Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-01T08:12:50.515Z Has data issue: false hasContentIssue false
  • English
  • Français

ESTIMATION OF A SEMIPARAMETRIC TRANSFORMATION MODEL IN THE PRESENCE OF ENDOGENEITY

Published online by Cambridge University Press:  07 May 2018

Anne Vanhems  and
Ingrid Van Keilegom
Anne Vanhems*
Affiliation:
Toulouse Business School
Ingrid Van Keilegom*
Affiliation:
KU Leuven
*
*Address correspondence to Anne Vanhems, Toulouse Business School, 1 place Jourdain, 31068 Toulouse, France; e-mail: [email protected]
†Ingrid Van Keilegom, ORSTAT, KU Leuven, Naamsestraat 69, 3000 Leuven, Belgium; e-mail: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a semiparametric transformation model, in which the regression function has an additive nonparametric structure and the transformation of the response is assumed to belong to some parametric family. We suppose that endogeneity is present in the explanatory variables. Using a control function approach, we show that the proposed model is identified under suitable assumptions, and propose a profile estimation method for the transformation. The proposed estimator is shown to be asymptotically normal under certain regularity conditions. A simulation study shows that the estimator behaves well in practice. Finally, we give an empirical example using the U.K. Family Expenditure Survey.

Type
ARTICLES
Information
Econometric Theory , Volume 35 , Issue 1 , February 2019 , pp. 73 - 110
Copyright
Copyright © Cambridge University Press 2018 

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 deeply thank Ying-Ying Lee for pointing out some incoherencies in a previous version of our article and for most stimulating discussions on the impact of a generated covariate on the asymptotic variance of our estimator. This research was supported by the European Research Council (2016–2021, Horizon 2020/ERC grant agreement No. 694409 and 295298), and by IAP Research Network P7/06 of the Belgian State.

