Instrumental variables estimation with many weak instruments using regularized JIVE

We consider instrumental variables regression in models where the number of available instruments may be larger than the sample size and consistent model selection in the first stage may not be possible. Such a situation may arise when there are many weak instruments. With many weak instruments, existing approaches to first-stage regularization can lead to a large bias relative to standard errors. We propose a jackknife instrumental variables estimator (JIVE) with regularization at each jackknife iteration that helps alleviate this bias. We derive the limiting behavior for a ridge-regularized JIV estimator (RJIVE), verifying that the RJIVE is consistent and asymptotically normal under conditions which allow for more instruments than observations and do not require consistent model selection. We provide simulation results that demonstrate the proposed RJIVE performs favorably in terms of size of tests and risk properties relative to other many-weak instrument estimation strategies in high-dimensional settings. We also apply the RJIVE to the Angrist and Krueger (1991) example where it performs favorably relative to other many-instrument robust procedures.