  Mendelian randomization is an instrumental variable method that utilizes
genetic information to investigate the causal effect of a modifiable exposure
on an outcome. In most cases, the exposure changes over time. Understanding the
time-varying causal effect of the exposure can yield detailed insights into
mechanistic effects and the potential impact of public health interventions.
Recently, a growing number of Mendelian randomization studies have attempted to
explore time-varying causal effects. However, the proposed approaches
oversimplify temporal information and rely on overly restrictive structural
assumptions, limiting their reliability in addressing time-varying causal
problems. This paper considers a novel approach to estimate time-varying
effects through continuous-time modelling by combining functional principal
component analysis and weak-instrument-robust techniques. Our method
effectively utilizes available data without making strong structural
assumptions and can be applied in general settings where the exposure
measurements occur at different timepoints for different individuals. We
demonstrate through simulations that our proposed method performs well in
estimating time-varying effects and provides reliable inference results when
the time-varying effect form is correctly specified. The method could
theoretically be used to estimate arbitrarily complex time-varying effects.
However, there is a trade-off between model complexity and instrument strength.
Estimating complex time-varying effects requires instruments that are
unrealistically strong. We illustrate the application of this method in a case
study examining the time-varying effects of systolic blood pressure on urea
levels.
