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

A major challenge in instrumental variables (IV) analysis is to find instruments that are valid, or have no direct effect on the outcome and are ignorable. Typically one is unsure whether all of the putative IVs are in fact valid. We propose a general inference procedure in the presence of invalid IVs, called Two-Stage Hard Thresholding (TSHT) with voting. TSHT uses two hard thresholding steps to select strong instruments and generate candidate sets of valid IVs. Voting takes the candidate sets and uses majority and plurality rules to determine the true set of valid IVs. In low dimensions with invalid instruments, our proposal (i) correctly selects valid IVs, (ii) consistently estimates the causal effect, (iii) produces valid confidence intervals for the causal effect, and (iv) has oracle-optimal width, even if the so-called 50% rule or the majority rule is violated. In high dimensions, we establish nearly identical results without oracle-optimality. In simulations, our proposal outperforms traditional and recent methods in the invalid IV literature. We also apply our method to re-analyze the causal effect of education on earnings.