Non-parametric efficient estimation of marginal structural models with continuous time-varying treatments

Summary Marginal structural models are a popular method for estimating causal effects in the presence of time-varying exposures. In spite of their popularity, no scalable nonparametric estimator exists for marginal structural models with multi-valued or continuous time-varying treatments. In this paper, we combine flexible, data-adaptive regression methods, including ensemble learning techniques, with recent developments in semiparametric efficiency theory for longitudinal studies to propose such an estimator. The proposed estimator is based on a study of the nonparametric identifying functional, including first-order von Mises expansions, as well as the efficient influence function and the efficiency bound. We show conditions under which the proposed estimators are efficient, asymptotically normal and sequentially doubly robust. We perform a simulation study to illustrate the properties of the estimators, and present the results of our motivating study on a COVID-19 dataset, studying the impact of mobility on the cumulative number of observed cases.