Hostname: page-component-797576ffbb-xmkxb Total loading time: 0 Render date: 2023-12-04T19:54:16.074Z Has data issue: false Feature Flags: { "corePageComponentGetUserInfoFromSharedSession": true, "coreDisableEcommerce": false, "useRatesEcommerce": true } hasContentIssue false


Published online by Cambridge University Press:  07 November 2022

Ruoyao Shi*
University of California, Riverside
Address correspondence to Ruoyao Shi, Department of Economics, University of California, Riverside, Riverside, CA, USA; e-mail:


In a two-step extremum estimation (M-estimation) framework with a finite-dimensional parameter of interest and a potentially infinite-dimensional first-step nuisance parameter, this paper proposes an averaging estimator that combines a semiparametric estimator based on a nonparametric first step and a parametric estimator which imposes parametric restrictions on the first step. The averaging weight is an easy-to-compute sample analog of an infeasible optimal weight that minimizes the asymptotic quadratic risk. Under Stein-type conditions, the asymptotic lower bound of the truncated quadratic risk difference between the averaging estimator and the semiparametric estimator is strictly less than zero for a class of data generating processes that includes both correct specification and varied degrees of misspecification of the parametric restrictions, and the asymptotic upper bound is weakly less than zero. The averaging estimator, along with an easy-to-implement inference method, is demonstrated in an example.

© The Author(s), 2022. Published by Cambridge University Press

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.)


The comments from the Editor (Peter C.B. Phillips), the Associate Editor (Patrik Guggenberger), and two anonymous referees were vastly helpful in improving this paper. The author also thanks Colin Cameron, Xu Cheng, Denis Chetverikov, Yanqin Fan, Jinyong Hahn, Bo Honoré, Toru Kitagawa, Zhipeng Liao, Hyungsik Roger Moon, Whitney Newey, Geert Ridder, Aman Ullah, Haiqing Xu, and the participants at various seminars and conferences for helpful comments. This project was generously supported by UC Riverside Regents’ Faculty Fellowship 2019–2020. Zhuozhen Zhao provided great research assistance. All remaining errors are the author’s.



