Quantile and Probability Curves without Crossing

Econometrica, Vol. 78, No. 3, pp. 1093-1125, May 2010

MIT Department of Economics, Research Paper No. 07-15

37 Pages Posted: 30 Apr 2007 Last revised: 9 Apr 2011

See all articles by Victor Chernozhukov

Victor Chernozhukov

Massachusetts Institute of Technology (MIT) - Department of Economics

Iván Fernández‐Val

Boston University - Department of Economics

Alfred Galichon

NYU, Department of Economics and Courant Institute

Date Written: April 27, 2007

Abstract

The most common approach to estimating conditional quantile curves is to fit a curve, typically linear, pointwise for each quantile. Linear functional forms, coupled with pointwise fitting, are used for a number of reasons including parsimony of the resulting approximations and good computational properties. The resulting fits, however, may not respect a logical monotonicity requirement - that the quantile curve be increasing as a function of probability. This paper studies the natural monotonization of these empirical curves induced by sampling from the estimated non-monotone model, and then taking the resulting conditional quantile curves that by construction are monotone in the probability. This construction of monotone quantile curves may be seen as a bootstrap and also as a monotonic rearrangement of the original non-monotone function. It is shown that the monotonized curves are closer to the true curves in finite samples, for any sample size. Under correct specification, the rearranged conditional quantile curves have the same asymptotic distribution as the original non-monotone curves. Under misspecification, however, the asymptotics of the rearranged curves may partially differ from the asymptotics of the original non-monotone curves. An analogous procedure is developed to monotonize the estimates of conditional distribution functions. The results are derived by establishing the compact (Hadamard) differentiability of the monotonized quantile and probability curves with respect to the original curves in discontinuous directions, tangentially to a set of continuous functions. In doing so, the compact differentiability of the rearrangement-related operators is established.

Keywords: Quantile regression, Monotonicity, Rearrangement, Approximation, Functional Delta Method, Hadamard Differentiability of Rearrangement Operators

JEL Classification: J02, E20, P20

Suggested Citation

Chernozhukov, Victor and Fernandez-Val, Ivan and Galichon, Alfred, Quantile and Probability Curves without Crossing (April 27, 2007). Econometrica, Vol. 78, No. 3, pp. 1093-1125, May 2010, MIT Department of Economics, Research Paper No. 07-15, Available at SSRN: https://ssrn.com/abstract=983158

Victor Chernozhukov (Contact Author)

Massachusetts Institute of Technology (MIT) - Department of Economics ( email )

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HOME PAGE: http://www.mit.edu/~vchern/

Ivan Fernandez-Val

Boston University - Department of Economics ( email )

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HOME PAGE: http://people.mit.edu/ivanf

Alfred Galichon

NYU, Department of Economics and Courant Institute ( email )

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New York, NY 10011
United States

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