Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-24T15:05:37.178Z Has data issue: false hasContentIssue false
  • English
  • Français

SEMIPARAMETRIC ESTIMATION OF RANDOM COEFFICIENTS IN STRUCTURAL ECONOMIC MODELS

Published online by Cambridge University Press:  02 November 2016

Stefan Hoderlein ,
Lars Nesheim  and
Anna Simoni
Stefan Hoderlein
Affiliation:
Boston College
Lars Nesheim*
Affiliation:
University College London
Anna Simoni
Affiliation:
CNRS-CREST
*
*Address correspondence to Lars Nesheim, Department of Economics, University College London, Gower Street, London WC1E 6BT, UK; e-mail: l.nesheim@ucl.ac.uk.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper discusses nonparametric estimation of the distribution of random coefficients in a structural model that is nonlinear in the random coefficients. We establish that the problem of recovering the probability density function (pdf) of random parameters falls into the class of convexly-constrained inverse problems. The framework offers an estimation method that separates computational solution of the structural model from estimation. We first discuss nonparametric identification. Then, we propose two alternative estimation procedures to estimate the density and derive their asymptotic properties. Our general framework allows us to deal with unobservable nuisance variables, e.g., measurement error, but also covers the case when there are no such nuisance variables. Finally, Monte Carlo experiments for several structural models are provided which illustrate the performance of our estimation procedure.

Type
ARTICLES
Information
Econometric Theory , Volume 33 , Issue 6 , December 2017 , pp. 1265 - 1305
Copyright
Copyright © Cambridge University Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

We thank the co-editor Yoon-Jae Whang and three anonymous referees whose comments greatly improved the paper. We have benefited from comments and discussions with Victor Aguirregabiria, Orazio Attanasio, Richard Blundell, Chris Carroll, Jeremy Fox, Emmanuel Guerre, Dirk Krueger, Arthur Lewbel, Enno Mammen, Rosa Matzkin, Peter Phillips, Jean-Marc Robin, Sami Stouli, Yuanyuan Wan, and Hal White. Lars Nesheim gratefully acknowledges financial support from the UK Economic and Social Research Council through the ESRC Centre for Microdata Methods and Practice grant RES-589-28-0001. Anna Simoni gratefully acknowledges financial support from the University of Mannheim through SFB 884 and from ANR-13-BSH1-0004, and ANR-11-LABEX-0047.

