  We propose a weak-instrument-robust subvector Lagrange multiplier test for
instrumental variables regression. We show that it is asymptotically
size-correct under a technical condition. This is the first
weak-instrument-robust subvector test for instrumental variables regression to
recover the degrees of freedom of the commonly used Wald test, which is not
robust to weak instruments. Additionally, we provide a closed-form solution for
subvector confidence sets obtained by inverting the subvector Anderson-Rubin
test. We show that they are centered around a k-class estimator. Also, we show
that the subvector confidence sets for single coefficients of the causal
parameter are jointly bounded if and only if Anderson's likelihood-ratio test
rejects the hypothesis that the first-stage regression parameter is of reduced
rank, that is, that the causal parameter is not identified. Finally, we show
that if a confidence set obtained by inverting the Anderson-Rubin test is
bounded and nonempty, it is equal to a Wald-based confidence set with a
data-dependent confidence level. We explicitly compute this Wald-based
confidence test.
