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LIKELIHOOD INFERENCE ON SEMIPARAMETRIC MODELS WITH GENERATED REGRESSORS

Published online by Cambridge University Press:  25 November 2019

Yukitoshi Matsushita
Affiliation:
Graduate School of Economics, Hitotsubashi University
Taisuke Otsu*
Affiliation:
London School of Economics
*
Address correspondence to Taisuke Otsu, Department of Economics, London School of Economics, Houghton Street, London, WC2A 2AE, UK; e-mail: [email protected].

Abstract

Hahn and Ridder (2013, Econometrica 81, 315–340) formulated influence functions of semiparametric three-step estimators where generated regressors are computed in the first step. This class of estimators covers several important examples for empirical analysis, such as production function estimators by Olley and Pakes (1996, Econometrica 64, 1263–1297) and propensity score matching estimators for treatment effects by Heckman, Ichimura, and Todd (1998, Review of Economic Studies 65, 261–294). The present article studies a nonparametric likelihood-based inference method for the parameters in such three-step estimation problems. In particular, we apply the general empirical likelihood theory of Bravo, Escanciano, and van Keilegom (2018, Annals of Statistics, forthcoming) to modify semiparametric moment functions to account for influences from plug-in estimates into the above important setup, and show that the resulting likelihood ratio statistic becomes asymptotically pivotal without undersmoothing in the first and second step nonparametric estimates.

Type
ARTICLES
Copyright
© Cambridge University Press 2019

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Footnotes

The authors would like to thank the Editor, Co-Editor, and anonymous referees for helpful comments. Financial support from the JSPS KAKENHI (26780133, 18K01541) (Matsushita) and the ERC Consolidator Grant (SNP 615882) are gratefully acknowledged (Otsu).

References

REFERENCES

Ahn, H. & Powell, J.L. (1993) Semiparametric estimation of censored selection models with a nonparametric selection mechanism. Journal of Econometrics 58, 329.CrossRefGoogle Scholar
Bravo, F., Chu, B., & Jacho-Chávez, D.T. (2017) Semiparametric estimation of moment condition models with weakly dependent data. Journal of Nonparametric Statistics 29, 108136.CrossRefGoogle Scholar
Bravo, F., Escanciano, J.C., & van Keilegom, I. (2018) Two-step semiparametric empirical likelihood inference. Annals of Statistics, forthcoming.Google Scholar
Camponovo, L. & Otsu, T. (2014) On Bartlett correctability of empirical likelihood in generalized power divergence family. Statistics and Probability Letters 86, 3843.CrossRefGoogle Scholar
Cattaneo, M. (2010) Efficient semiparametric estimation of multi-valued treatment effects under ignorability. Journal of Econometrics 155, 138154.CrossRefGoogle Scholar
Hahn, J. & Ridder, G. (2013) Asymptotic variance of semiparametric estimators with generated regressors. Econometrica 81, 315340.Google Scholar
Hansen, B.E. (2008) Uniform convergence rates for kernel estimation with dependent data. Econometric Theory 24, 726748.CrossRefGoogle Scholar
Heckman, J.J., Ichimura, H., & Todd, P. (1998) Matching as an econometric evaluation estimator. Review of Economic Studies 65, 261294.CrossRefGoogle Scholar
Hirano, K., Imbens, G.W., & Ridder, G. (2003) Efficient estimation of average treatment effects using the estimated propensity score. Econometrica 71, 11611189.CrossRefGoogle Scholar
Ichimura, H. (1993) Estimation of single index models. Journal of Econometrics 58, 71120.CrossRefGoogle Scholar
Ichimura, H. & Linton, O. (2005) Asymptotic expansions for some semiparametric program evaluation estimators. In Andrews, D. & Stock, J. (eds.), Identification and Inference for Econometric Models, pp.149170, Cambridge University Press, NY.CrossRefGoogle Scholar
Kitamura, Y. (1997) Empirical likelihood methods with weakly dependent process. Annals of Statistics 25, 20842102.CrossRefGoogle Scholar
Linton, O. (2002) Edgeworth approximations for semiparametric instrumental variable estimators and test statistics. Journal of Econometrics 106, 325368.CrossRefGoogle Scholar
Mammen, E., Rothe, C., & Schienle, M. (2016) Semiparametric estimation with generated covariates. Econometric Theory 32, 11401177.CrossRefGoogle Scholar
Newey, W.K. (1994) The asymptotic variance of semiparametric estimators. Econometrica 62, 13491382.CrossRefGoogle Scholar
Newey, W.K. (2009) Two-step series estimation of sample selection models. Econometrics Journal 12, S217S229.CrossRefGoogle Scholar
Newey, W.K., Hsieh, F., & Robins, J.M. (2004) Twicing kernels and small bias property of semiparametric estimators. Econometrica 72, 947962.CrossRefGoogle Scholar
Newey, W.K. & Smith, R.J. (2004) Higher order properties of GMM and generalized empirical likelihood estimators. Econometrica 72, 219255.CrossRefGoogle Scholar
Nishiyama, Y. & Robinson, P.M. (2000) Edgeworth expansions for semiparametric averaged derivatives. Econometrica 68, 931979.CrossRefGoogle Scholar
Olley, G.S. & Pakes, A. (1996) The dynamics of productivity in the telecommunications equipment industry. Econometrica 64, 12631297.CrossRefGoogle Scholar
Pagan, A. (1984) Econometric issues in the analysis of regressions with generated regressors. International Economic Review 25, 221247.CrossRefGoogle Scholar
Powell, J.L. (2001) Semiparametric estimation of censored selection models. In Hsiao, C., Morimune, K., & Powell, J. (eds.), Nonlinear Statistical Modeling, pp. 165-196, Cambridge University Press.CrossRefGoogle Scholar
Qin, J. & Lawless, J. (1995) Estimating equations, empirical likelihood and constraints on parameters. Canadian Journal of Statistics 23, 145159.CrossRefGoogle Scholar
Rothe, C. & Firpo, S. (2016) Properties of Doubly Robust Estimators when Nuisance Functions are Estimated Nonparametrically. Working paper.Google Scholar
Schennach, S.M. (2007) Point estimation with exponentially tilted empirical likelihood. Annals of Statistics 35, 634672.CrossRefGoogle Scholar
Smith, R.J. (1997) Alternative semi-parametric likelihood approaches to generalised method of moments estimation. Economic Journal 107, 503519.CrossRefGoogle Scholar
Xue, L. & Xue, D. (2011) Empirical likelihood for semiparametric regression model with missing response data. Journal of Multivariate Analysis 102, 723740.CrossRefGoogle Scholar
Zhu, L., Lin, L., Cui, X., & Li, G. (2010) Bias-corrected empirical likelihood in a multi-link semiparametric model. Journal of Multivariate Analysis 101, 850-868.CrossRefGoogle Scholar
Zhu, L. & Xue, L. (2006) Empirical likelihood confidence regions in a partially linear single- index model. Journal of the Royal Statistical Society, B 68, 549570.CrossRefGoogle Scholar