Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-26T07:26:38.440Z Has data issue: false hasContentIssue false
  • English
  • Français

SIMPLE SEMIPARAMETRIC ESTIMATION OF ORDERED RESPONSE MODELS

Published online by Cambridge University Press:  22 July 2022

Ruixuan Liu  and
Zhengfei Yu
Ruixuan Liu
Affiliation:
Chinese University of Hong Kong
Zhengfei Yu*
Affiliation:
University of Tsukuba
*
Address correspondence to Zhengfei Yu, Faculty of Humanities and Social Sciences, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8571, Japan; e-mail: [email protected].
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We propose two simple semiparametric estimation methods for ordered response models with an unknown error distribution. The proposed methods do not require users to choose any tuning parameters, and they automatically incorporate the monotonicity restriction of the unknown distribution function. Fixing finite-dimensional parameters in the model, we construct nonparametric maximum likelihood estimates for the error distribution based on the related binary choice data or the entire ordered response data. We then obtain estimates for finite-dimensional parameters based on moment conditions given the estimated distribution function. Our semiparametric approaches deliver root-n consistent and asymptotically normal estimators of the regression coefficient and threshold parameter. We also develop valid bootstrap procedures for inference. The advantages of our methods are borne out in simulation studies and a real data application.

Type
ARTICLES
Information
Econometric Theory , Volume 40 , Issue 1 , February 2024 , pp. 1 - 36
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Footnotes

We are grateful to the Editor (Peter C.B. Phillips), the Co-Editor (Simon Lee), and two anonymous referees for their comments and suggestions that greatly improved the paper. We would like to thank Jason Abrevaya, Bo Honoré, Hidehiko Ichimura, Hiroyuki Kasahara, Toru Kitagawa, Brendan Kline, Zhongjian Lin, Oliver Linton, Essie Maasoumi, Taisuke Otsu, Aureo de Paula, Adam Rosen, Katsumi Shimotsu, and Haiqing Xu; seminar participants at UT Austin and SUNY Albany; and participants at the 2018 North American Econometric Society Meetings, the 2018 CEME Conference for Young Econometricians, and the Workshop on Advances in Econometrics 2018 for their insightful comments and suggestions. Yu gratefully acknowledges the support of JSPS KAKENHI Grant Numbers 19K13666 and 21K01419. All errors are ours.

