We study instrumental-variable designs where policy reforms strongly shift the distribution of an endogenous variable but only weakly move its mean. We formalize this by introducing distributional relevance: instruments may be purely distributional. Within a triangular model, distributional relevance suffices for nonparametric identification of average structural effects via a control function. We then propose Quantile Least Squares (Q-LS), which aggregates conditional quantiles of X given Z into an optimal mean-square predictor and uses this projection as an instrument in a linear IV estimator. We establish consistency, asymptotic normality, and the validity of standard 2SLS variance formulas, and we discuss regularization across quantiles. Monte Carlo designs show that Q-LS delivers well-centered estimates and near-correct size when mean-based 2SLS suffers from weak instruments. In Health and Retirement Study data, Q-LS exploits Medicare Part D-induced distributional shifts in out-of-pocket risk to sharpen estimates of its effects on depression.

翻译：本文研究工具变量设计，其中政策改革显著改变内生变量的分布，但对其均值影响微弱。我们通过引入分布相关性对此进行形式化：工具变量可以是纯粹分布性的。在三角模型框架内，分布相关性足以通过控制函数实现平均结构效应的非参数识别。随后我们提出分位数最小二乘法（Q-LS），该方法将给定Z时X的条件分位数聚合为最优均方预测器，并将该投影作为线性工具变量估计中的工具变量使用。我们证明了该方法的相合性、渐近正态性以及标准两阶段最小二乘方差公式的有效性，并讨论了跨分位数的正则化问题。蒙特卡洛模拟表明，当基于均值的两阶段最小二乘法面临弱工具变量问题时，Q-LS能提供中心化良好的估计量和接近正确的检验水平。在健康与退休研究数据中，Q-LS利用医疗保险D部分引发的自付风险分布变化，从而更精确地估计了其对抑郁症状的影响效应。