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Adaptive estimation in the nonparametric random coefficients binary choice model by needlet thresholding

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  • Eric Gautier

    ()
    (CREST - Centre de Recherche en Économie et Statistique - INSEE - École Nationale de la Statistique et de l'Administration Économique, ENSAE - École Nationale de la Statistique et de l'Administration Économique - ENSAE ParisTech)

  • Erwan Le Pennec

    ()
    (INRIA Saclay - Ile de France - SELECT - INRIA - Université Paris Sud - Paris XI - CNRS : UMR, LM-Orsay - Laboratoire de Mathématiques d'Orsay - CNRS : UMR8628 - Université Paris Sud - Paris XI)

Abstract

In this article we consider the estimation of the joint distribution of the random coefficients and error term in the nonparametric random coefficients binary choice model. In this model from economics, each agent has to choose between two mutually exclusive alternatives based on the observation of attributes of the two alternatives and of the agents, the random coefficients account for unobserved heterogeneity of preferences. Because of the scale invariance of the model, we want to estimate the density of a random vector of Euclidean norm 1. If the regressors and coefficients are independent, the choice probability conditional on a vector of $d-1$ regressors is an integral of the joint density on half a hyper-sphere determined by the regressors. Estimation of the joint density is an ill-posed inverse problem where the operator that has to be inverted in the so-called hemispherical transform. We derive lower bounds on the minimax risk under $\xL^p$ losses and smoothness expressed in terms of Besov spaces on the sphere $\mathbb{S}^{d-1}$. We then consider a needlet thresholded estimator with data-driven thresholds and obtain adaptivity for $\xL^p$ losses and Besov ellipsoids under assumptions on the random design.

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Bibliographic Info

Paper provided by HAL in its series Working Papers with number inria-00601274.

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Date of creation: Jun 2011
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Handle: RePEc:hal:wpaper:inria-00601274

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Keywords: Discrete choice models; random coefficients; inverse problems; minimax rate optimality; adaptation; needlets; data-driven thresholding.;

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  1. Eric Gautier & Yuichi Kitamura, 2009. "Nonparametric Estimation in Random Coefficients Binary Choice Models," Cowles Foundation Discussion Papers 1721, Cowles Foundation for Research in Economics, Yale University.
  2. Eric Gautier & Stefan Soderlein, 2011. "Estimating the Distribution of Treatment Effects," Working Papers 2011-25, Centre de Recherche en Economie et Statistique.
  3. Hoderlein, Stefan & Klemelä, Jussi & Mammen, Enno, 2010. "Analyzing The Random Coefficient Model Nonparametrically," Econometric Theory, Cambridge University Press, vol. 26(03), pages 804-837, June.
  4. Ichimura, H. & Thompson, S., 1993. "Maximum Likelihood Estimation of a Binary Choice Model with Random Coefficients of Unknown Distributions," Papers 268, Minnesota - Center for Economic Research.
  5. 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.
  6. 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.
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Cited by:
  1. Eric Gautier & Stefan Hoderlein, 2012. "A Triangular Treatment Effect Model With Random Coefficients In The Selection Equation," Boston College Working Papers in Economics 838, Boston College Department of Economics.
  2. Andrew Chesher & Adam Rosen, 2012. "An instrumental variable random coefficients model for binary outcomes," CeMMAP working papers CWP34/12, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  3. Fabian Dunker & Stefan Hoderlein & Hiroaki Kaido, 2013. "Random coefficients in static games of complete information," CeMMAP working papers CWP12/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.

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