Nonparametric Estimation in Random Coefficients Binary Choice Models


  • Eric Gautier,

    1. CREST (ENSAE), 3 avenue Pierre Larousse, 92245 Malakoff Cedex, France;
    Search for more papers by this author
  • Yuichi Kitamura

    1. Cowles Foundation for Research in Economics, Yale University, New Haven, CT 06520, U.S.A.;
    Search for more papers by this author
    • We thank Whitney Newey and two anonymous referees for comments that greatly improved this paper. We also thank seminar participants at Chicago, CREST, Harvard/MIT, the Henri Poincaré Institute, Hitotsubashi, LSE, Mannheim, Minnesota, Northwestern, NYU, Paris 6, Princeton, Rochester, Simon Fraser, Tilburg, Toulouse 1 University, UBC, UCL, UCLA, UCSD, the Tinbergen Institute, and the University of Tokyo, and participants of the 2008 Cowles summer econometrics conference, EEA/ESEM, FEMES, Journées STAR, and SETA and 2009 CIRM Rencontres de Statistiques Mathématiques for helpful comments. Yuhan Fang and Xiaoxia Xi provided excellent research assistance. Kitamura acknowledges financial support from the National Science Foundation via Grants SES-0241770, SES-0551271, and SES-0851759. Gautier is grateful for support from the Cowles Foundation as this research was initiated during his visit as a postdoctoral associate.


This paper considers random coefficients binary choice models. The main goal is to estimate the density of the random coefficients nonparametrically. This is an ill-posed inverse problem characterized by an integral transform. A new density estimator for the random coefficients is developed, utilizing Fourier–Laplace series on spheres. This approach offers a clear insight on the identification problem. More importantly, it leads to a closed form estimator formula that yields a simple plug-in procedure requiring no numerical optimization. The new estimator, therefore, is easy to implement in empirical applications, while being flexible about the treatment of unobserved heterogeneity. Extensions including treatments of nonrandom coefficients and models with endogeneity are discussed.