Quantile and Probability Curves without Crossing

This paper proposes a method to address the longstanding problem of lack of monotonicity in estimation of conditional and structural quantile functions, also known as the quantile crossing problem. The method consists in sorting or monotone rearranging the original estimated non-monotone curve into a monotone rearranged curve. We show that the rearranged curve is closer to the true quantile curve in finite samples than the original curve, establish a functional delta method for rearrangement-related operators, and derive functional limit theory for the entire rearranged curve and its functionals. We also establish validity of the bootstrap for estimating the limit law of the the entire rearranged curve and its functionals. Our limit results are generic in that they apply to every estimator of a monotone econometric function, provided that the estimator satisfies a functional central limit theorem and the function satisfies some smoothness conditions. Consequently, our results apply to estimation of other econometric functions with monotonicity restrictions, such as demand, production, distribution, and structural distribution functions. We illustrate the results with an application to estimation of structural quantile functions using data on Vietnam veteran status and earnings. JEL Classification: C10, C50 AMS Classification: 62J02; 62E20, 62P20 Date: This version of the paper is of July 25, 2009. Previous, more extended, versions (September 2006, April 2007) are available at www.mit.edu/∼vchern/www and www.ArXiv.org. The method developed in this paper has now been incorporated in the package quantreg (Koenker, 2007) in R. The title of this paper is (partially) borrowed from the work of Xuming He (1997), to whom we are grateful for the inspiration and formulation of the problem. We would like to thank the editor Oliver Linton, three anonymous referees, Alberto Abadie, Josh Angrist, Andrew Chesher, Phil Cross, James Durbin, Ivar Ekeland, Brigham Frandsen, Raymond Guiteras, Xuming He, Roger Koenker, Joonhwan Lee, Vadim Marmer, Ilya Molchanov, Francesca Molinari, Whitney Newey, Steve Portnoy, Shinichi Sakata, Art Shneyerov, Alp Simsek, and participants at BU, CEMFI, CEMMAP Measurement Matters Conference, Columbia Conference on Optimal Transportation, Columbia, Cornell, Cowles Foundation 75th Anniversary Conference, Duke-Triangle, Ecole Polytechnique, Frontiers of Microeconometrics in Tokyo, Georgetown, Harvard-MIT, MIT, Northwestern, UBC, UCL, UIUC, University of Alicante, and University of Gothenburg Conference “Nonsmooth Inference, Analysis, and Dependence,” for comments that helped us to considerably improve the paper. We are grateful to Alberto Abadie for providing us the data for the empirical example. The authors gratefully acknowledge research support from the National Science Foundation and chaire X-Dauphine “Finance et Développement Durable”. † Massachusetts Institute of Technology, Department of Economics and Operations Research Center, University College London, CEMMAP. E-mail: [email protected]. § Boston University, Department of Economics. E-mail: [email protected]. ‡ Ecole Polytechnique, Département d’Economie. E-mail: [email protected]. 1