The Incredible Shrinking Instruments: Using Empirical Bayes to Increase Efficiency in IV Designs with Many Instruments

In instrumental variables (IV) estimation with many instruments, such as judge fixed effects designs, the precision of jackknife-estimated first-stage fitted values can vary widely across observations. We propose a Shrunken JIVE estimator (SJIVE) that shrinks leave-one-out first-stage predictions toward overall or conditional means, with the degree of shrinkage depending on first-stage precision. We derive the variance-minimizing shrinkage rule and show that SJIVE has weakly lower asymptotic variance than JIVE. Under exclusion and monotonicity conditions, SJIVE identifies a proper weighted average of individual treatment effects. We illustrate the method in an application studying the effect of patent approval on the follow-on innovation of startup firms, where SJIVE produces substantially smaller standard errors than UJIVE.