  Mendelian randomization (MR) has become a popular approach to study causal
effects by using genetic variants as instrumental variables. We propose a new
MR method, GENIUS-MAWII, which simultaneously addresses the two salient
phenomena that adversely affect MR analyses: many weak instruments and
widespread horizontal pleiotropy. Similar to MR GENIUS (Tchetgen Tchetgen et
al., 2021), we achieve identification of the treatment effect by leveraging
heteroscedasticity of the exposure. We then derive the class of influence
functions of the treatment effect, based on which, we construct a continuous
updating estimator and establish its consistency and asymptotic normality 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. We demonstrate in
simulations that GENIUS-MAWII has clear advantages in the presence of
directional or correlated horizontal pleiotropy compared to other methods. We
apply our method to study the effect of body mass index on systolic blood
pressure using UK Biobank.
