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Shape Regressions

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Abstract

Learning about the shape of a probability distribution, not just about its location or dispersion, is often an important goal of empirical analysis. Given a continuous random variable Y and a random vector X defined on the same probability space, the conditional distribution function (CDF) and the conditional quantile function (CQF) offer two equivalent ways of describing the shape of the conditional distribution of Y given X. To these equivalent representations correspond two alternative approaches to shape regression. One approach - distribution regression - is based on direct estimation of the conditional distribution function (CDF); the other approach - quantile regression - is instead based on direct estimation of the conditional quantile function (CQF). Since the CDF and the CQF are generalized inverses of each other, indirect estimates of the CQF and the CDF may be obtained by taking the generalized inverse of the direct estimates obtained from either approach, possibly after rearranging to guarantee monotonicity of estimated CDFs and CQFs. The equivalence between the two approaches holds for standard nonparametric estimators in the unconditional case. In the conditional case, when modeling assumptions are introduced to avoid curse-of-dimensionality problems, this equivalence is generally lost as a convenient parametric model for the CDF need not imply a convenient parametric model for the CQF, and vice versa. Despite the vast literature on the quantile regression approach, and the recent attention to the distribution regression approach, no systematic comparison of the two has been carried out yet. Our paper fills-in this gap by comparing the asymptotic properties of estimators obtained from the two approaches, both when the assumed parametric models on which they are based are correctly specified and when they are not.

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  • Franco Peracchi & Samantha Leorato, 2015. "Shape Regressions," Working Papers gueconwpa~15-15-06, Georgetown University, Department of Economics.
  • Handle: RePEc:geo:guwopa:gueconwpa~15-15-06
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    1. Samantha Leorato & Franco Peracchi, 2015. "Comparing Distribution and Quantile Regression," EIEF Working Papers Series 1511, Einaudi Institute for Economics and Finance (EIEF), revised Oct 2015.
    2. Victor Chernozhukov & Iv·n Fern·ndez-Val & Alfred Galichon, 2010. "Quantile and Probability Curves Without Crossing," Econometrica, Econometric Society, vol. 78(3), pages 1093-1125, May.
    3. Hall, Peter & Wolff, Rodney C. L. & Yao, Qiwei, 1999. "Methods for estimating a conditional distribution function," LSE Research Online Documents on Economics 6631, London School of Economics and Political Science, LSE Library.
    4. Koenker, Roger W & Bassett, Gilbert, Jr, 1978. "Regression Quantiles," Econometrica, Econometric Society, vol. 46(1), pages 33-50, January.
    5. Christoph Rothe, 2012. "Partial Distributional Policy Effects," Econometrica, Econometric Society, vol. 80(5), pages 2269-2301, September.
    6. Christoph Rothe & Dominik Wied, 2013. "Misspecification Testing in a Class of Conditional Distributional Models," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 108(501), pages 314-324, March.
    7. Roger Koenker & Samantha Leorato & Franco Peracchi, 2013. "Distributional vs. Quantile Regression," CEIS Research Paper 300, Tor Vergata University, CEIS, revised 17 Dec 2013.
    8. Horowitz, Joel L & Manski, Charles F, 1995. "Identification and Robustness with Contaminated and Corrupted Data," Econometrica, Econometric Society, vol. 63(2), pages 281-302, March.
    9. Koenker,Roger, 2005. "Quantile Regression," Cambridge Books, Cambridge University Press, number 9780521845731, August.
    10. Joshua Angrist & Victor Chernozhukov & Iván Fernández-Val, 2006. "Quantile Regression under Misspecification, with an Application to the U.S. Wage Structure," Econometrica, Econometric Society, vol. 74(2), pages 539-563, March.
    11. Peracchi, Franco, 2002. "On estimating conditional quantiles and distribution functions," Computational Statistics & Data Analysis, Elsevier, vol. 38(4), pages 433-447, February.
    12. Holger Dette & Stanislav Volgushev, 2008. "Non-crossing non-parametric estimates of quantile curves," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 70(3), pages 609-627.
    13. repec:spo:wpecon:info:hdl:2441/5rkqqmvrn4tl22s9mc4b6ga2g is not listed on IDEAS
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    1. Samantha Leorato & Franco Peracchi, 2015. "Comparing Distribution and Quantile Regression," EIEF Working Papers Series 1511, Einaudi Institute for Economics and Finance (EIEF), revised Oct 2015.

    More about this item

    Keywords

    Distribution regression; quantile regression; functional delta-method; non-separable models; influence function;

    JEL classification:

    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General
    • C21 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Cross-Sectional Models; Spatial Models; Treatment Effect Models
    • C25 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Discrete Regression and Qualitative Choice Models; Discrete Regressors; Proportions; Probabilities

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