Ackerberg, D., Chen, X., & Hahn, J. (2012) A practical asymptotic variance estimator for two-step semiparametric estimators. Review of Economics and Statistics 94(2), 481P498.CrossRefGoogle Scholar
Ackerberg, D., Chen, X., Hahn, J., & Liao, Z. (2014) Asymptotic efficiency of semiparametric two-step GMM. Review of Economic Studies 81(3), 919943.CrossRefGoogle Scholar
Ahn, H., Ichimura, H., & Powell, J.L. (1996) Simple estimators for monotone index models. Manuscript, Department of Economics, UC Berkeley.Google Scholar
Ahn, H. & Powell, J.L. (1993) Semiparametric estimation of censored selection models with a nonparametric selection mechanism. Journal of Econometrics 58(1–2), 329.CrossRefGoogle Scholar
Altonji, J.G., Elder, T.E., & Taber, C.R. (2005) Selection on observed and unobserved variables: Assessing the effectiveness of catholic schools. Journal of Political Economy 113(1), 151184.CrossRefGoogle Scholar
Andrews, D.W. (1994) Asymptotics for semiparametric econometric models via stochastic equicontinuity. Econometrica 62(1), 4372.Google Scholar
Andrews, D.W., Cheng, X., & Guggenberger, P. (2020) Generic results for establishing the asymptotic size of confidence sets and tests. Journal of Econometrics 218(2), 496531.CrossRefGoogle Scholar
Andrews, D.W. & Guggenberger, P. (2009) Validity of subsampling and “plug-in asymptotic” inference for parameters defined by moment inequalities. Econometric Theory 25(3), 669709.CrossRefGoogle Scholar
Andrews, D.W. & Guggenberger, P. (2010) Asymptotic size and a problem with subsampling and with the m out of n bootstrap. Econometric Theory 26(2), 426468.Google Scholar
Andrews, I., Gentzkow, M., & Shapiro, J.M. (2017) Measuring the sensitivity of parameter estimates to estimation moments. The Quarterly Journal of Economics 132(4), 15531592.CrossRefGoogle Scholar
Armstrong, T.B. & Kolesár, M. (2021) Sensitivity analysis using approximate moment condition models. Quantitative Economics 12(1), 77108.CrossRefGoogle Scholar
Bang, H. & Robins, J.M. (2005) Doubly robust estimation in missing data and causal inference models. Biometrics 61(4), 962973.CrossRefGoogle ScholarPubMed
Bickel, P.J., Klaassen, C.A., Ritov, J., & Wellner, J.A. (1993) Efficient and Adaptive Estimation for Semiparametric Models . Johns Hopkins University Press.Google Scholar
Bickel, P.J. & Ritov, Y. (2003) Nonparametric estimators which can be “plugged-in”. Annals of Statistics 31(4), 10331053.CrossRefGoogle Scholar
Bierens, H.J. (1990) A consistent conditional moment test of functional form. Econometrica 58(6), 14431458.CrossRefGoogle Scholar
Blundell, R. & Powell, J.L. (2003) Endogeneity in nonparametric and semiparametric regression models. In Advances in Economics and Econometrics . Cambridge University Press.Google Scholar
Blundell, R.W. & Powell, J.L. (2004) Endogeneity in semiparametric binary response models. The Review of Economic Studies 71(3), 655679.CrossRefGoogle Scholar
Bonhomme, S. & Weidner, M. (2021) Minimizing sensitivity to model misspecification. Preprint, arXiv:1807.02161.Google Scholar
Buchholz, N., Shum, M., & Xu, H. (2021) Semiparametric estimation of dynamic discrete choice models. Journal of Econometrics 223(2), 312327.CrossRefGoogle Scholar
Cao, W., Tsiatis, A.A., & Davidian, M. (2009) Improving efficiency and robustness of the doubly robust estimator for a population mean with incomplete data. Biometrika 96(3), 723734.CrossRefGoogle ScholarPubMed
Chen, X., Linton, O., & Van Keilegom, I. (2003) Estimation of semiparametric models when the criterion function is not smooth. Econometrica 71(5), 15911608.CrossRefGoogle Scholar
Cheng, X., Liao, Z., & Shi, R. (2019) On uniform asymptotic risk of averaging GMM estimators. Quantitative Economics 10(3), 931979.Google Scholar
Chernozhukov, V., Chetverikov, D., Demirer, M., Duflo, E., Hansen, C., Newey, W., & Robins, J. (2018) Double/debiased machine learning for treatment and structural parameters. The Econometrics Journal 21(1), C1C68.CrossRefGoogle Scholar
