A standard assumption for causal inference about the joint effects of time-varying treatment is that one has measured sufficient covariates to ensure that within covariate strata, subjects are exchangeable across observed treatment values, also known as sequential randomization assumption (SRA). SRA is often criticized as it requires one to accurately measure all confounders. Realistically, measured covariates can rarely capture all confounders with certainty. Often covariate measurements are at best proxies of confounders, thus invalidating inferences under SRA. In this paper, we extend the proximal causal inference (PCI) framework of Miao et al. (2018) to the longitudinal setting under a semiparametric marginal structural mean model (MSMM). PCI offers an opportunity to learn about joint causal effects in settings where SRA based on measured time-varying covariates fails, by formally accounting for the covariate measurements as imperfect proxies of underlying confounding mechanisms. We establish nonparametric identification with a pair of time-varying proxies and provide a corresponding characterization of regular and asymptotically linear estimators of the parameter indexing the MSMM, including a rich class of doubly robust estimators, and establish the corresponding semiparametric efficiency bound for the MSMM. Extensive simulation studies and a data application illustrate the finite sample behavior of proposed methods.
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