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A major challenge in instrumental variables (IV) analysis is to findinstruments that are valid, or have no direct effect on the outcome and areignorable. Typically one is unsure whether all of the putative IVs are in factvalid. We propose a general inference procedure in the presence of invalid IVs,called Two-Stage Hard Thresholding (TSHT) with voting. TSHT uses two hardthresholding steps to select strong instruments and generate candidate sets ofvalid IVs. Voting takes the candidate sets and uses majority and pluralityrules to determine the true set of valid IVs. In low dimensions, if thesufficient and necessary identification condition under invalid instruments ismet, which is more general than the so-called 50% rule or the majority rule,our proposal (i) correctly selects valid IVs, (ii) consistently estimates thecausal effect, (iii) produces valid confidence intervals for the causal effect,and (iv) has oracle-optimal width. In high dimensions, we establish nearlyidentical results without oracle-optimality. In simulations, our proposaloutperforms traditional and recent methods in the invalid IV literature. Wealso apply our method to re-analyze the causal effect of education on earnings.

Confidence Intervals for Causal Effects with Invalid Instruments using Two-Stage Hard Thresholding with Voting