# Quantile Regression under Misspecification, with an Application to the U.S. Wage Structure

https://doi.org/10.1111/j.1468-0262.2006.00671.x
p. 539-563

Joshua Angrist, Victor Chernozhukov, Iván Fernández‐Val

Quantile regression (QR) fits a linear model for conditional quantiles just as ordinary least squares (OLS) fits a linear model for conditional means. An attractive feature of OLS is that it gives the minimum mean‐squared error linear approximation to the conditional expectation function even when the linear model is misspecified. Empirical research using quantile regression with discrete covariates suggests that QR may have a similar property, but the exact nature of the linear approximation has remained elusive. In this paper, we show that QR minimizes a weighted mean‐squared error loss function for specification error. The weighting function is an average density of the dependent variable near the true conditional quantile. The weighted least squares interpretation of QR is used to derive an omitted variables bias formula and a partial quantile regression concept, similar to the relationship between partial regression and OLS. We also present asymptotic theory for the QR process under misspecification of the conditional quantile function. The approximation properties of QR are illustrated using wage data from the U.S. census. These results point to major changes in inequality from 1990 to 2000.

## Supplemental Material

### Proofs of Theorems and Corollaries for "Quantile Regression under Misspecification, with an Application to the U.S. Wage Structure"

This supplement to the paper ?Quantile Regression under Misspecification, with an Application to the U.S. Wage Structure? provides added technical details related to the proofs of the theorems and corollaries not included in the main text. In particular, it contains the proofs for Theorem 3 on the uniform consistency and asymptotic Gaussianity of the sample QR process and the proofs for the Corollaries of this Theorem, along with the proofs of uniform consistency for the estimators of the components of the covariance kernel.

### Supplement To ?Quantile Regression under Misspecification, with an Application to the U.S. Wage Structure?: figures 1

This file contains figures 1, 2a and 2b.

### Supplement To ?Quantile Regression under Misspecification, with an Application to the U.S. Wage Structure?: Variable Definitions, Data, And Programs

This supplement provides added technical details related to the data, variable definitions, and estimation. The paper has two empirical components: estimation of quantile regression weighting schemes and robust inference on the quantile regression process for earnings equations. Both rely on Census microdata for 1980, 1990, and 2000. The original raw data are available from the Integrated Public Use Microdata Series (IPUMS) web site and our Stata extracts are available here. In addition to a description of the data and variables, this supplement includes all Stata and R (version 2.0.1) command files used to construct Figures 1 and 2, and Table I.

### Supplement To "Quantile Regression under Misspecification, with an Application to the U.S. Wage Structure?": Data

This zip file contains the replication files for the manuscript.