Chernozhukov, V., Escanciano, J.C., Ichimura, H., Newey, W.K., & Robins, J.M. (2022) Locally robust semiparametric estimation. Econometrica 90(4), 15011535.CrossRefGoogle Scholar
Claeskens, G. & Hjort, N.L. (2008) Model Selection and Model Averaging . Cambridge University Press.Google Scholar
Crepon, B., Kramarz, F., & Trognon, A. (1997) Parameters of interest, nuisance parameters and orthogonality conditions. An application to autoregressive error component models. Journal of Econometrics 82(1), 135156.Google Scholar
DiTraglia, F.J. (2016) Using invalid instruments on purpose: Focused moment selection and averaging for GMM. Journal of Econometrics 195(2), 187208.CrossRefGoogle Scholar
Donald, S.G. & Newey, W.K. (1994) Series estimation of semilinear models. Journal of Multivariate Analysis 50(1), 3040.CrossRefGoogle Scholar
Fan, Y. & Ullah, A. (1999) Asymptotic normality of a combined regression estimator. Journal of Multivariate Analysis 71(2), 191240.CrossRefGoogle Scholar
Fessler, P. & Kasy, M. (2019) How to use economic theory to improve estimators: Shrinking toward theoretical restrictions. Review of Economics and Statistics 101(4), 681698.CrossRefGoogle Scholar
Firpo, S. (2007) Efficient semiparametric estimation of quantile treatment effects. Econometrica 75(1), 259276.CrossRefGoogle Scholar
Fourdrinier, D., Strawderman, W.E., & Wells, M.T. (2018) Shrinkage Estimation . Springer.CrossRefGoogle Scholar
Gallant, A.R. & Nychka, D.W. (1987) Semi-nonparametric maximum likelihood estimation. Econometrica 55(2), 363390.CrossRefGoogle Scholar
Hahn, J. & Liao, Z. (2021) Bootstrap standard error estimates and inference. Econometrica 89(4), 19631977.CrossRefGoogle Scholar
Han, A.K. (1987) Non-parametric analysis of a generalized regression model: The maximum rank correlation estimator. Journal of Econometrics 35(2–3), 303316.CrossRefGoogle Scholar
Hansen, B.E. (2007) Least squares model averaging. Econometrica 75(4), 11751189.CrossRefGoogle Scholar
Hansen, B.E. (2014) Model averaging, asymptotic risk, and regressor groups. Quantitative Economics 5(3), 495530.CrossRefGoogle Scholar
Hansen, B.E. (2016) Efficient shrinkage in parametric models. Journal of Econometrics 190(1), 115132.CrossRefGoogle Scholar
Hansen, B.E. (2017) Stein-like 2SLS estimator. Econometric Reviews 36(6–9), 840852.CrossRefGoogle Scholar
Hansen, B.E. & Racine, J.S. (2012) Jackknife model averaging. Journal of Econometrics 167(1), 3846.CrossRefGoogle Scholar
Heckman, J.J. (1976) The common structure of statistical models of truncation, sample selection and limited dependent variables and a simple estimator for such models. In Annals of Economic and Social Measurement , vol. 5, pp. 475492. National Bureau of Economic Research.Google Scholar
Heckman, J.J. (1979) Sample selection bias as a specification error. Econometrica 47(1), 153161.CrossRefGoogle Scholar
Hirano, K., Imbens, G.W., & Ridder, G. (2003) Efficient estimation of average treatment effects using the estimated propensity score. Econometrica 71(4), 11611189.CrossRefGoogle Scholar
Hjort, N.L. & Claeskens, G. (2003) Frequentist model average estimators. Journal of the American Statistical Association 98(464), 879899.CrossRefGoogle Scholar
Hjort, N.L. & Claeskens, G. (2006) Focused information criteria and model averaging for the Cox hazard regression model. Journal of the American Statistical Association 101(476), 14491464.CrossRefGoogle Scholar
Honoré, B.E. (1992) Trimmed LAD and least squares estimation of truncated and censored regression models with fixed effects. Econometrica 60(3), 533565.CrossRefGoogle Scholar
Hotz, V.J. & Miller, R.A. (1993) Conditional choice probabilities and the estimation of dynamic models. The Review of Economic Studies 60(3), 497529.CrossRefGoogle Scholar
Ichimura, H. & Lee, S. (2010) Characterization of the asymptotic distribution of semiparametric M-estimators. Journal of Econometrics 159(2), 252266.Google Scholar
Ichimura, H. & Newey, W. (2017) The Influence Function of Semiparametric Estimators. CEMMAP Working paper CWP06/17, The Institute for Fiscal Studies, Department of Economics, University College London.Google Scholar
