We develop IV Fréchet regression (IVFR), an instrumental-variable (IV) method for settings where the outcome is an entire distribution. Framing the problem as an IV regression in 2-Wasserstein space, IVFR extends global Fréchet regression to the case with endogenous covariates. IVFR projects IV-weighted quantile curves onto the space of valid distributions and then recovers the corresponding regression coefficient functions. The projection provably reduces the estimation error in finite samples and guarantees valid fitted distributions. We show that the IVFR estimator converges weakly to a mean-zero Gaussian process and establish the validity of a multiplier bootstrap procedure for uniform inference. In simulations, the projection reduces the integrated mean squared error (IMSE) by up to 63% relative to existing methods. Revisiting the effects of Chinese import competition on the wage distribution within commuting zones, the proposed method produces 9-10% narrower confidence bands than existing methods. Using our novel uniform confidence bands, we find no evidence that import competition reduced wages at the very bottom of the distribution, but only between the 10th and 35th quantile. We also revisit the effect of county food stamp programs on the county's birth weight distribution and find no significant effects.

翻译：我们提出了IV Fréchet回归（IVFR），这是一种适用于结果变量为完整分布情形的工具变量（IV）方法。通过将该问题建模为2-Wasserstein空间中的IV回归，IVFR将全局Fréchet回归扩展到存在内生协变量的情形。IVFR将IV加权分位曲线投影到有效分布空间，继而恢复对应的回归系数函数。该投影可证明地在有限样本中降低估计误差，并保证拟合分布的有效性。我们证明了IVFR估计量弱收敛于零均值高斯过程，并建立了用于统一样本外推断的乘子自助法有效性。模拟结果显示，相较于现有方法，该投影使综合均方误差（IMSE）最多降低63%。在重新审视中国进口竞争对通勤区工资分布影响的研究中，本方法产出的置信带较现有方法窄9-10%。通过我们提出的新型统一样本外置信带，未发现进口竞争压低工资分布最底端水平的证据，仅发现其对10%至35%分位数区间存在影响。我们亦重新评估了县级食品券计划对新生儿体重分布的影响，未发现显著效应。