References

REFERENCES

Ai, C. & Chen, X. (2003) Efficient estimation of models with conditional moment restrictions containing unknown functions. Econometrica 71(6), 1795–1843.Google Scholar
Attanasio, O.P. & Weber, G. (2010) Consumption and saving: Models of intertemporal allocation and their implications for public policy. Journal of Economic Literature 48(3), 693–751.Google Scholar
Beran, R., Feuerverger, A., & Hall, P. (1996) On nonparametric estimation of intercept and slope distributions in random coefficient regression. The Annals of Statistics 24(6), 2569–2592.Google Scholar
Beran, R. & Millar, P.W. (1994) Minimum distance estimation in random coefficient regression models. The Annals of Statistics 22(4), 1976–1992.CrossRefGoogle Scholar
Blundell, R., Chen, X., & Kristensen, D. (2007) Semi-nonparametric IV estimation of shape-invariant engel curves. Econometrica 75(6), 1613–1669.CrossRefGoogle Scholar
Canay, I.A., Santos, A., & Shaikh, A.M. (2013) On the testability of identification in some nonparametric models with endogeneity. Econometrica 81(6), 2535–2559.Google Scholar
Carrasco, M. & Florens, J.-P. (2000) Generalization of GMM to a continuum of moment conditions. Econometric Theory 16(6), 797–834.Google Scholar
Carrasco, M. & Florens, J.-P. (2011) A spectral method for deconvolving a density. Econometric Theory 27, 546–581.Google Scholar
Carrasco, M., Florens, J.-P., & Renault, E. (2007) Linear inverse problems in structural econometrics estimation based on spectral decomposition and regularization. In Heckman, J.J. & Leamer, E.E. (eds.), Handbook of Econometrics, vol. 6, Part B, pp. 5633–5751. Elsevier.CrossRefGoogle Scholar
Chen, X. & Pouzo, D. (2012) Estimation of nonparametric conditional moment models with possibly nonsmooth generalized residuals. Econometrica 80(1), 277–321.Google Scholar
Chen, X. & Reiss, M. (2011) On rate optimality for ill-posed inverse problems in econometrics. Econometric Theory 27(6), 497–521.CrossRefGoogle Scholar
Darolles, S., Fan, Y., Florens, J.P., & Renault, E. (2011) Nonparametric instrumental regression. Econometrica 79(5), 1541–1565.Google Scholar
Dunker, F., Hoderlein, S., & Kaido, K. (2013) Random Coefficients in Static Games of Complete Information. Technical report, Cemmap Working paper, CWP12/13.Google Scholar
Engl, H., Hanke, M., & Neubauer, A. (2000) Regularization of Inverse Problems. Kluwer Academic.Google Scholar
Florens, J.-P. (2003) Inverse problems and structural econometrics: The example of instrumental variables. In Dewatripont, M., Hansen, L.P., & Turnovsky, S.J. (eds.), Advances in Economics and Econometrics, vol. 2, pp. 284–311. Cambridge University Press.CrossRefGoogle Scholar
Florens, J., Mouchart, M., & Rolin, J.-M. (1990) Elements of Bayesian Statistics. Dekker.Google Scholar
Florens, J.-P. & Simoni, A. (2012) Nonparametric estimation of an instrumental regression: A quasi-bayesian approach based on regularized posterior. Journal of Econometrics 170(2), 458–475.CrossRefGoogle Scholar
Florens, J.-P. & Simoni, A. (2016) Regularizing priors for linear inverse problems. Econometric Theory 32, 71–121.CrossRefGoogle Scholar
Gajek, L. (1986) On improving density estimators which are not bona fide functions. The Annals of Statistics 14(4), 1612–1618.CrossRefGoogle Scholar
Gautier, E. & Hoderlein, S. (2015) A Triangular Treatment Effect Model with Random Coefficients in the Selection Equation. Technical report, Toulouse School of Economics.Google Scholar
Gautier, E. & Kitamura, Y. (2013) Nonparametric estimation in random coefficients binary choice models. Econometrica 81(2), 581–607.Google Scholar
Hall, P. & Horowitz, J.L. (2005) Nonparametric methods for inference in the presence of instrumental variables. The Annals of Statistics 33(6), 2904–2929.CrossRefGoogle Scholar
Hansen, P. (1988) Computation of the singular value expansion. Computing 40(3), 185–199.CrossRefGoogle Scholar
Heckman, J. & Singer, B. (1984) A method for minimizing the impact of distributional assumptions in econometric models for duration data. Econometrica 52(2), 271–320.Google Scholar
Henry, M., Kitamura, Y., & Salanié, B. (2014) Partial identification of finite mixtures in econometric models. Quantitative Economics 5(1), 123–144.CrossRefGoogle Scholar
Hoderlein, S. (2011) How many consumers are rational? Journal of Econometrics 164(2), 294–309.CrossRefGoogle Scholar
Hoderlein, S., Klemelä, J., & Mammen, E. (2010) Analyzing the random coefficient model nonparametrically. Econometric Theory 26, 804–837.Google Scholar
Hoderlein, S., Nesheim, L., & Simoni, A. (2012) Heterogeneous Euler Equations: A Semiparametric Sructural Approach. Technical report, University College London.Google Scholar
Hohage, T. (2000) Regularization of exponentially ill-posed problems. Numerical Functional Analysis and Optimization 21(3–4), 439–464.CrossRefGoogle Scholar
Horowitz, J.L. (2007) Asymptotic normality of a nonparametric instrumental variables estimator. International Economic Review 48(4), 1329–1349.Google Scholar
Hu, Y. & Schennach, S.M. (2008) Instrumental variable treatment of nonclassical measurement error models. Econometrica 76(1), 195–216.Google Scholar
Ichimura, H. & Thompson, T. (1998) Maximum likelihood estimation of a binary choice model with random coefficients of unknown distribution. Journal of Econometrics 86(2), 269–295.CrossRefGoogle Scholar
Johannes, J., Simoni, A., & Schenk, R. (2015) Adaptive Bayesian estimation in indirect Gaussian sequence space models. ArXiv e-prints 1502.00184.Google Scholar
Kasahara, H. & Shimotsu, K. (2009) Nonparametric identification of finite mixture models of dynamic discrete choices. Econometrica 77(1), 135–175.Google Scholar
Lewbel, A. & Pendakur, K. (2016) Unobserved preference heterogeneity in demand using generalized random coefficients. Journal of Political Economy, forthcoming.Google Scholar
Mammen, E., Rothe, C., & Schienle, M. (2012) Nonparametric regression with nonparametrically generated covariates. The Annals of Statistics 40(2), 1132–1170.CrossRefGoogle Scholar
Mandelbaum, A. & Ruschendorf, L. (1987) Complete and symmetrically complete families of distributions. The Annals of Statistics 15(3), 1229–1244.Google Scholar
Masten, M. (2014) Random Coefficients on Endogenous Variables in Simultaneous Equations Models. Technical report, Cemmap Working paper, CWP01/14.Google Scholar
Matzkin, R.L. (2007) Nonparametric identification. In Heckman, J.J. & Leamer, E.E. (eds.), Handbook of Econometrics, vol. 6, Part B, pp. 5307–5368. Elsevier.CrossRefGoogle Scholar
Morozov, V. (1993) Regularization Methods for Ill-Posed Problems. CRC Press.Google Scholar
Neubauer, A. (1986) Tikhonov Regularization of Ill-Posed Linear Operator Equations on Closed Convex Sets. Johannes Kepler-Universität Linz.Google Scholar
Neubauer, A. (1988) Tikhonov-regularization of ill-posed linear operator equations on closed convex sets. Journal of Approximation Theory 53(3), 304–320.Google Scholar
Newey, W.K. & Powell, J.L. (2003) Instrumental variable estimation of nonparametric models. Econometrica 71(5), 1565–1578.CrossRefGoogle Scholar
Rosenblatt, M. (1969) Conditional probability density and regression estimators. In Shnaiah, P. (ed.), Multivariate Analysis II, pp. 25–31. Academic Press.Google Scholar
van der Vaart, A.W. (1998) Asymptotic Statistics. Cambridge University Press.CrossRefGoogle Scholar
Supplementary material: File

Hoderlein supplementary material S1

Hoderlein supplementary material

Download Hoderlein supplementary material S1(File)
File 53.8 KB
Supplementary material: PDF

Hoderlein supplementary material S2

Hoderlein supplementary material

Download Hoderlein supplementary material S2(PDF)
PDF 288.4 KB