  Weak instruments present a major setback to empirical work. This paper
introduces an estimator that admits weak, uncorrelated, or mean-independent
instruments that are non-independent of endogenous covariates. Relative to
conventional instrumental variable methods, the proposed estimator weakens the
relevance condition considerably without imposing a stronger exclusion
restriction. Identification mainly rests on (1) a weak conditional median
exclusion restriction imposed on pairwise differences in disturbances and (2)
non-independence between covariates and instruments. Under mild conditions, the
estimator is consistent and asymptotically normal. Monte Carlo experiments
showcase an excellent performance of the estimator, and two empirical examples
illustrate its practical utility.
