On the Estimation of Heterogeneous Functional Treatment Effects

This work addresses the estimation of the Functional Conditional Average Treatment Effect (F-CATE ) – that is, the causal effect of a treatment on a functional outcome, conditional on covariates – in the framework of Functional Data Analysis (FDA). The objective is to uncover how treatment effects evolve over time and vary across levels of observed characteristics, a central challenge in modern causal inference with functional responses. To tackle this problem, we propose a novel class of functional meta-learners, specifically adapting R- and DR-learners to the functional setting, and emphasizing doubly robust (DR) approaches that combine outcome regression and propensity modeling to obtain unbiased estimators, even under partial mis-specification. Our methodology is grounded in a rigorous theoretical analysis, providing convergence guarantees and mathematical justification for the robustness properties of the estimators. We conduct extensive simulation studies to assess performance under various nuisance function mis- specification scenarios, illustrating the double robustness of the DR-learner: accurate recovery of F-CATE is achieved as long as either the propensity model or the outcome model is correctly specified. Finally, we apply our estimators to real-world data from the Survey of Health, Ageing and Retirement in Europe (SHARE), evaluating the causal effect of chronic medical conditions on the quality-of-life trajectories of older adults. The analysis controls for a rich set of covariates and provides visualizations of treatment effects across different subpopulations. Results demonstrate the practical value of functional meta-learners in capturing treatment heterogeneity over time.