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Quantile and Probability Curves without Crossing

Author

Listed:
  • Victor Chernozhukov

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

  • Ivan Fernandez-Val

    (Department of Economics - Tilburg University [Netherlands])

  • Alfred Galichon

    (ECON - Département d'économie (Sciences Po) - Sciences Po - Sciences Po - CNRS - Centre National de la Recherche Scientifique)

Abstract
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.

Suggested Citation

  • Victor Chernozhukov & Ivan Fernandez-Val & Alfred Galichon, 2010. "Quantile and Probability Curves without Crossing," SciencePo Working papers Main hal-01052958, HAL.
    • Handle: RePEc:hal:spmain:hal-01052958
    Note: View the original document on HAL open archive server: https://sciencespo.hal.science/hal-01052958
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