IDEAS home Printed from https://ideas.repec.org/p/hal/wpaper/hal-02130472.html
   My bibliography  Save this paper

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

Author

Listed:
  • Christophe Gaillac

    (TSE-R - Toulouse School of Economics - UT Capitole - Université Toulouse Capitole - UT - Université de Toulouse - INRA - Institut National de la Recherche Agronomique - EHESS - École des hautes études en sciences sociales - CNRS - Centre National de la Recherche Scientifique, CREST - Centre de Recherche en Économie et Statistique - ENSAI - Ecole Nationale de la Statistique et de l'Analyse de l'Information [Bruz] - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - ENSAE Paris - École Nationale de la Statistique et de l'Administration Économique - CNRS - Centre National de la Recherche Scientifique)

  • Eric Gautier

    (TSE-R - Toulouse School of Economics - UT Capitole - Université Toulouse Capitole - UT - Université de Toulouse - INRA - Institut National de la Recherche Agronomique - EHESS - École des hautes études en sciences sociales - CNRS - Centre National de la Recherche Scientifique, UT Capitole - Université Toulouse Capitole - UT - Université de Toulouse)

Abstract

We consider a linear model where the coefficients - intercept and slopes - are random with a law in a nonparametric class and independent from the regressors. Identification often requires the regressors to have a support which is the whole space. This is hardly ever the case in practice. Alternatively, the coefficients can have a compact support but this is not compatible with unbounded error terms as usual in regression models. In this paper, the regressors can have a support which is a proper subset but the slopes (not the intercept) do not have heavy-tails. Lower bounds on the supremum risk for the estimation of the joint density of the random coefficients density are obtained for a wide range of smoothness, where some allow for polynomial and nearly parametric rates of convergence. We present a minimax optimal estimator, a data-driven rule for adaptive estimation, and made available a R package.

Suggested Citation

  • Christophe Gaillac & Eric Gautier, 2020. "Adaptive estimation in the linear random coefficients model when regressors have limited variation," Working Papers hal-02130472, HAL.
    • Handle: RePEc:hal:wpaper:hal-02130472
    Note: View the original document on HAL open archive server: https://hal.science/hal-02130472v4
    as

