Doubly-Robust Functional Average Treatment Effect Estimation
This paper addresses causal inference for functional outcomes—such as time-series or spatial curves—in observational studies, proposing a robust estimation framework for the Functional Average Treatment Effect (FATE). We introduce DR-FoS, the first doubly robust estimator for functional outcomes: it achieves consistency if either the outcome regression model or the propensity score model is correctly specified. Leveraging functional data analysis, semiparametric causal inference, and the functional central limit theorem, we establish its asymptotic convergence to a Gaussian process, enabling construction of simultaneous confidence bands over the entire domain. Simulation studies demonstrate that DR-FoS substantially outperforms existing methods in finite samples. Applied to the Survey of Health, Ageing and Retirement in Europe (SHARE), it detects statistically significant dynamic causal effects on functional health trajectories. The proposed framework provides both rigorous theoretical guarantees and practical efficacy for functional causal inference.