  Mendelian randomization (MR) is a widely-used method to estimate the causal
relationship between a risk factor and disease. A fundamental part of any MR
analysis is to choose appropriate genetic variants as instrumental variables.
Genome-wide association studies often reveal that hundreds of genetic variants
may be robustly associated with a risk factor, but in some situations
investigators may have greater confidence in the instrument validity of only a
smaller subset of variants. Nevertheless, the use of additional instruments may
be optimal from the perspective of mean squared error even if they are slightly
invalid; a small bias in estimation may be a price worth paying for a larger
reduction in variance. For this purpose, we consider a method for "focused"
instrument selection whereby genetic variants are selected to minimise the
estimated asymptotic mean squared error of causal effect estimates. In a
setting of many weak and locally invalid instruments, we propose a novel
strategy to construct confidence intervals for post-selection focused
estimators that guards against the worst case loss in asymptotic coverage. In
empirical applications to: (i) validate lipid drug targets; and (ii)
investigate vitamin D effects on a wide range of outcomes, our findings suggest
that the optimal selection of instruments does not involve only a small number
of biologically-justified instruments, but also many potentially invalid
instruments.
