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I construct and justify confidence intervals for longitudinal causalparameters estimated with machine learning. Longitudinal parameters includelong term, dynamic, and mediated effects. I provide a nonasymptotic theorem forany longitudinal causal parameter in a general class, estimated with anymachine learning algorithm that satisfies a few simple conditions. The mainresult encompasses local parameters defined for specific demographics as wellas proximal parameters defined in the presence of unobserved confounding. Iprove consistency, Gaussian approximation, and semiparametric efficiency. Therate of Gaussian approximation is $n^{-1/2}$ for global parameters, and itdegrades gracefully for local parameters. I articulate a simple set ofconditions to translate mean square rates into statistical inference, andverify that they hold for adversarial estimators over generic function spaces.A key feature of the main result is a new multiple robustness to ill posednessfor proximal causal inference in longitudinal settings. Of independentinterest, I provide what appears to be the first mean square rate for nestednonparametric instrumental variable regression.

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