Skip to main content


  • Anne Vanhems (a1) and Ingrid Van Keilegom (a2)

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.

Corresponding author
*Address correspondence to Anne Vanhems, Toulouse Business School, 1 place Jourdain, 31068 Toulouse, France; e-mail:
Ingrid Van Keilegom, ORSTAT, KU Leuven, Naamsestraat 69, 3000 Leuven, Belgium; e-mail:
Hide All

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.

Hide All
Bickel, P.J. & Doksum, K. (1981) An analysis of transformations revisited. Journal of the American Statistical Association 76, 296311.
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, 168191.
Blundell, R., Chen, X., & Kristensen, D. (2007) Semi-nonparametric IV estimation of shape-invariant Engel curves. Econometrica 75, 16131669.
Box, G.E.P. & Cox, D.R. (1964) An analysis of transformations. Journal of the Royal Statistical Society - Series B 26, 211252.
Carroll, R.J. & Ruppert, D. (1988) Transformation and Weighting in Regression. Chapman and Hall.
Chen, X., Linton, O.B., & Van Keilegom, I. (2003) Estimation of semiparametric models when the criterion function is not smooth. Econometrica 71, 15911608.
Chen, X. & Pouzo, D. (2009) Efficient estimation of semiparametric conditional moment models with possibly nonsmooth residuals. Journal of Econometrics 152, 4660.
Cheng, G. (2015) Moment consistency of the exchangeably weighted bootstrap for semiparametric M-estimation. Scandinavian Journal of Statistics 42, 665684.
Cheng, G. & Huang, J.Z. (2010) Bootstrap consistency for general semiparametric M-estimation. Annals of Statistics 38, 28842915.
Cheng, G. & Kosorok, M.R. (2008) General frequentist properties of the posterior profile distribution. Annals of Statistics 36, 18191853.
Cheng, G. & Pillai, N. (2012) Semiparametric Model Based Bootstrap. Working paper.
Chiappori, P.-A., Komunjer, I., & Kristensen, D. (2015) Nonparametric identification and estimation of transformation models. Journal of Econometrics 188, 2239.
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, 118.
Colling, B. & Van Keilegom, I. (2016) Goodness-of-fit tests in semiparametric transformation models. TEST 25, 291308.
Delsol, L. & Van Keilegom, I. (2014) Semiparametric M-estimation with Non-smooth Criterion Functions. Technical report, Available at, DP2011/41.
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, 426443.
Fève, F. & Florens, J.P. (2010) The practice of nonparametric estimation by solving inverse problems: The example of transformation models. Econometrics Journal 13, S1S27.
Florens, J.-P., Johannes, J., & Van Bellegem, S. (2012) Instrumental regression in partially linear models. Econometrics Journal 15, 304324.
Florens, J.P. & Sokullu, S. (2017) Nonparametric estimation of semiparametric transformation models. Econometric Theory 33, 839873.
Härdle, W. & Mammen, E. (1993) Comparing nonparametric versus parametric regression fits. Annals of Statistics 21, 19261947.
Hayashi, F. (2000) Econometrics. Princeton University Press.
Heuchenne, C., Samb, R., & Van Keilegom, I. (2015) Estimating the residual distribution in semiparametric transformation models. Electronic Journal of Statistics 9, 23912419.
Horowitz, J.L. (1996) Semiparametric estimation of a regression model with an unknown transformation of the dependent variable. Econometrica 64, 103137.
Horowitz, J.L. (2001) Nonparametric estimation of a generalized additive model with an unknown link function. Econometrica 69, 499513.
Imbens, G. & Newey, W. (2009) Identification and estimation of triangular simultaneous equations models without additivity. Econometrica 77, 14811512.
Imbens, G.W. & Rubin, D.B. (2015) Causal Inference in Statistics, Social, and Biomedical Sciences: An Introduction. Cambridge University Press.
Jacho-Chavez, D., Lewbel, A., & Linton, O. (2010) Identification and nonparametric estimation of a transformed additively separable model. Journal of Econometrics 156, 392407.
Lee, Y. (2015) Partial Mean Processes with Generated Regressors: Continuous Treatment Effects and Nonseparable Models. Working paper.
Linton, O.B. & Nielsen, J.P. (1995) A kernel method of estimating structured nonparametric regression using marginal integration. Biometrika 82, 93100.
Linton, O., Sperlich, S., & Van Keilegom, I. (2008) Estimation on a semiparametric transformation model. Annals of Statistics 36, 686718.
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, 14431490.
Mammen, E. & Park, B.U. (2005) Bandwidth selection for smooth backfitting in additive models. Annals of Statistics 33, 12601294.
Mammen, E., Rothe, C., & Schienle, M. (2012) Nonparametric regression with nonparametrically generated covariates. Annals of Statistics 40, 11321170.
Mammen, E., Rothe, C., & Schienle, M. (2016) Semiparametric estimation with generated covariates. Econometric Theory 32, 11401177.
Moon, J.M. (2013) Sieve extremum estimation of transformation models. Technical report, UCSD, Working papers.
Neumeyer, N., Noh, H., & Van Keilegom, I. (2016) Heteroscedastic semiparametric transformation models: Estimation and testing for validity. Statistica Sinica 26, 925954.
Newey, W.K., Powell, J.L., & Vella, F. (1999) Nonparametric estimation of triangular simultaneous equation models. Econometrica 67, 565603.
Pakes, A. & Pollard, D. (1989) Simulation and the asymptotics of optimization estimators. Econometrica 57, 10271057.
Rivers, D. & Vuong, Q.H. (1988) Limited information estimators and exogeneity tests for simultaneous probit models. Journal of Econometrics 39, 347366.
Sakia, R.M. (1992) The Box-Cox transformation technique: A review. The Statistician 41, 169178.
Sherman, R. (1994) Maximal inequalities for degenerate U-processes with applications to optimization estimators. Annals of Statistics 22, 439459.
Su, L. & Ullah, A. (2008) Local polynomial estimation of nonparametric simultaneous equations models. Journal of Econometrics 144, 193218.
Van der Vaart, A.W. & Wellner, J.A. (1996) Weak Convergence and Empirical Processes. Springer-Verlag.
Wooldridge, J. (2008) Introductory Econometrics: A Modern Approach. South-Western College Publishing.
Zellner, A. & Revankar, N.S. (1969) Generalized production functions. Review of Economic Studies 36, 241250.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Econometric Theory
  • ISSN: 0266-4666
  • EISSN: 1469-4360
  • URL: /core/journals/econometric-theory
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed