We introduce the Multiplicative Quasi-Instrumental Variable (MQIV) model, a framework for causal inference with unmeasured confounding that leverages an instrument that may be imperfectly exogenous. We allow the candidate quasi-instrument to have a direct effect on the outcome not mediated by the treatment, thus violating the standard IV exclusion restriction. We establish nonparametric identification of the population average treatment effect on the treated (ATT) under a treatment model that is multiplicative with respect to the quasi-IV and the hidden confounder (Hernan and Robins, 2006). Such a multiplicative treatment model may arise naturally either when treatment occurs only if two independent instrument-driven and confounder-driven causal mechanisms are present; or alternatively, when an instrument's effect on treatment uptake is inherently heterogeneous and scales with a person's latent propensity, best capturing settings in which it is challenging for a given instrument to overcome a person's inherent lack of preference for the treatment in view. Importantly, as we establish, the MQIV model is simultaneously agnostic to treatment-effect heterogeneity with respect to hidden confounders and violation of the core IV exclusion restriction condition. Identification is achieved via a modified Wald ratio estimand, which corrects the bias due to the exclusion restriction violation, and we propose a new class of estimators that are multiply robust and semiparametric efficient. Finally, we evaluate the approach in extensive simulations and an application to evaluate the causal effect of having three or more children on mothers' labor-market engagement.

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