  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 for a single endogenous regressor.
We discuss how to assess the strength of the instruments and how to conduct
weak identification-robust inference under heteroskedasticity. We establish new
results for a jack-knifed version of the Lagrange Multiplier (LM) test
statistic. Furthermore, 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 correct size
and good power properties.
