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BOUNDED SUPPORT IN LINEAR RANDOM COEFFICIENT MODELS: IDENTIFICATION AND VARIABLE SELECTION

Published online by Cambridge University Press:  26 March 2024

Philipp Hermann  and
Hajo Holzmann
Philipp Hermann
Affiliation:
Philipps-Universität Marburg
Hajo Holzmann*
Affiliation:
Philipps-Universität Marburg
*
Address correspondence to Hajo Holzmann, Department of Mathematics and Computer Science, Philipps-Universität Marburg, Marburg, Germany, holzmann@mathematik.uni-marburg.de.
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Abstract

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We consider linear random coefficient regression models, where the regressors are allowed to have a finite support. First, we investigate identification, and show that the means and the variances and covariances of the random coefficients are identified from the first two conditional moments of the response given the covariates if the support of the covariates, excluding the intercept, contains a Cartesian product with at least three points in each coordinate. We also discuss identification of higher-order mixed moments, as well as partial identification in the presence of a binary regressor. Next, we show the variable selection consistency of the adaptive LASSO for the variances and covariances of the random coefficients in finite and moderately high dimensions. This implies that the estimated covariance matrix will actually be positive semidefinite and hence a valid covariance matrix, in contrast to the estimate arising from a simple least squares fit. We illustrate the proposed method in a simulation study.

Information

Type
ARTICLES
Information
Econometric Theory , Volume 41 , Issue 4 , August 2025 , pp. 947 - 976
Copyright
© The Author(s), 2024. Published by Cambridge University Press

References

REFERENCES

Arellano, M., & Bonhomme, S. (2012). Identifying distributional characteristics in random coefficients panel data models. Review of Economic Studies , 79(3), 987–1020.CrossRefGoogle Scholar
Beran, R., Feuerverger, A., & Hall, P. (1996). On nonparametric estimation of intercept and slope distributions in random coefficient regression. Annals of Statistics , 24(6), 2569–2592.CrossRefGoogle Scholar
Beran, R., & Hall, P. (1992). Estimating coefficient distributions in random coefficient regressions. Annals of Statistics , 20(4), 1970–1984.CrossRefGoogle Scholar
Breunig, C., & Hoderlein, S. (2018). Specification testing in random coefficient models. Quantitative Economics , 9(3), 1371–1417.CrossRefGoogle Scholar
Dunker, F., Eckle, K., Proksch, K., & Schmidt-Hieber, J. (2019). Tests for qualitative features in the random coefficients model. Electronic Journal of Statistics , 13(2), 2257–2306.CrossRefGoogle Scholar
Gaillac, C., & Gautier, E. (2022). Adaptive estimation in the linear random coefficients model when regressors have limited variation. Bernoulli , 28, 504–524.CrossRefGoogle Scholar
Gautier, E., & Kitamura, Y. (2013). Nonparametric estimation in random coefficients binary choice models. Econometrica , 81(2), 581–607.Google Scholar
Hildreth, C., & Houck, J. P. (1968). Some estimators for a linear model with random coefficients. Journal of the American Statistical Association , 63(322), 584–595.CrossRefGoogle Scholar
Hoderlein, S., Holzmann, H., & Meister, A. (2017). The triangular model with random coefficients. Journal of Econometrics , 201(1), 144–169.CrossRefGoogle Scholar
Hoderlein, S., Klemelä, J., & Mammen, E. (2010). Analyzing the random coefficient model nonparametrically. Econometric Theory , 26, 804–837.CrossRefGoogle Scholar
Holzmann, H., & Meister, A. (2020). Rate-optimal nonparametric estimation for random coefficient regression models. Bernoulli , 26(4), 2790–2814.CrossRefGoogle Scholar
Huang, J., Ma, S., & Zhang, C.-H. (2008). Adaptive lasso for sparse high-dimensional regression models. Statistica Sinica , 18, 1603–1618.Google Scholar
Ichimura, H., & Thompson, T. S. (1998). Maximum likelihood estimation of a binary choice model with random coefficients of unknown distribution. Journal of Econometrics , 86(2), 269–295.CrossRefGoogle Scholar
Leeb, H., & Pötscher, B. (2003). The finite-sample distribution of post-model-selection estimators and uniform versus nonuniform approximations Econometric Theory , 19(1), 100–142.CrossRefGoogle Scholar
Lewbel, A. (2005). Modeling heterogeneity. Working Papers in Economics, Boston College, 402.Google Scholar
Lewbel, A., & Pendakur, K. (2017). Unobserved preference heterogeneity in demand using generalized random coefficients. Journal of Political Economy , 125(4), 1100–1148.CrossRefGoogle Scholar
Li, S., Cai, T. T., & Li, H. (2022). Inference for high-dimensional linear mixed-effects models: A quasi-likelihood approach. Journal of the American Statistical Association , 117, 1835–1846.CrossRefGoogle ScholarPubMed
Loh, P.-L., & Wainwright, M. J. (2017). Support recovery without incoherence: A case for nonconvex regularization. Annals of Statistics , 45(6), 2455–2482.CrossRefGoogle Scholar
Masten, M. A. (2018). Random coefficients on endogenous variables in simultaneous equations models. Review of Economic Studies , 85(2), 1193–1250.CrossRefGoogle Scholar
Rigollet, P., & Hütter, J.-C. (2019). High dimensional statistics. Lecture notes for course 18S997, MIT OpenCourseWare, Cambridge.Google Scholar
Schelldorfer, J., Bühlmann, P., & van de Geer, S. (2011). Estimation for high-dimensional linear mixed-effects models using ${\ell}_1$ -penalization. Scandinavian Journal of Statistics , 38(2), 197–214.CrossRefGoogle Scholar
Swamy, P. (1970). Efficient inference in a random coefficients model. Econometrica , 38, 311–324.CrossRefGoogle Scholar
Thomas, E. G. (2014). A polarization identity for multilinear maps. Indagationes Mathematicae , 25(3), 468–474.CrossRefGoogle Scholar
van der Vaart, A. W. (1998). Asymptotic statistics . Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press.CrossRefGoogle Scholar
Vandenberghe, L., & Boyd, S. (1996). Semidefinite programming. SIAM Review , 38(1), 49–95.CrossRefGoogle Scholar
Vershynin, R. (2018). High-dimensional probability: An introduction with applications in data science , volume 47. Cambridge University Press.Google Scholar
Wagener, J., & Dette, H. (2013). The adaptive lasso in high-dimensional sparse heteroscedastic models. Mathematical Methods of Statistics , 22(2), 137–154.CrossRefGoogle Scholar
Wainwright, M. J. (2019). High-dimensional statistics: A non-asymptotic viewpoint , volume 48. Cambridge University Press.Google Scholar
Zou, H. (2006). The adaptive lasso and its oracle properties. Journal of the American Statistical Association , 101(476), 1418–1429.CrossRefGoogle Scholar
Zou, H., & Zhang, H. H. (2009). On the adaptive elastic-net with a diverging number of parameters. Annals of Statistics, 37(4), 1733.CrossRefGoogle ScholarPubMed
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