Skip to main content


  • Jinyong Hahn (a1), Zhipeng Liao (a1) and Geert Ridder (a2)

This article studies two-step sieve M estimation of general semi/nonparametric models, where the second step involves sieve estimation of unknown functions that may use the nonparametric estimates from the first step as inputs, and the parameters of interest are functionals of unknown functions estimated in both steps. We establish the asymptotic normality of the plug-in two-step sieve M estimate of a functional that could be root-n estimable. The asymptotic variance may not have a closed form expression, but can be approximated by a sieve variance that characterizes the effect of the first-step estimation on the second-step estimates. We provide a simple consistent estimate of the sieve variance, thereby facilitating Wald type inferences based on the Gaussian approximation. The finite sample performance of the two-step estimator and the proposed inference procedure are investigated in a simulation study.

Corresponding author
*Address correspondence to Zhipeng Liao, Department of Economics, UCLA, Los Angeles, CA 90095-1477, USA; e-mail:
Hide All

We gratefully acknowledge insightful comments from Xiaohong Chen, who was a co-author of the initial version. We appreciate useful suggestions from Liangjun Su, the coeditor and three anonymous referees. The outstanding editorial input by the Editor, Professor Phillips, in our last version of the manuscript is greatly appreciated. All errors are the responsibility of the authors.

Hide All
Ackerberg, D., Chen, X., & Hahn, J. (2012) A practical asymptotic variance estimator for two-step semiparametric estimators. Review of Economics and Statistics 94, 481498.
Altonji, J. & Matzkin, R. (2005) Cross section and panel data estimators for nonseparable models with endogenous regressors. Econometrica 73, 10531102.
Blundell, R. & Powell, J.L. (2004) Endogeneity in semiparametric binary response models. Review of Economic Studies 71, 655679.
Blundell, R. & Powell, J.L. (2007) Censored regression quantiles with endogenous regressors. Journal of Econometrics 141, 6583.
Chen, X. (2007) Large sample sieve estimation of semi-nonparametric models. In Heckman, J.J. & Leamer, E.E. (eds.), Handbook of Econometrics, vol. 6B, pp. 55495632. Elsevier.
Chen, X. & Liao, Z. (2014) Sieve M inference of irregular parameters. Journal of Econometrics 182(1), 7086.
Chen, X., Liao, Z., & Sun, Y. (2014) Sieve inference on possibly misspecified semi-nonparametric time series models. Journal of Econometrics 178(3), 639658.
Chen, X., Linton, O., & van Keilegom, I. (2003) Estimation of semiparametric models when the criterion function is not smooth. Econometrica 71, 15911608.
Chen, X. & Shen, X. (1998) Sieve extremum estimates for weakly dependent data. Econometrica 66, 289314.
Chernozhukov, V., Escanciano, J., Ichimura, H., & Newey, W. (2016) Locally Robust Semiparametric Estimation. Working paper, MIT.
Chesher, A. (2003) Identification in nonseparable models. Econometrica 71(5), 14051414
Das, M., Newey, W., & Vella, F. (2003) Nonparametric estimation of sample selection models. Review of Economic Studies 70, 3358.
Escanciano, J., Jacho-Chávez, D., & Lewbel, A. (2014) Uniform convergence of weighted sums of non and semiparametric residuals for estimation and testing. Journal of Econometrics 178, 426443.
Florens, J., Heckman, J., Meghir, C., & Vytlacil, E. (2008) Identification of treatment effects using control functions in models with continuous, endogenous treatment and heterogeneous effects. Econometrica 76, 11911206.
Hahn, J. & Ridder, G. (2013) The asymptotic variance of semi-parametric estimators with generated regressors. Econometrica 81, 315340.
Ichimura, H. & Lee, S. (2010) Characterization of the asymptotic distribution of semiparametric M estimators. Journal of Econometrics 159, 252266.
Imbens, G. & Newey, W. (2009) Identification and estimation of triangular simultaneous equations models without additivity. Econometrica 77, 14811512.
Lee, S. (2007) Endogeneity in quantile regression models: A control function approach. Journal of Econometrics 141, 11311158.
Lee, Y. (2015) Partial Mean Processes with Generated Regressors: Continuous Treatment Effects and Nonseparable Models. Working paper, UC-Irvine.
Lewbel, A. & Linton, O. (2002) Nonparametric censored and truncated regression. Econometrica 70, 765779.
Li, Q. & Wooldridge, M. (2002) Semiparametric estimation of partially linear models for dependent data with generated regressors. Econometric Theory 18, 625645.
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(5), 11401177.
Murphy, K. & Topel, R. (1985) Estimation and inference in two-step econometric models. Journal of Business and Economic Statistics 3, 370379.
Newey, W. (1984) A method of moments interpretation of sequential estimators. Economics Letters 14, 201206.
Newey, W. (1994) The asymptotic variance of semiparametric estimators. Econometrica 62, 13491382.
Newey, W. (2009) Two-step series estimation of sample selection models. Econometrics Journal 12, S217S229.
Newey, W., Powell, J., & Vella, F. (1999) Nonparametric estimation of triangular simultaneous equations models. Econometrica 67, 565603.
Olley, G. & Pakes, A. (1996) The dynamics of productivity in the telecommunications equipment industry. Econometrica 64, 12631297.
Shen, X. (1997) On methods of sieves and penalization. Annals of Statistics 25, 25552591.
Wooldridge, J.M. (2002) Econometric Analysis of Cross Section and Panel Data. MIT Press.
Wooldridge, J.M. (2015) Control function methods in applied econometrics. Journal of Human Resources 50, 420445.
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? *
Type Description Title
Supplementary materials

Hahn et al. supplementary material
Hahn et al. supplementary material 1

 Unknown (245 KB)
245 KB
Supplementary materials

Hahn et al. supplementary material
Hahn et al. supplementary material 2

 Unknown (244 KB)
244 KB
Supplementary materials

Hahn et al. supplementary material
Hahn et al. supplementary material 3

 PDF (1.3 MB)
1.3 MB
Supplementary materials

Hahn et al. supplementary material
Hahn et al. supplementary material 4

 Unknown (234 KB)
234 KB


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