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Adaptive estimation in the linear random coefficients model when regressors have limited variation

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  • Gautier, Eric
  • Gaillac, Christophe

Abstract

We consider a linear model where the coecients - intercept and slopes - are random and independent from regressors which support is a proper subset. When the slopes do not have heavy tails, the joint density of the random coecients is identied. Lower bounds on the supremum risk for the estimation of the density are derived for this model and a related white noise model. We present an estimator, its rates of convergence, and a data-driven rule which delivers adaptive estimators.

Suggested Citation

  • Gautier, Eric & Gaillac, Christophe, 2019. "Adaptive estimation in the linear random coefficients model when regressors have limited variation," TSE Working Papers 19-1026, Toulouse School of Economics (TSE).
    • Handle: RePEc:tse:wpaper:123181
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    References listed on IDEAS

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    1. Chen, Xiaohong & Reiss, Markus, 2011. "On Rate Optimality For Ill-Posed Inverse Problems In Econometrics," Econometric Theory, Cambridge University Press, vol. 27(3), pages 497-521, June.
    2. Christoph Breunig & Stefan Hoderlein, 2018. "Specification testing in random coefficient models," Quantitative Economics, Econometric Society, vol. 9(3), pages 1371-1417, November.
    3. Lacour, C. & Massart, P., 2016. "Minimal penalty for Goldenshluger–Lepski method," Stochastic Processes and their Applications, Elsevier, vol. 126(12), pages 3774-3789.
    4. Gautier, Eric & Gaillac, Christophe, 2019. "Estimates for the SVD of the Truncated Fourier Transform on L2(cosh(b.)) and Stable Analytic Continuation," TSE Working Papers 19-1013, Toulouse School of Economics (TSE).
    5. Lacour, Claire, 2008. "Nonparametric estimation of the stationary density and the transition density of a Markov chain," Stochastic Processes and their Applications, Elsevier, vol. 118(2), pages 232-260, February.
    6. Hoderlein, Stefan & Holzmann, Hajo & Meister, Alexander, 2017. "The triangular model with random coefficients," Journal of Econometrics, Elsevier, vol. 201(1), pages 144-169.
    7. Eric Gautier & Yuichi Kitamura, 2013. "Nonparametric Estimation in Random Coefficients Binary Choice Models," Econometrica, Econometric Society, vol. 81(2), pages 581-607, March.
    8. Fabian Dunker & Konstantin Eckle & Katharina Proksch & Johannes Schmidt-Hieber, 2017. "Tests for qualitative features in the random coefficients model," Papers 1704.01066, arXiv.org, revised Mar 2018.
    9. Lacour, Claire, 2008. "Adaptive estimation of the transition density of a particular hidden Markov chain," Journal of Multivariate Analysis, Elsevier, vol. 99(5), pages 787-814, May.
    10. 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.
    11. Hoderlein, Stefan & Klemelä, Jussi & Mammen, Enno, 2010. "Analyzing The Random Coefficient Model Nonparametrically," Econometric Theory, Cambridge University Press, vol. 26(3), pages 804-837, June.
    12. Gautier, Eric & Hoderlein, Stefan, 2011. "A triangular treatment effect model with random coefficients in the selection equation," TSE Working Papers 15-598, Toulouse School of Economics (TSE), revised 25 Aug 2015.
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    Cited by:

    1. Michael Jansson & Demian Pouzo, 2017. "Towards a General Large Sample Theory for Regularized Estimators," Papers 1712.07248, arXiv.org, revised Jul 2020.
    2. Gautier, Eric & Gaillac, Christophe, 2019. "Estimates for the SVD of the Truncated Fourier Transform on L2(cosh(b.)) and Stable Analytic Continuation," TSE Working Papers 19-1013, Toulouse School of Economics (TSE).
    3. Gaillac, Christophe & Gautier, Eric, 2021. "Non Parametric Classes for Identification in Random Coefficients Models when Regressors have Limited Variation," TSE Working Papers 21-1218, Toulouse School of Economics (TSE).
    4. Wang, Ao, 2023. "Sieve BLP: A semi-nonparametric model of demand for differentiated products," Journal of Econometrics, Elsevier, vol. 235(2), pages 325-351.
    5. Éric Gautier, 2021. "Relaxing Monotonicity in Endogenous Selection Models and Application to Surveys," Post-Print hal-03306234, HAL.
    6. Wang, Ao, 2020. "Identifying the Distribution of Random Coefficients in BLP Demand Models Using One Single Variation in Product Characteristics," The Warwick Economics Research Paper Series (TWERPS) 1304, University of Warwick, Department of Economics.

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