Hostname: page-component-745bb68f8f-kw2vx Total loading time: 0 Render date: 2025-01-16T16:58:13.790Z Has data issue: false hasContentIssue false
  • English
  • Français

NONPARAMETRIC ESTIMATION OF SEMIPARAMETRIC TRANSFORMATION MODELS

Published online by Cambridge University Press:  06 June 2016

Jean-Pierre Florens  and
Senay Sokullu
Jean-Pierre Florens
Affiliation:
Toulouse School of Economics
Senay Sokullu*
Affiliation:
University of Bristol
*
*Address correspondence to Senay Sokullu, Department of Economics, University of Bristol, 8 Woodland Road, Bristol BS8 1TN, United Kingdom; e-mail: Senay.Sokullu@bristol.ac.uk.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we develop a nonparametric estimation technique for semiparametric transformation models of the form: H (Y) = φ(Z) + X′β + U where H,φ are unknown functions, β is an unknown finite-dimensional parameter vector and the variables (Y,Z) are endogenous. Identification of the model and asymptotic properties of the estimator are analyzed under the mean independence assumption between the error term and the instruments. We show that the estimators are consistent, and a $\sqrt N$-convergence rate and asymptotic normality for $\hat \beta$ can be attained. The simulations demonstrate that our nonparametric estimates fit the data well.

Type
ARTICLES
Information
Econometric Theory , Volume 33 , Issue 4 , August 2017 , pp. 839 - 873
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.)

