Linear instrumental variable regressions are widely used to estimate causal effects. Many instruments arise from the use of “technical” instruments and more recently from the empirical strategy of “judge design”. This paper surveys and summarizes ideas from recent literature on estimation and statistical inferences with many instruments. We discuss how to assess the strength of the instruments and how to conduct weak identification-robust inference under heteroscedasticity. We establish new results for a jack-knifed version of the Lagrange Multiplier (LM) test statistic. Many exogenous regressors arise often in practice to ensure the validity of the instruments. We extend the weak-identification-robust tests to settings with both many exogenous regressors and many instruments. We propose a test that properly partials out many exogenous regressors while preserving the re-centering property of the jack-knife. The proposed tests have uniformly correct size and good power properties.

Empirical research typically involves an efficiency-robustness tradeoff. A researcher seeking to estimate a scalar parameter can invoke strong assumptions to motivate a restricted estimator that is precise but may be heavily biased if the assumptions are violated, or they can relax some of these assumptions to motivate a more variable unrestricted estimator that is asymptotically unbiased. When a bound on the bias of the restricted estimator is available, it is optimal to shrink the unrestricted estimator towards the restricted estimator. For settings where a bound is not known, or when that bound may not be sharp, we propose shrinkage estimators that are adaptive: they minimize the percentage increase in worst case risk relative to an oracle that knows the magnitude of the restricted estimator’s bias. We show how to compute the adaptive estimator by solving for a least favorable prior in a weighted convex minimax problem. A simple lookup table is provided for computing the adaptive estimates from the restricted and unrestricted estimates, their standard errors, and their correlation. We revisit five influential empirical papers and study how estimates of economic parameters change when adapting to misspecification.