  Most of the existing Mendelian randomization (MR) methods are limited by the
assumption of linear causality between exposure and outcome, and the
development of new non-linear MR methods is highly desirable. We introduce
two-stage prediction estimation and control function estimation from
econometrics to MR and extend them to non-linear causality. We give conditions
for parameter identification and theoretically prove the consistency and
asymptotic normality of the estimates. We compare the two methods theoretically
under both linear and non-linear causality. We also extend the control function
estimation to a more flexible semi-parametric framework without detailed
parametric specifications of causality. Extensive simulations numerically
corroborate our theoretical results. Application to UK Biobank data reveals
non-linear causal relationships between sleep duration and systolic/diastolic
blood pressure.
