a functional causal parameter, the Functional Average Treatment Effect (FATE), and propose a semi-parametric outcome regression estimator. Quantifying the uncertainty in the estimation presents a challenge since existing inferential techniques developed for univariate outcomes cannot satisfactorily address the multiple comparison problem induced by the functional nature of the causal parameter. We show how to obtain valid inference on the FATE using simultaneous confidence bands, which cover the FATE with a given probability over the entire domain. Simulation experiments illustrate the empirical coverage of the simultaneous confidence bands in finite samples. Finally, we use the methods to infer the effect of early adult location on subsequent income development for one Swedish birth cohort.

翻译：本文提出了利用观测数据研究二元处理对功能性结果的因果效应的方法。我们定义了一个功能性因果参数——功能性平均处理效应（FATE），并提出了一种半参数结果回归估计量。由于现有为单变量结果开发的推断技术无法令人满意地解决由因果参数的功能性所引发的多重比较问题，因此量化估计中的不确定性构成了一项挑战。我们展示了如何利用同时置信带对FATE进行有效推断，这些置信带在整个定义域内以给定概率覆盖FATE。模拟实验说明了在有限样本中同时置信带的经验覆盖情况。最后，我们使用这些方法推断瑞典某出生队列个体早期成年居住地对后续收入发展的影响。