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Quantile And Probability Curves Without Crossing

  • Victor Chernozhukov

    ()

    (MIT, Department of Economics & Operations Research Center, University College London and The University of Chicago)

  • Ivan Fernandez-Val

    ()

    (Department of Economics, Boston University)

  • Alfred Galichon

    ()

    (Harvard University, Department of Economics)

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.

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Paper provided by Boston University - Department of Economics in its series Boston University - Department of Economics - Working Papers Series with number WP2007-011.

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Length: 36 pages
Date of creation: Apr 2007
Date of revision:
Handle: RePEc:bos:wpaper:wp2007-011
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