QUANTILE AND PROBABILITY CURVES WITHOUT CROSSING

By Victor Chernozhukov, Iv'an Fern'andez-Val, Alfred Galichon. 2007

569 citations recorded by the source corpus.

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.

Subfields: Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/a7b9ddcc0d316b21f0f8ab101d340ca2fa872164

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Citations in the corpus

Vector Quantile Regression (2014, methodology): “There exist procedures to repair the crossing problem, see [7].”
The Big Data Newsvendor: Practical Insights from Machine Learning (2013, background): “(2006), Chernozhukov and Hansen (2008), Chernozhukov et al. (2010), Belloni and Chernozhukov (2011) and references therein for more recent works on high-dimensionality.”
Constrained Regression for Interval-Valued Data (2013, background): “75], and the data are available in a classic point-valued format, then quantile regression with monotonicity restrictions could be implemented as proposed by Chernozhukov, Fernandez-Val, and Galichon (2010).”
Optimal Bandwidth Selection for Nonparametric Conditional Distribution and Quantile Functions (2013, unknown): “…problem one could use the “rearrangement” method to generate a new quantile estimator, say, q̂∗α(x) (obtained based on rearrangement of q̂α(x)), and that q̂∗α(x) is monotone in α (see Chernozhukov, Fernndez-Val, and Galichon 2010, p. 1097 for a detailed description as how to generate q̂∗α(x)).”
Quantile Selection Models with an Application to Understanding Changes in Wage Inequality (2017, unknown): “[21] Chernozhukov, V.”
Unconditional Quantile Treatment Effects Under Endogeneity (2007, unknown): “We suggest using the method of Chernozhukov, Fernandez-Val, and Galichon (2010) to monotonize the estimated distribution functions F̂Y 0|c(u) and F̂Y 1|c(u) via rearrangement.”
Earnings and Consumption Dynamics: A Nonlinear Panel Data Framework (2015, background): “We can rewrite (16) as f(a1|y) = E [f(a1|ηi1) | yi = y] , (17) where the expectation is taken with respect to the density of ηi1 given yi, for a fixed value a1.”
Nonlinear Panel Data Estimation via Quantile Regressions (2015, methodology): “For example, when h(y) = |y| model (20) is a panel data version of the CAViaR model of Engle and Manganelli (2004).”
Inference on breakdown frontiers (2017, methodology): “It follows from the remainder of the proof in Chernozhukov, Fernández-Val, and Galichon (2010) that sup c∈C ∣∣∣∣ [ φ (1) 3 (θ0 + tnhn) ] (c)− [φ(1)3 (θ0)](c) tn − [φ(1)′3 θ0(h)](c) ∣∣∣∣= o(1) where ‖ · ‖e is the Euclidean norm, and whereφ(1)′3 θ0 is defined in equation (30).”