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SMOOTHED ESTIMATING EQUATIONS FOR INSTRUMENTAL VARIABLES QUANTILE REGRESSION

Published online by Cambridge University Press:  22 January 2016

David M. Kaplan  and
Yixiao Sun
David M. Kaplan*
Affiliation:
University of Missouri
Yixiao Sun
Affiliation:
University of California, San Diego
*
*Address correspondence to David M. Kaplan, Department of Economics, University of Missouri, 118 Professional Bldg, 909 University Ave, Columbia, MO 65211-6040; e-mail: kaplandm@missouri.edu.
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Abstract

The moment conditions or estimating equations for instrumental variables quantile regression involve the discontinuous indicator function. We instead use smoothed estimating equations (SEE), with bandwidth h. We show that the mean squared error (MSE) of the vector of the SEE is minimized for some h > 0, leading to smaller asymptotic MSE of the estimating equations and associated parameter estimators. The same MSE-optimal h also minimizes the higher-order type I error of a SEE-based χ2 test and increases size-adjusted power in large samples. Computation of the SEE estimator also becomes simpler and more reliable, especially with (more) endogenous regressors. Monte Carlo simulations demonstrate all of these superior properties in finite samples, and we apply our estimator to JTPA data. Smoothing the estimating equations is not just a technical operation for establishing Edgeworth expansions and bootstrap refinements; it also brings the real benefits of having more precise estimators and more powerful tests.

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Type
ARTICLES
Information
Econometric Theory , Volume 33 , Issue 1 , February 2017 , pp. 105 - 157
Copyright
Copyright © Cambridge University Press 2016 

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Footnotes

Thanks to Victor Chernozhukov (co-editor) and an anonymous referee for insightful comments and references, and thanks to Peter C. B. Phillips (editor) for additional editorial help. Thanks to Xiaohong Chen, Brendan Beare, Andres Santos, and active seminar and conference participants for insightful questions and comments.

