Identification and Robust Inference for Multiple Treatment Effects with Possibly Invalid Instruments

Abstract

The instrumental variable (IV) method is widely used to infer causal effects in observational studies with unmeasured confounding, but invalid instruments can compromise both population identification and finite-sample inference. This paper studies linear IV models with multiple endogenous treatments and possibly invalid instruments. Identification is more delicate than in the single-treatment setting because a single instrument no longer identifies a scalar candidate effect; instead, each relevant instrument defines a hyperplane in the multidimensional effect space. For identification of multiple treatment effects, we introduce generalized plurality and majority rules which require a sufficiently large number of IVs to be valid. For inference, data-dependent instrument selection may fail to separate certain invalid IVs from valid ones, leading to undercoverage of confidence intervals when these invalid instruments are mistakenly selected as valid. We propose a sampling confidence interval for each treatment effect, which is robust to IV selection errors. We establish asymptotic coverage and parametric-rate length of our sampling confidence interval under regularity conditions and illustrate this method in a Mendelian randomization application.