Mendelian randomization (MR) addresses causal questions using genetic variants as instrumental variables. We propose a new MR method, G-Estimation under No Interaction with Unmeasured Selection (GENIUS)-MAny Weak Invalid IV, which simultaneously addresses the 2 salient challenges in MR: many weak instruments and widespread horizontal pleiotropy. Similar to MR-GENIUS, we use heteroscedasticity of the exposure to identify the treatment effect. We derive influence functions of the treatment effect, and then we construct a continuous updating estimator and establish its asymptotic properties under a many weak invalid instruments asymptotic regime by developing novel semiparametric theory. We also provide a measure of weak identification, an overidentification test, and a graphical diagnostic tool.