References

REFERENCES

Abadie, A., Angrist, J., & Imbens, G. (2002) Instrumental variables estimates of the effect of subsidized training on the quantiles of trainee earnings. Econometrica 70, 91–117.CrossRefGoogle Scholar
Bera, A.K., Bilias, Y., & Simlai, P. (2006) Estimating functions and equations: An essay on historical developments with applications to econometrics. In Mills, T.C. & Patterson, K. (eds.), Palgrave Handbook of Econometrics: Volume 1 Econometric Theory, pp. 427–476. Palgrave MacMillan.Google Scholar
Breiman, L. (1994) Bagging predictors. Technical Report 421, Department of Statistics, University of California, Berkeley.Google Scholar
Cattaneo, M.D., Crump, R.K., & Jansson, M. (2012) Optimal inference for instrumental variables regression with non-Gaussian errors. Journal of Econometrics 167, 1–15.CrossRefGoogle Scholar
Chamberlain, G. (1987) Asymptotic efficiency in estimation with conditional moment restrictions. Journal of Econometrics 34, 305–334.CrossRefGoogle Scholar
Chen, X. & Pouzo, D. (2009) Efficient estimation of semiparametric conditional moment models with possibly nonsmooth residuals. Journal of Econometrics 152, 46–60.CrossRefGoogle Scholar
Chen, X. & Pouzo, D. (2012) Estimation of nonparametric conditional moment models with possibly nonsmooth moments. Econometrica 80, 277–322.Google Scholar
Chernozhukov, V., Hansen, C., & Jansson, M. (2009) Finite sample inference for quantile regression models. Journal of Econometrics 152, 93–103.CrossRefGoogle Scholar
Chernozhukov, V. & Hansen, C. (2005) An IV model of quantile treatment effects. Econometrica 73, 245–261.CrossRefGoogle Scholar
Chernozhukov, V. & Hansen, C. (2006) Instrumental quantile regression inference for structural and treatment effect models. Journal of Econometrics 132, 491–525.CrossRefGoogle Scholar
Chernozhukov, V. & Hansen, C. (2008) Instrumental variable quantile regression: A robust inference approach. Journal of Econometrics 142, 379–398.CrossRefGoogle Scholar
Chernozhukov, V. & Hansen, C. (2013) Quantile models with endogeneity. Annual Review of Economics 5, 57–81.Google Scholar
Chernozhukov, V. & Hong, H. (2003) An MCMC approach to classical estimation. Journal of Econometrics 115, 293–346.Google Scholar
Fan, J. & Liao, Y. (2014) Endogeneity in high dimensions. Annals of Statistics 42, 872–917.CrossRefGoogle ScholarPubMed
Galvao, A.F. (2011) Quantile regression for dynamic panel data with fixed effects. Journal of Econometrics 164, 142–157.CrossRefGoogle Scholar
Hall, P. (1992) Bootstrap and Edgeworth Expansion. Springer Series in Statistics. Springer-Verlag.CrossRefGoogle Scholar
Heyde, C.C. (1997) Quasi-Likelihood and Its Application: A General Approach to Optimal Parameter Estimation. Springer Series in Statistics. Springer.CrossRefGoogle Scholar
Horowitz, J.L. (1992) A smoothed maximum score estimator for the binary response model. Econometrica 60, 505–531.CrossRefGoogle Scholar
Horowitz, J.L. (1998) Bootstrap methods for median regression models. Econometrica 66, 1327–1351.CrossRefGoogle Scholar
Horowitz, J.L. (2002) Bootstrap critical values for tests based on the smoothed maximum score estimator. Journal of Econometrics 111, 141–167.CrossRefGoogle Scholar
Huber, P.J. (1964) Robust estimation of a location parameter. The Annals of Mathematical Statistics 35, 73–101.CrossRefGoogle Scholar
Hwang, J. & Sun, Y. (2015) Should we go one step further? An accurate comparison of one-step and two-step procedures in a generalized method of moments framework. Working paper, Department of Economics, UC San Diego.Google Scholar
Jun, S.J. (2008) Weak identification robust tests in an instrumental quantile model. Journal of Econometrics 144, 118–138.Google Scholar
Kinal, T.W. (1980) The existence of moments of k-class estimators. Econometrica 48, 241–249.CrossRefGoogle Scholar
Koenker, R. & Bassett, G. Jr. (1978) Regression quantiles. Econometrica 46, 33–50.CrossRefGoogle Scholar
Kwak, D.W. (2010) Implementation of instrumental variable quantile regression (IVQR) methods Working paper, Michigan State University.Google Scholar
Liang, K.-Y. & Zeger, S. (1986) Longitudinal data analysis using generalized linear models. Biometrika 73, 13–22.CrossRefGoogle Scholar
MaCurdy, T. & Hong, H. (1999) Smoothed quantile regression in generalized method of moments. Working paper, Stanford University.Google Scholar
Müller, H.-G. (1984) Smooth optimum kernel estimators of densities, regression curves and modes. The Annals of Statistics 12, 766–774.CrossRefGoogle Scholar
Nagar, A.L. (1959) The bias and moment matrix of the general k-class estimators of the parameters in simultaneous equations. Econometrica 27, 573–595.CrossRefGoogle Scholar
Newey, W.K. (1990) Efficient instrumental variables estimation of nonlinear models. Econometrica 58, 809–837.CrossRefGoogle Scholar
Newey, W.K. (2004) Efficient semiparametric estimation via moment restrictions. Econometrica 72, 1877–1897.CrossRefGoogle Scholar
Newey, W.K. & Powell, J.L. (1990) Efficient estimation of linear and type I censored regression models under conditional quantile restrictions. Econometric Theory 6, 295–317.CrossRefGoogle Scholar
Otsu, T. (2008) Conditional empirical likelihood estimation and inference for quantile regression models. Journal of Econometrics 142, 508–538.CrossRefGoogle Scholar
Phillips, P.C.B. (1982) Small sample distribution theory in econometric models of simultaneous equations. Cowles Foundation Discussion Paper 617, Yale University.Google Scholar
Ruppert, D. & Carroll, R.J. (1980) Trimmed least squares estimation in the linear model. Journal of the American Statistical Association 75, 828–838.CrossRefGoogle Scholar
van der Vaart, A.W. (1998) Asymptotic Statistics. Cambridge University Press.CrossRefGoogle Scholar
Whang, Y.-J. (2006) Smoothed empirical likelihood methods for quantile regression models. Econometric Theory 22, 173–205.CrossRefGoogle Scholar
Zhou, Y., Wan, A.T.K., & Yuan, Y. (2011) Combining least-squares and quantile regressions. Journal of Statistical Planning and Inference 141, 3814–3828.CrossRefGoogle Scholar