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 the jackknife-estimated first stage can vary widely across observations. When such variability exists, we show that the precision of JIVE second-stage estimates is meaningfully improved by shrinking judge propensities towards their conditional means, where the shrinkage factor depends on the precision of the first-stage fitted value. Doing so requires no further assumptions and identifies the same local average treatment effect as the usual (unshrunken) JIVE estimator. We illustrate the precision gains from using a Shrunken JIVE estimator (SJIVE) in an application from the literature studying pre-trial detention of defendants in criminal cases.