Confidence Intervals for Causal Effects with Invalid In- struments using Two-Stage Hard Thresholding

The instrumental variable (IV) method is commonly used to estimate the causal effect of a treatment on an outcome by using IVs that satisfy the assumptions of association with treatment, no direct effect on the outcome and ignorability. A major challenge in IV analysis is to find said IVs, but typically one is unsure of whether all of the putative IVs are in fact valid (i.e. satisfy the assumptions). We propose a general inference procedure that provides honest inference in the presence of invalid IVs, even after controlling for a large number of covariates. The key step of our method is a novel selection procedure, which we call Two-Stage Hard Thresholding (TSHT), where we use hard thresholding to select the set of non-redundant instruments in the first stage and subsequently use hard thresholding to select the valid instruments in the second stage using the thresholding from the first stage. TSHT allows us to not only select invalid IVs, but also provides honest confidence intervals of the treatment effect at √ n rate. We establish asymptotic properties of our procedure and demonstrate that our procedure performs well in simulation studies compared to traditional IV methods, especially when the instruments are invalid.