IV regression with distribution-valued outcomes

We develop IV Fr\'echet 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 Fr\'echet 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.