  Robins (1998) introduced marginal structural models (MSMs), a general class
of counterfactual models for the joint effects of time-varying treatment
regimes in complex longitudinal studies subject to time-varying confounding. He
established identification of MSM parameters under a sequential randomization
assumption (SRA), which essentially rules out unmeasured confounding of
treatment assignment over time. In this technical report, we consider
sufficient conditions for identification of MSM parameters with the aid of a
time-varying instrumental variable, when sequential randomization fails to hold
due to unmeasured confounding. Our identification conditions essentially
require that no unobserved confounder predicts compliance type for the
time-varying treatment, the longitudinal generalization of the identifying
condition of Wang and Tchetgen Tchetgen (2018). Under this assumption, We
derive a large class of semiparametric estimators that extends standard
inverse-probability weighting (IPW), the most popular approach for estimating
MSMs under SRA, by incorporating the time-varying IV through a modified set of
weights. The set of influence functions for MSM parameters is derived under a
semiparametric model with sole restriction on observed data distribution given
by the MSM, and is shown to provide a rich class of multiply robust estimators,
including a local semiparametric efficient estimator.