References

REFERENCES

Bickel, P.J. & Doksum, K. (1981) An analysis of transformations revisited. Journal of the American Statistical Association 76, 296–311.CrossRefGoogle Scholar
Birke, M., Van Bellegem, S., & Van Keilegom, I. (2017) Semi-parametric estimation in a single-index model with endogenous variables. Scandinavian Journal of Statistics 44, 168–191.CrossRefGoogle Scholar
Blundell, R., Chen, X., & Kristensen, D. (2007) Semi-nonparametric IV estimation of shape-invariant Engel curves. Econometrica 75, 1613–1669.CrossRefGoogle Scholar
Box, G.E.P. & Cox, D.R. (1964) An analysis of transformations. Journal of the Royal Statistical Society - Series B 26, 211–252.Google Scholar
Carroll, R.J. & Ruppert, D. (1988) Transformation and Weighting in Regression. Chapman and Hall.CrossRefGoogle Scholar
Chen, X., Linton, O.B., & Van Keilegom, I. (2003) Estimation of semiparametric models when the criterion function is not smooth. Econometrica 71, 1591–1608.CrossRefGoogle Scholar
Chen, X. & Pouzo, D. (2009) Efficient estimation of semiparametric conditional moment models with possibly nonsmooth residuals. Journal of Econometrics 152, 46–60.CrossRefGoogle Scholar
Cheng, G. (2015) Moment consistency of the exchangeably weighted bootstrap for semiparametric M-estimation. Scandinavian Journal of Statistics 42, 665–684.CrossRefGoogle Scholar
Cheng, G. & Huang, J.Z. (2010) Bootstrap consistency for general semiparametric M-estimation. Annals of Statistics 38, 2884–2915.CrossRefGoogle Scholar
Cheng, G. & Kosorok, M.R. (2008) General frequentist properties of the posterior profile distribution. Annals of Statistics 36, 1819–1853.CrossRefGoogle Scholar
Cheng, G. & Pillai, N. (2012) Semiparametric Model Based Bootstrap. Working paper.Google Scholar
Chiappori, P.-A., Komunjer, I., & Kristensen, D. (2015) Nonparametric identification and estimation of transformation models. Journal of Econometrics 188, 22–39.CrossRefGoogle Scholar
Colling, B., Heuchenne, C., Samb, R., & Van Keilegom, I. (2015) Estimation of the error density in a semiparametric transformation model. Annals of the Institute of Statistical Mathematics 67, 1–18.CrossRefGoogle Scholar
Colling, B. & Van Keilegom, I. (2016) Goodness-of-fit tests in semiparametric transformation models. TEST 25, 291–308.CrossRefGoogle Scholar
Delsol, L. & Van Keilegom, I. (2014) Semiparametric M-estimation with Non-smooth Criterion Functions. Technical report, Available at http://www.uclouvain.be/en-369695.html, DP2011/41.Google Scholar
Escanciano, J.C., Jacho-Chvez, D.T., & Lewbel, A. (2014) Uniform convergence of weighted sums of non and semiparametric residuals for estimation and testing. Journal of Econometrics 178, 426–443.CrossRefGoogle Scholar
Fève, F. & Florens, J.P. (2010) The practice of nonparametric estimation by solving inverse problems: The example of transformation models. Econometrics Journal 13, S1–S27.CrossRefGoogle Scholar
Florens, J.-P., Johannes, J., & Van Bellegem, S. (2012) Instrumental regression in partially linear models. Econometrics Journal 15, 304–324.CrossRefGoogle Scholar
Florens, J.P. & Sokullu, S. (2017) Nonparametric estimation of semiparametric transformation models. Econometric Theory 33, 839–873.CrossRefGoogle Scholar
Härdle, W. & Mammen, E. (1993) Comparing nonparametric versus parametric regression fits. Annals of Statistics 21, 1926–1947.CrossRefGoogle Scholar
Hayashi, F. (2000) Econometrics. Princeton University Press.Google Scholar
Heuchenne, C., Samb, R., & Van Keilegom, I. (2015) Estimating the residual distribution in semiparametric transformation models. Electronic Journal of Statistics 9, 2391–2419.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. (2001) Nonparametric estimation of a generalized additive model with an unknown link function. Econometrica 69, 499–513.CrossRefGoogle Scholar
Imbens, G. & Newey, W. (2009) Identification and estimation of triangular simultaneous equations models without additivity. Econometrica 77, 1481–1512.Google Scholar
Imbens, G.W. & Rubin, D.B. (2015) Causal Inference in Statistics, Social, and Biomedical Sciences: An Introduction. Cambridge University Press.CrossRefGoogle Scholar
Jacho-Chavez, D., Lewbel, A., & Linton, O. (2010) Identification and nonparametric estimation of a transformed additively separable model. Journal of Econometrics 156, 392–407.CrossRefGoogle Scholar
Lee, Y. (2015) Partial Mean Processes with Generated Regressors: Continuous Treatment Effects and Nonseparable Models. Working paper.Google Scholar
Linton, O.B. & Nielsen, J.P. (1995) A kernel method of estimating structured nonparametric regression using marginal integration. Biometrika 82, 93–100.CrossRefGoogle Scholar
Linton, O., Sperlich, S., & Van Keilegom, I. (2008) Estimation on a semiparametric transformation model. Annals of Statistics 36, 686–718.CrossRefGoogle Scholar
Mammen, E., Linton, O.B., & Nielsen, J.P. (1999) The existence and asymptotic properties of a backfitting projection algorithm under weak conditions. Annals of Statistics 27, 1443–1490.Google Scholar
Mammen, E. & Park, B.U. (2005) Bandwidth selection for smooth backfitting in additive models. Annals of Statistics 33, 1260–1294.CrossRefGoogle Scholar
Mammen, E., Rothe, C., & Schienle, M. (2012) Nonparametric regression with nonparametrically generated covariates. Annals of Statistics 40, 1132–1170.CrossRefGoogle Scholar
Mammen, E., Rothe, C., & Schienle, M. (2016) Semiparametric estimation with generated covariates. Econometric Theory 32, 1140–1177.CrossRefGoogle Scholar
Moon, J.M. (2013) Sieve extremum estimation of transformation models. Technical report, UCSD, Working papers.Google Scholar
Neumeyer, N., Noh, H., & Van Keilegom, I. (2016) Heteroscedastic semiparametric transformation models: Estimation and testing for validity. Statistica Sinica 26, 925–954.Google Scholar
Newey, W.K., Powell, J.L., & Vella, F. (1999) Nonparametric estimation of triangular simultaneous equation models. Econometrica 67, 565–603.CrossRefGoogle Scholar
Pakes, A. & Pollard, D. (1989) Simulation and the asymptotics of optimization estimators. Econometrica 57, 1027–1057.CrossRefGoogle Scholar
Rivers, D. & Vuong, Q.H. (1988) Limited information estimators and exogeneity tests for simultaneous probit models. Journal of Econometrics 39, 347–366.CrossRefGoogle Scholar
Sakia, R.M. (1992) The Box-Cox transformation technique: A review. The Statistician 41, 169–178.CrossRefGoogle Scholar
Sherman, R. (1994) Maximal inequalities for degenerate U-processes with applications to optimization estimators. Annals of Statistics 22, 439–459.CrossRefGoogle Scholar
Su, L. & Ullah, A. (2008) Local polynomial estimation of nonparametric simultaneous equations models. Journal of Econometrics 144, 193–218.CrossRefGoogle Scholar
Van der Vaart, A.W. & Wellner, J.A. (1996) Weak Convergence and Empirical Processes. Springer-Verlag.CrossRefGoogle Scholar
Wooldridge, J. (2008) Introductory Econometrics: A Modern Approach. South-Western College Publishing.Google Scholar
Zellner, A. & Revankar, N.S. (1969) Generalized production functions. Review of Economic Studies 36, 241–250.CrossRefGoogle Scholar