Quantile Regression under Misspecification, with an Application

to the U.S. Wage Structure

Joshua Angrist, Victor Chernozhukov, and Iv´an Fern´andez-Val

December 21, 2004

Abstract

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 itgives the minimum mean square 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 US census. These results point to major changes in inequality from 1990-2000.Acknowledgment. We thank David Autor, Gary Chamberlain, George Deltas, Bernd Fitzenberger, Jinyong Hahn, Jerry Hausman, Frank Kleibergen, Roger Koenker, Rafael Lalive, Tony Lancaster, Art Lewbel, and Whitney Newey for helpful discussions, and seminar participants at Berkeley, BYU, Brown, Duke, the University of Michigan, Michigan State University, the Harvard-MIT Econometrics Workshop, the University of Toronto, the University of Illinois at Urbana-Champaign, and the 2001 and 2004 Winter Econometric Society Meetings for comments.

[此贴子已经被作者于2005-8-4 12:35:01编辑过]