References

REFERENCES

Abrevaya, J. & Huang, J. (2005) On the bootstrap of the maximum score estimator. Econometrica 73(4), 1175–1204.CrossRefGoogle Scholar
Aitchison, J. & Silvey, S. (1957) The generalization of probit analysis to the case of multiple responses. Biometrika 44, 131–140.CrossRefGoogle Scholar
Ayer, M., Brunk, H., Ewing, G., Reid, W., & Silverman, E. (1955) An empirical distribution function for sampling with incomplete information. Annals of Mathematical Statistics 26, 641–647.CrossRefGoogle Scholar
Balabdaoui, F., Groeneboom, P., & Hendrickx, K. (2019) Score estimation in the monotone single-index model. Scandinavian Journal of Statistics 46(2), 517–544.CrossRefGoogle Scholar
Beare, B.K. & Fang, Z. (2017) Weak convergence of the least concave majorant of estimators for a concave distribution function. Electronic Journal of Statistics 11(2), 3841–3870.CrossRefGoogle Scholar
Bellemare, C., Melenberg, B., & Van Soest, A. (2002) Semi-parametric models for satisfaction with income. Portuguese Economic Journal 1, 181–203.CrossRefGoogle Scholar
Cavanagh, C. & Sherman, R.P. (1998) Rank estimators for monotonic index models. Journal of Econometrics 84, 351–381.CrossRefGoogle Scholar
Chen, S. (2002) Rank estimation of transformation models. Econometrica 70, 1683–1697.CrossRefGoogle Scholar
Chen, S. & Khan, S. (2003) Rates of convergence for estimating regression coefficients in heteroskedastic discrete response models. Journal of Econometrics 117, 245–278.CrossRefGoogle Scholar
Chen, X., Lin, Q., & Sen, B. (2020) On degrees of freedom of projection estimators with applications to multivariate nonparametric regression. Journal of the American Statistical Association 115, 173–186.CrossRefGoogle Scholar
Chen, X., Linton, O., & Van Keilegom, I. (2003) Estimation of semiparametric models when the criterion function is not smooth. Econometrica 71, 1591–1608.CrossRefGoogle Scholar
Coppejans, M. (2007) On efficient estimation of the ordered response model. Journal of Econometrics 137, 577–614.CrossRefGoogle Scholar
Cosslett, S.R. (1983) Distribution-free maximum likelihood estimator of the binary choice model. Econometrica 51, 765–782.CrossRefGoogle Scholar
Gentleman, R. & Vandal, A. (2018) Icens: NPMLE for censored and truncated data. R package version 1.53.0.Google Scholar
Geskus, R. & Groeneboom, P. (1996) Asymptotically optimal estimation of smooth functionals for interval censoring, I. Statistica Neerlandica 50, 69–88.CrossRefGoogle Scholar
Geskus, R. & Groeneboom, P. (1997) Asymptotically optimal estimation of smooth functionals for interval censoring, II. Statistica Neerlandica 51, 201–219.CrossRefGoogle Scholar
Geskus, R. & Groeneboom, P. (1999) Asymptotically optimal estimation of smooth functionals for interval censoring, case 2. Annals of Statistics 27, 627–674.CrossRefGoogle Scholar
Greene, W.H. & Hensher, D.A. (2010) Modeling Ordered Choices: A Primer . Cambridge University Press.CrossRefGoogle Scholar
Groeneboom, P. & Hendrickx, K. (2017) The nonparametric bootstrap for the current status model. Electronic Journal of Statistics 11, 3446–3483.CrossRefGoogle Scholar
Groeneboom, P. & Hendrickx, K. (2018) Current status linear regression. Annals of Statistics 46, 1415–1444.CrossRefGoogle Scholar
Groeneboom, P. & Hendrickx, K. (2019) Estimation in monotone single-index models. Statistica Neerlandica 73(1), 78–99.CrossRefGoogle Scholar
Groeneboom, P. & Jongbloed, G. (2014) Nonparametric Estimation Under Shape Constraints . Cambridge University Press.CrossRefGoogle Scholar
Groeneboom, P. & Wellner, J.A. (1992) Information Bounds and Nonparametric Maximum Likelihood Estimation . Birkhauser.CrossRefGoogle Scholar
Han, A.K. (1987) Non-parametric analysis of a generalized regression model: The maximum rank correlation estimator. Journal of Econometrics 35, 303–316.CrossRefGoogle Scholar
Honoré, B.E. & de Paula, Á. (2010) Interdependent durations. Review of Economic Studies 77, 1138–1163.CrossRefGoogle Scholar
Honoré, B.E. & de Paula, Á. (2018) A new model for interdependent durations with an application to joint retirement. Quantitative Economics 9, 1299–1333.CrossRefGoogle Scholar
Horowitz, J.L. (1992) A smoothed maximum score estimator for the binary response model. Econometrica , 505–531.CrossRefGoogle Scholar
