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 significant 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. This paper illustrates that the existing confidence intervals may substantially undercover due to the post-selection problem. To address this, we construct uniformly valid confidence intervals for the causal effect, robust to the mistakes in separating valid and invalid instruments. Our proposal is to search for a range of effect values that lead to sufficiently many valid instruments. We further devise a novel sampling method, which, together with searching, leads to a more precise confidence interval. Our proposed searching and sampling confidence intervals are uniformly valid and achieve the parametric length 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 education on earnings.