Imbens, G.W. (2003) Sensitivity to exogeneity assumptions in program evaluation. American Economic Review 93(2), 126132.CrossRefGoogle Scholar
James, W. & Stein, C. (1961) Estimation with quadratic loss. In Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability , vol. 1, pp. 361379. University of California Press.Google Scholar
Judge, G.G. & Mittelhammer, R.C. (2004) A semiparametric basis for combining estimation problems under quadratic loss. Journal of the American Statistical Association 99(466), 479487.Google Scholar
Judge, G.G. & Mittelhammer, R.C. (2007) Estimation and inference in the case of competing sets of estimating equations. Journal of Econometrics 138(2), 513531.CrossRefGoogle Scholar
Keane, M.P. & Wolpin, K.I. (1997) The career decisions of young men. Journal of Political Economy 105(3), 473522.CrossRefGoogle Scholar
Kitagawa, T. & Muris, C. (2016) Model averaging in semiparametric estimation of treatment effects. Journal of Econometrics 193(1), 271289.Google Scholar
Klein, R.W. & Spady, R.H. (1993) An efficient semiparametric estimator for binary response models. Econometrica 61(2), 387421.CrossRefGoogle Scholar
Le Cam, L. (1972) Limits of experiments. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability , vol. 1, pp. 245261. University of California Press.Google Scholar
Leamer, E.E. (1985) Sensitivity analyses would help. The American Economic Review 75(3), 308313.Google Scholar
Lee, L.-F. (1982) Some approaches to the correction of selectivity bias. The Review of Economic Studies 49(3), 355372.CrossRefGoogle Scholar
Leeb, H. & Pötscher, B.M. (2005) Model selection and inference: Facts and fiction. Econometric Theory 21(1), 2159.CrossRefGoogle Scholar
Leeb, H. & Pötscher, B.M. (2008) Sparse estimators and the oracle property, or the return of Hodges’ estimator. Journal of Econometrics 142(1), 201211.CrossRefGoogle Scholar
Liu, C.-A. (2015) Distribution theory of the least squares averaging estimator. Journal of Econometrics 186(1), 142159.Google Scholar
Lu, X. & Su, L. (2015) Jackknife model averaging for quantile regressions. Journal of Econometrics 188(1), 4058.CrossRefGoogle Scholar
Magnus, J.R., Powell, O., & Prüfer, P. (2010) A comparison of two model averaging techniques with an application to growth empirics. Journal of Econometrics 154(2), 139153.Google Scholar
Mittelhammer, R.C. & Judge, G.G. (2005) Combining estimators to improve structural model estimation and inference under quadratic loss. Journal of Econometrics 128(1), 129.CrossRefGoogle Scholar
Mukhin, Y. (2018) Sensitivity of regular estimators. Preprint, arXiv:1805.08883.Google Scholar
Nelson, F.D. (1984) Efficiency of the two-step estimator for models with endogenous sample selection. Journal of Econometrics 24, 181196.CrossRefGoogle Scholar
Newey, W.K. (1990) Semiparametric efficiency bounds. Journal of Applied Econometrics 5(2), 99135.CrossRefGoogle Scholar
Newey, W.K. (1994) The asymptotic variance of semiparametric estimators. Econometrica 62(6), 13491382.CrossRefGoogle Scholar
Newey, W.K. (2009) Two-step series estimation of sample selection models. The Econometrics Journal 12, S217S229.CrossRefGoogle Scholar
Newey, W.K. & McFadden, D. (1994) Large sample estimation and hypothesis testing. In Handbook of Econometrics , vol. 4, pp. 21112245. Elsevier.Google Scholar
Newey, W.K. & Powell, J.L. (1993) Efficiency bounds for some semiparametric selection models. Journal of Econometrics 58(1–2), 169184.CrossRefGoogle Scholar
Newey, W.K. & Powell, J.L. (1999) Two-Step Estimation, Optimal Moment Conditions, and Sample Selection Models. Working paper 99-06, Department of Economics, Massachusetts Institute of Technology.Google Scholar
Newey, W.K., Powell, J.L., & Walker, J.R. (1990) Semiparametric estimation of selection models: Some empirical results. The American Economic Review 80(2), 324328.Google Scholar
Neyman, J. (1959) Optimal asymptotic tests of composite hypotheses. In Probability and Statsitics , pp. 213234. Wiley.Google Scholar