Horowitz, J.L. (2009) Semiparametric and Nonparametric Methods in Econometrics . Springer.CrossRefGoogle Scholar
Ichimura, H. (1993) Semiparametric least squares (SLS) and weighted SLS estimation of single-index models. Journal of Econometrics 58, 71–120.CrossRefGoogle Scholar
Ichimura, H. & Lee, S. (2010) Characterization of the asymptotic distribution of semiparametric M-estimators. Journal of Econometrics 159, 252–266.CrossRefGoogle Scholar
Ichimura, H. & Todd, P.E. (2007) Implementing nonparametric and semiparametric estimators. In J.J. Heckman, E. Leamer (eds.), Handbook of Econometrics , vol. 6B, Elsevier. pp. 5369–5468.Google Scholar
Kiefer, J. & Wolfowitz, J. (1956) Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters. The Annals of Mathematical Statistics 27(4), 887–906.CrossRefGoogle Scholar
Klein, R.W. & Sherman, R.P. (2002) Shift restrictions and semiparametric estimation in ordered response models. Econometrica 70(2), 663–691.CrossRefGoogle Scholar
Klein, R.W. & Spady, R.H. (1993) An efficient semiparametric estimator for binary response models. Econometrica 61(2), 387–421.CrossRefGoogle Scholar
La Cruz, W., Martínez, J., & Raydan, M. (2006) Spectral residual method without gradient information for solving large-scale nonlinear systems of equations. Mathematics of Computation 75(255), 1429–1448.CrossRefGoogle Scholar
Lee, M.-J. (1992) Median regression for ordered discrete response. Journal of Econometrics 51, 59–77.CrossRefGoogle Scholar
Lewbel, A. (2000) Semiparametric qualitative response model estimation with unknown heteroscedasticity or instrumental variables. Journal of Econometrics 97, 145–177.CrossRefGoogle Scholar
Lewbel, A. (2002) Ordered Response Threshold Estimation. Working paper, Boston College.Google Scholar
Lewbel, A. & Schennach, S.M. (2007) A simple ordered data estimator for inverse density weighted expectations. Journal of Econometrics 136(1), 189–211.CrossRefGoogle Scholar
Manski, C.F. (1985) Semiparametric analysis of discrete response: Asymptotic properties of the maximum score estimator. Journal of Econometrics 27, 313–333.CrossRefGoogle Scholar
Melenberg, B. & Van Soest, A. (1996) Measuring the costs of children: Parametric and semiparametric estimators 1. Statistica Neerlandica 50, 171–192.CrossRefGoogle Scholar
Mukherjee, R. & Sen, B. (2019) On efficiency of the plug-in principle for estimating smooth integrated functionals of a nonincreasing density. Electronic Journal of Statistics 13, 4416–4448.CrossRefGoogle Scholar
Nan, B. & Wellner, J.A. (2013) A general semiparametric Z-estimation approach for case-cohort studies. Statistica Sinica 23(3), 1155–1180.Google ScholarPubMed
Newey, W.K. (1994) The asymptotic variance of semiparametric estimators. Econometrica 62, 1349–1382.CrossRefGoogle Scholar
Robertson, T., Wright, F., & Dykstra, R. (1988) Order Restricted Statistical Inference . Wiley.Google Scholar
Sherman, R.P. (1993) The limiting distribution of the maximum rank correlation estimator. Econometrica 61, 123–137.CrossRefGoogle Scholar
Van de Geer, S. (1993) Hellinger-consistency of certain nonparametric maximum likelihood estimators. Annals of Statistics 21, 14–44.CrossRefGoogle Scholar
Van de Geer, S. (1995) Asymptotic normality in mixture models. ESAIM: Probability and Statistics 1, 17–33.CrossRefGoogle Scholar
Van de Geer, S. (2000) Empirical Processes in M-Estimation . Cambridge University Press.Google Scholar
Van Der Vaart, A. (1991) On differentiable functionals. Annals of Statistics 19, 178–204.Google Scholar
Van Der Vaart, A. (1998) Asymptotic Statistics . Cambridge University Press.CrossRefGoogle Scholar
Van Der Vaart, A. & Wellner, J.A. (1996) Weak Convergence and Empirical Processes . Springer.CrossRefGoogle Scholar
Varadhan, R. & Gilbert, P. (2009) BB: An R package for solving a large system of nonlinear equations and for optimizing a high-dimensional nonlinear objective function. Journal of Statistical Software 32(4), 1–26.CrossRefGoogle Scholar
Wellner, J.A. & Zhan, Y. (1997) A hybrid algorithm for computation of the nonparametric maximum likelihood estimator from censored data. Journal of the American Statistical Association 92, 945–959.CrossRefGoogle Scholar
Supplementary material: PDF

Liu and Yu supplementary material

Liu and Yu supplementary material

Download Liu and Yu supplementary material(PDF)
PDF 579.7 KB