  This paper presents methods to study the causal effect of a binary treatment
on a functional outcome with observational data. We define a Functional Average
Treatment Effect and develop an outcome regression estimator. 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 how the simultaneous confidence bands take the multiple
comparison problem into account. Finally, we use the methods to infer the
effect of early adult location on subsequent income development for one Swedish
birth cohort.
