Rectified Linear Unit Regression

Abstract

This paper develops a regression framework for analyzing integrated conditional distribution and quantile functions. The proposed method, termed rectified linear unit (ReLU) regression, projects the ReLU-transformed outcome onto covariates and admits a closed-form estimator. Its population regression function is the best linear approximation to the integrated conditional distribution function of the outcome, and the corresponding convex conjugate, obtained via the Legendre-Fenchel transform, approximates the integrated conditional quantile function. Both the regression and its conjugate require only mild distributional assumptions and accommodate non-continuous outcomes. We establish the asymptotic distribution of the estimator and develop inference for the conjugate functional via the delta method for Hadamard directionally differentiable maps. Building on these results, we establish identification and inference for average quantile treatment effects over arbitrary subintervals of probability levels. This broadens the set of distributional parameters available to empirical work.