Causal inference with a functional outcome

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.