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Nonparametric Estimation in Random Coefficients Binary Choice Models

  • Eric Gautier

    (Crest)

  • Yuichi Kitamura

    (Crest)

This paper considers nonparametric estimation of the joint density of the random coe±-cients in binary choice models. Nonparametric inference allows to be °exible about the treatment ofunobserved heterogeneity. This is an ill-posed inverse problem characterized by an integral transform,namely the hemispherical transform. The kernel is boxcar and the operator is a convolution operatoron the sphere. Utilizing Fourier-Laplace expansions o®ers a clear insight on the identi¯cation problem.We present a new class of density estimators for the random coe±cients relying on estimates for thechoice probability. Characterizing the degree of ill-posedness we are able to relate the rate of conver-gence of the estimation of the density of the random coe±cient with the rate of convergence of theestimation of the choice probability. We present a particular estimate for the choice probability and itsasymptotic properties. The corresponding estimate of the density of the random coe±cient takes a sim-ple closed form. It is easy to implement in empirical applications. We obtain rates of consistency in allLp spaces and prove asymptotic normality. Extensions including estimation of marginals, treatmentsof non-random coe±cients, models with endogeneity and multiple alternatives are discussed.

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Paper provided by Centre de Recherche en Economie et Statistique in its series Working Papers with number 2008-15.

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Length: 51
Date of creation: 2008
Date of revision:
Handle: RePEc:crs:wpaper:2008-15
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  1. Briesch, Richard A. & Chintagunta, Pradeep K. & Matzkin, Rosa L., 2010. "Nonparametric Discrete Choice Models With Unobserved Heterogeneity," Journal of Business & Economic Statistics, American Statistical Association, vol. 28(2), pages 291-307.
  2. Heckman, James & Singer, Burton, 1984. "A Method for Minimizing the Impact of Distributional Assumptions in Econometric Models for Duration Data," Econometrica, Econometric Society, vol. 52(2), pages 271-320, March.
  3. Elbers, Chris & Ridder, Geert, 1982. "True and Spurious Duration Dependence: The Identifiability of the Proportional Hazard Model," Review of Economic Studies, Wiley Blackwell, vol. 49(3), pages 403-09, July.
  4. Stephane Hess & Denis Bolduc & John Polak, 2010. "Random covariance heterogeneity in discrete choice models," Transportation, Springer, vol. 37(3), pages 391-411, May.
  5. Patrick Bajari & Jeremy T. Fox & Stephen P. Ryan, 2007. "Linear Regression Estimation of Discrete Choice Models with Nonparametric Distributions of Random Coefficients," American Economic Review, American Economic Association, vol. 97(2), pages 459-463, May.
  6. Chesher, Andrew & Santos Silva, J M C, 2002. "Taste Variation in Discrete Choice Models," Review of Economic Studies, Wiley Blackwell, vol. 69(1), pages 147-68, January.
  7. Susan Athey & Guido W. Imbens, 2007. "Discrete Choice Models With Multiple Unobserved Choice Characteristics," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 48(4), pages 1159-1192, November.
  8. Carrasco, Marine & Florens, Jean-Pierre & Renault, Eric, 2007. "Linear Inverse Problems in Structural Econometrics Estimation Based on Spectral Decomposition and Regularization," Handbook of Econometrics, in: J.J. Heckman & E.E. Leamer (ed.), Handbook of Econometrics, edition 1, volume 6, chapter 77 Elsevier.
  9. Ichimura, Hidehiko & Thompson, T. Scott, 1998. "Maximum likelihood estimation of a binary choice model with random coefficients of unknown distribution," Journal of Econometrics, Elsevier, vol. 86(2), pages 269-295, June.
  10. Brownstone, David & Train, Kenneth, 1999. "Forecasting new product penetration with flexible substitution patterns," University of California Transportation Center, Working Papers qt3tb6j874, University of California Transportation Center.
  11. Klemelä, Jussi, 2000. "Estimation of Densities and Derivatives of Densities with Directional Data," Journal of Multivariate Analysis, Elsevier, vol. 73(1), pages 18-40, April.
  12. Steven T. Berry & Philip A. Haile, 2009. "Nonparametric Identification of Multinomial Choice Demand Models with Heterogeneous Consumers," Cowles Foundation Discussion Papers 1718, Cowles Foundation for Research in Economics, Yale University, revised Mar 2010.
  13. P. Groeneboom & G. Jongbloed, 2003. "Density estimation in the uniform deconvolution model," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 57(1), pages 136-157.
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