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

Instrumental variable method is among the most commonly used causal inference approaches for analyzing observational studies with unmeasured confounders. Despite its popularity, the instruments' invalidity is a major concern for practical applications and a fast-growing area of research is inference for the causal effect with possibly invalid instruments. In this paper, we construct uniformly valid confidence intervals for the causal effect when the instruments are possibly invalid. We illustrate the post-selection problem of existing inference methods relying on instrument selection. Our proposal is to search for the value of the treatment effect such that a sufficient amount of candidate instruments are 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.