Causal Inference with Invalid Instruments: Post-selection Problems and A Solution Using Searching and Sampling

Instrumental variable methods are among the most commonly used causal inference approaches to account for unmeasured confounders in observational studies. The presence of invalid instruments is a major concern for practical applications and a fast-growing area of research is inference for the causal effect with possibly invalid instruments. The existing inference methods rely on correctly separating valid and invalid instruments in a data dependent way. In this paper, we illustrate post-selection problems of these existing methods. We construct uniformly valid confidence intervals for the causal effect, which are robust to the mistakes in separating valid and invalid instruments. Our proposal is to search for the causal effect such that a sufficient amount of candidate instruments can be taken as valid. We further devise a novel sampling method, which, together with searching, lead to a more precise confidence interval. Our proposed searching and sampling confidence intervals are shown to be uniformly valid under the finite-sample majority and plurality rules. We compare our proposed methods with existing inference methods over a large set of simulation studies and apply them to study the effect of the triglyceride level on the glucose level over a mouse data set.