    Download full text from publisher

    File URL: https://hal.science/hal-02130472v4/document
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Hoderlein, Stefan & Holzmann, Hajo & Meister, Alexander, 2017. "The triangular model with random coefficients," Journal of Econometrics, Elsevier, vol. 201(1), pages 144-169.
    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. 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.
    4. 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.
    5. 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.
    6. Christoph Breunig & Stefan Hoderlein, 2018. "Specification testing in random coefficient models," Quantitative Economics, Econometric Society, vol. 9(3), pages 1371-1417, November.
    7. Lacour, C. & Massart, P., 2016. "Minimal penalty for Goldenshluger–Lepski method," Stochastic Processes and their Applications, Elsevier, vol. 126(12), pages 3774-3789.
    8. 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).
    9. 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.
    10. Eric Gautier & Yuichi Kitamura, 2013. "Nonparametric Estimation in Random Coefficients Binary Choice Models," Econometrica, Econometric Society, vol. 81(2), pages 581-607, March.
    11. Fabian Dunker & Konstantin Eckle & Katharina Proksch & Johannes Schmidt-Hieber, 2017. "Tests for qualitative features in the random coefficients model," Courant Research Centre: Poverty, Equity and Growth - Discussion Papers 225, Courant Research Centre PEG.
    12. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Christophe Gaillac & Eric Gautier, 2021. "Nonparametric classes for identification in random coefficients models when regressors have limited variation," Working Papers hal-03231392, HAL.
    2. Michael Jansson & Demian Pouzo, 2017. "Towards a General Large Sample Theory for Regularized Estimators," Papers 1712.07248, arXiv.org, revised Jul 2020.
    3. 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).
    4. 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.
    5. Éric Gautier, 2021. "Relaxing Monotonicity in Endogenous Selection Models and Application to Surveys," Post-Print hal-03306234, HAL.
    6. Wang, Ao, 2023. "Sieve BLP: A semi-nonparametric model of demand for differentiated products," Journal of Econometrics, Elsevier, vol. 235(2), pages 325-351.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Eric Gautier & Erwann Le Pennec, 2011. "Adaptive Estimation in the Nonparametric Random Coefficients Binary Choice Model by Needlet Thresholding," Working Papers 2011-20, Center for Research in Economics and Statistics.
    2. Zhan Gao & M. Hashem Pesaran, 2023. "Identification and estimation of categorical random coefficient models," Empirical Economics, Springer, vol. 64(6), pages 2543-2588, June.
    3. Arthur Lewbel & Krishna Pendakur, 2017. "Unobserved Preference Heterogeneity in Demand Using Generalized Random Coefficients," Journal of Political Economy, University of Chicago Press, vol. 125(4), pages 1100-1148.
    4. Giovanni Compiani & Yuichi Kitamura, 2016. "Using mixtures in econometric models: a brief review and some new results," Econometrics Journal, Royal Economic Society, vol. 19(3), pages 95-127, October.
    5. Christophe Gaillac & Eric Gautier, 2021. "Nonparametric classes for identification in random coefficients models when regressors have limited variation," Working Papers hal-03231392, HAL.
    6. Fabian Dunker & Konstantin Eckle & Katharina Proksch & Johannes Schmidt-Hieber, 2017. "Tests for qualitative features in the random coefficients model," Courant Research Centre: Poverty, Equity and Growth - Discussion Papers 225, Courant Research Centre PEG.
    7. Éric Gautier, 2021. "Relaxing Monotonicity in Endogenous Selection Models and Application to Surveys," Post-Print hal-03306234, HAL.
    8. Escanciano, Juan Carlos, 2023. "Irregular identification of structural models with nonparametric unobserved heterogeneity," Journal of Econometrics, Elsevier, vol. 234(1), pages 106-127.
    9. Juan Carlos Escanciano, 2020. "Irregular Identification of Structural Models with Nonparametric Unobserved Heterogeneity," Papers 2005.08611, arXiv.org.
    10. Christoph Breunig & Stefan Hoderlein, 2016. "Nonparametric Specification Testing in Random Parameter Models," Boston College Working Papers in Economics 897, Boston College Department of Economics.
    11. Christoph Breunig & Stefan Hoderlein, 2018. "Specification testing in random coefficient models," Quantitative Economics, Econometric Society, vol. 9(3), pages 1371-1417, November.
    12. D’Haultfœuille, Xavier & Hoderlein, Stefan & Sasaki, Yuya, 2024. "Testing and relaxing the exclusion restriction in the control function approach," Journal of Econometrics, Elsevier, vol. 240(2).
    13. Fabian Dunker & Stefan Hoderlein & Hiroaki Kaido, 2023. "Nonparametric identification of random coefficients in aggregate demand models for differentiated products," The Econometrics Journal, Royal Economic Society, vol. 26(2), pages 279-306.
    14. Breunig, Christoph, 2021. "Varying random coefficient models," Journal of Econometrics, Elsevier, vol. 221(2), pages 381-408.
    15. Hoderlein, Stefan & Nesheim, Lars & Simoni, Anna, 2017. "Semiparametric Estimation Of Random Coefficients In Structural Economic Models," Econometric Theory, Cambridge University Press, vol. 33(6), pages 1265-1305, December.
    16. Joel L. Horowitz, 2013. "Ill-posed inverse problems in economics," CeMMAP working papers 37/13, Institute for Fiscal Studies.
    17. 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.
    18. Eric Gautier & Stefan Soderlein, 2011. "Estimating the Distribution of Treatment Effects," Working Papers 2011-25, Center for Research in Economics and Statistics.
    19. Dunker, Fabian & Hoderlein, Stefan & Kaido, Hiroaki & Sherman, Robert, 2018. "Nonparametric identification of the distribution of random coefficients in binary response static games of complete information," Journal of Econometrics, Elsevier, vol. 206(1), pages 83-102.
    20. Joel L. Horowitz, 2013. "Ill-posed inverse problems in economics," CeMMAP working papers CWP37/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.

    More about this item

    Keywords

    Minimax; Random Coefficients; Adaptation; Ill-posed Inverse Problem;
    All these keywords.

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:hal:wpaper:hal-02130472. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: CCSD (email available below). General contact details of provider: https://hal.archives-ouvertes.fr/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.