References

REFERENCES

Ai, C. & Chen, X. (2003) Efficient estimation of models with conditional moment restrictions containing unknown functions. Econometrica 71(6), 1795–1843.CrossRefGoogle Scholar
Amemiya, T. (1986) Advanced Econometrics. Basil Blackwell.Google Scholar
Andrews, D.W. (2011) Examples of L 2 -Complete and Boundedly-Complete Distributions. Cowles Foundation Discussion Papers 1801 Cowles Foundation for Research in Economics, Yale University.Google Scholar
Bass, F.M. (1969) A new product growth for model consumer durables. Management Science 15, 215–227.CrossRefGoogle Scholar
Berry, S.T. & Haile, P.A. (2014) Identification in differentiated products markets using market level data. Econometrica 82(5), 1749–1797.Google Scholar
Blundell, R., Chen, X., & Kristensen, D. (2007) Semi-nonparametric IV estimation of shape-invariant engel curves. Econometrica 75(6), 1613–1669.CrossRefGoogle Scholar
Bontemps, C., Simioni, M., & Surry, Y. (2008) Semiparametric hedonic price models: assessing the effects of agricultural nonpoint source pollution. Journal of Applied Econometrics 23(6), 825–842.CrossRefGoogle Scholar
Box, G.E.P. & Cox, D.R. (1964) An analysis of transformations. Journal of the Royal Statistical Society 26(2), 211–252.Google Scholar
Breunig, C. & Johannes, J. (2009) On rate optimal local estimation in nonparametric instrumental regression. Tech. rep. Heidelberg University.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. and Leamer, E. (eds.), Handbook of Econometrics, vol. 6, chap. 77. Elsevier.Google Scholar
Chen, X. & Reiss, M. (2011) On rate optimality for ill-posed inverse problems in econometrics. Econometric Theory 27(03), 497–521.CrossRefGoogle Scholar
Chiappori, P.A., Komunjer, I., & Kristensen, D. (2011) Nonparametric Identification and Estimation of Transformation Models. CAM Working Papers, 2011-01, University of Copenhagen, Department of Economics, Centre for Applied Microeconometrics.Google Scholar
Darolles, S., Fan, Y., Florens, J.P., & Renault, E. (2011) Nonparametric instrumental regression. Econometrica 79(5), 1541–1565.Google Scholar
Das, M. (2005) Instrumental variables estimators of nonparametric models with discrete endogenous regressors. Journal of Econometrics 124(2), 335–361.CrossRefGoogle Scholar
D’ Haultfoeuille, X. (2011) On the completeness condition in nonparametric instrumental problems. Econometric Theory 27(03), 460–471.CrossRefGoogle Scholar
Engl, H.W., Hanke, M., & Neubauer, A. (1996) Regularization of Inverse Problems. Kluwer Academic Publications.CrossRefGoogle Scholar
Engle, R.F., Granger, C.W.J., Rice, J., & Weiss, A. (1986) Semiparametric estimates of the relation between weather and electricity sales. Journal of the American Statistical Association 81(394), 310–20.CrossRefGoogle Scholar
Feve, F. & Florens, J.P. (2010) The practice of non parametric estimation by solving inverse problems: The example of transformation models. Econometrics Journal 13(3), 1–27.CrossRefGoogle Scholar
Florens, J.P., Heckman, J., Meghir, C., & Vytlacil, E. (2002) Instrumental variables, local instrumental variables and control functions. Tech. rep. Centre for Microdata Methods and Practice, Institute for Fiscal Studies. CeMMAP working papers. CWP15/02. https://ideas.repec.org/p/ifs/cemmap/15-02.html.CrossRefGoogle Scholar
Florens, J.P., Heckman, J.J., Meghir, C., & Vytlacil, E. (2008) Identification of treatment effects using control functions in models with continuous, endogenous treatment and heterogeneous effects. Econometrica 76(5), 1191–1206.Google Scholar
Florens, J.P., Johannes, J., & Bellegem, S.V. (2012) Instrumental regression in partially linear models. Econometrics Journal 15(2), 304–324.CrossRefGoogle Scholar
Florens, J.P., Mouchart, M., & Rolin, J.M. (1990) Elements of Bayesian Statistics. M. Dekker.Google Scholar
Freyberger, J. & Horowitz, J. (2013) Identification and shape restrictions in nonparametric insturmental variables estimation. Tech. rep. Centre for Microdata Methods and Practice, Institute for Fiscal Studies. CeMMAP working papers. CWP31/13. https://ideas.repec.org/p/ifs/cemmap/31-13.html.CrossRefGoogle Scholar
Hall, P. & Horowitz, J.L. (2005) Nonparametric methods for inference in the presence of instrumental variables. Annals of Statistics 32, 2904–2929.Google Scholar
Hansen, B.E. (2008) Uniform convergence rates for kernel estimation with dependent data. Econometric Theory 24(03), 726–748.CrossRefGoogle Scholar
Horowitz, J.L. (1996) Semiparametric estimation of a regression model with an unknown transformation of the dependent variable. Econometrica 64, 103–137.CrossRefGoogle Scholar
Horowitz, J.L. (2011) Applied nonparametric instrumental variables estimation. Econometrica 79(2), 347–397.Google Scholar
Hu, Y. & Schennach, S.M. (2008) Instrumental variable treatment of nonclassical measurement error models. Econometrica 76(1), 195–216.CrossRefGoogle Scholar
Hu, Y. & Shiu, J.L. (2011) Nonparametric Identification Using Instrumental Variables: Sufficient Conditions for Completeness. CeMMAP working papers CWP25/11 Centre for Microdata Methods and Practice, Institute for Fiscal Studies.Google Scholar
Ichimura, H. & Todd, P.E. (2007) Implementing nonparametric and semiparametric estimators. In Heckman, J. and Leamer, E. (eds.), Handbook of Econometrics, vol. 6, chap. 74, Elsevier.Google Scholar
Kaiser, U. & Song, M. (2009) Do media consumers really dislike advertising? an empirical assessment of the role of advertising in print media markets. International Journal of Industrial Organization 27(2), 292–301.CrossRefGoogle Scholar
Kaiser, U. & Wright, J. (2006) Price structure in two-sided markets: Evidence from the magazine industry. International Journal of Industrial Organization 24(1), 1–28.CrossRefGoogle Scholar
Linton, O., Sperlich, S., & Keilegom, I.V. (2008) Estimation of a semiparametric transformation model. Annals of Statistics 36(2), 686–718.CrossRefGoogle Scholar
Morozov, V.A. (1993) Regularization Methods for Ill-Posed Problems. CRC Press.Google Scholar
Newey, W.K. & Powell, J.L. (2003) Instrumental variable estimation of nonparametric models. Econometrica 71(5), 1565–1578.CrossRefGoogle Scholar
Pollard, D. (1984) Convergence of Stochastic Processes. Springer-Verlag.CrossRefGoogle Scholar
Rothe, C. (2010) Nonparametric estimation of distributional policy effects. Journal of Econometrics 155(1), 56–70.CrossRefGoogle Scholar
Santos, A. (2012) Inference in nonparametric instrumental variables with partial identification. Econometrica 80, 213–275.Google Scholar
Sokullu, S. (2015) A semi-parametric analysis of two-sided markets: An application to the local daily newspapers in the U.S. Journal of Applied Econometrics. to appear. doi: 10.1002/jae.2464.CrossRefGoogle Scholar