A Finite Sample Theorem for Longitudinal Causal Inference with Machine Learning: Long Term, Dynamic, and Mediated Effects

I construct and justify confidence intervals for longitudinal causal parameters estimated with machine learning. Longitudinal parameters include long term, dynamic, and mediated effects. I provide a nonasymptotic theorem for any longitudinal causal parameter in a general class, estimated with any machine learning algorithm that satisfies a few simple conditions. The main result encompasses local parameters defined for specific demographics as well as proximal parameters defined in the presence of unobserved confounding. I prove consistency, Gaussian approximation, and semiparametric efficiency. The rate of Gaussian approximation is $n^{-1/2}$ for global parameters, and it degrades gracefully for local parameters. I articulate a simple set of conditions to translate mean square rates into statistical inference, and verify that they hold for adversarial estimators over generic function spaces. A key feature of the main result is a new multiple robustness to ill posedness for proximal causal inference in longitudinal settings. Of independent interest, I provide what appears to be the first mean square rate for nested nonparametric instrumental variable regression.