Oster, E. (2019) Unobservable selection and coefficient stability: Theory and evidence. Journal of Business & Economic Statistics 37(2), 187204.CrossRefGoogle Scholar
Pakes, A. & Olley, S. (1995) A limit theorem for a smooth class of semiparametric estimators. Journal of Econometrics 65(1), 295332.CrossRefGoogle Scholar
Peng, J. & Yang, Y. (2022) On improvability of model selection by model averaging. Journal of Econometrics 229(2), 246262.CrossRefGoogle Scholar
Powell, J.L. (1986) Symmetrically trimmed least squares estimation for Tobit models. Econometrica 54(6), 14351460.CrossRefGoogle Scholar
Powell, J.L. (1994) Estimation of semiparametric models. Handbook of Econometrics 4, 24432521.CrossRefGoogle Scholar
Powell, J.L. (2001) Semiparametric estimation of censored selection models. In Hsiao, C., Morimune, K., & Powell, J. (eds.), Nonlinear Statistical Modeling: Proceedings of the Thirteenth International Symposium in Economic Theory and Econometrics: Essays in Honor of Takeshi Amemiya , vol. 13, pp. 165196. Cambridge University Press.CrossRefGoogle Scholar
Robinson, P.M. (1988) Root-N-consistent semiparametric regression. Econometrica 56(4), 931954.CrossRefGoogle Scholar
Robinson, P.M. (1989) Hypothesis testing in semiparametric and nonparametric models for econometric time series. The Review of Economic Studies 56(4), 511534.CrossRefGoogle Scholar
Rosenbaum, P.R. & Rubin, D.B. (1983) Assessing sensitivity to an unobserved binary covariate in an observational study with binary outcome. Journal of the Royal Statistical Society: Series B (Methodological) 45(2), 212218.Google Scholar
Rubin, D.B. & van der Laan, M.J. (2008) Empirical efficiency maximization: Improved locally efficient covariate adjustment in randomized experiments and survival analysis. The International Journal of Biostatistics 4(1), Article no. 5.CrossRefGoogle ScholarPubMed
Scharfstein, D.O., Rotnitzky, A., & Robins, J.M. (1999) Adjusting for nonignorable drop-out using semiparametric nonresponse models. Journal of the American Statistical Association 94(448), 10961120.CrossRefGoogle Scholar
Shao, J. (1992) Bootstrap variance estimators with truncation. Statistics & Probability Letters 15(2), 95101.CrossRefGoogle Scholar
Sherman, R.P. (1993) The limiting distribution of the maximum rank correlation estimator. Econometrica 61(1), 123137.CrossRefGoogle Scholar
Stein, C. (1956) Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics , pp. 197206. University of California Press.Google Scholar
Tsiatis, A.A., Davidian, M., & Cao, W. (2011) Improved doubly robust estimation when data are monotonely coarsened, with application to longitudinal studies with dropout. Biometrics 67(2), 536545.CrossRefGoogle ScholarPubMed
Van der Vaart, A.W. (2000) Asymptotic Statistics . Cambridge University Press.Google Scholar
Wales, T.J. & Woodland, A.D. (1980) Sample selectivity and the estimation of labor supply functions. International Economic Review 21(2), 437468.CrossRefGoogle Scholar
Wan, A.T., Zhang, X., & Zou, G. (2010) Least squares model averaging by mallows criterion. Journal of Econometrics 156(2), 277283.CrossRefGoogle Scholar
Wasserman, L. (2006) All of Nonparametric Statistics . Springer Science & Business Media.Google Scholar
Yang, Y. (2001) Adaptive regression by mixing. Journal of the American Statistical Association 96(454), 574588.CrossRefGoogle Scholar
Yang, Y. (2003) Regression with multiple candidate models: Selecting or mixing? Statistica Sinica 13(3), 783809.Google Scholar
Yang, Y. (2005) Can the strengths of AIC and BIC be shared? A conflict between model indentification and regression estimation. Biometrika 92(4), 937950.CrossRefGoogle Scholar
Zhang, X. & Liang, H. (2011) Focused information criterion and model averaging for generalized additive partial linear models. Annals of Statistics 39(1), 174200.CrossRefGoogle Scholar
Supplementary material: PDF

Shi supplementary material

Shi supplementary material

Download Shi supplementary material(PDF)