Skip to main content Accessibility help


  • David M. Kaplan (a1) and Yixiao Sun (a2)


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 χ 2 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.


Corresponding author

*Address correspondence to David M. Kaplan, Department of Economics, University of Missouri, 118 Professional Bldg, 909 University Ave, Columbia, MO 65211-6040; e-mail:


Hide All
Abadie, A., Angrist, J., & Imbens, G. (2002) Instrumental variables estimates of the effect of subsidized training on the quantiles of trainee earnings. Econometrica 70, 91117.
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. 427476. Palgrave MacMillan.
Breiman, L. (1994) Bagging predictors. Technical Report 421, Department of Statistics, University of California, Berkeley.
Cattaneo, M.D., Crump, R.K., & Jansson, M. (2012) Optimal inference for instrumental variables regression with non-Gaussian errors. Journal of Econometrics 167, 115.
Chamberlain, G. (1987) Asymptotic efficiency in estimation with conditional moment restrictions. Journal of Econometrics 34, 305334.
Chen, X. & Pouzo, D. (2009) Efficient estimation of semiparametric conditional moment models with possibly nonsmooth residuals. Journal of Econometrics 152, 4660.
Chen, X. & Pouzo, D. (2012) Estimation of nonparametric conditional moment models with possibly nonsmooth moments. Econometrica 80, 277322.
Chernozhukov, V., Hansen, C., & Jansson, M. (2009) Finite sample inference for quantile regression models. Journal of Econometrics 152, 93103.
Chernozhukov, V. & Hansen, C. (2005) An IV model of quantile treatment effects. Econometrica 73, 245261.
Chernozhukov, V. & Hansen, C. (2006) Instrumental quantile regression inference for structural and treatment effect models. Journal of Econometrics 132, 491525.
Chernozhukov, V. & Hansen, C. (2008) Instrumental variable quantile regression: A robust inference approach. Journal of Econometrics 142, 379398.
Chernozhukov, V. & Hansen, C. (2013) Quantile models with endogeneity. Annual Review of Economics 5, 5781.
Chernozhukov, V. & Hong, H. (2003) An MCMC approach to classical estimation. Journal of Econometrics 115, 293346.
Fan, J. & Liao, Y. (2014) Endogeneity in high dimensions. Annals of Statistics 42, 872917.
Galvao, A.F. (2011) Quantile regression for dynamic panel data with fixed effects. Journal of Econometrics 164, 142157.
Hall, P. (1992) Bootstrap and Edgeworth Expansion. Springer Series in Statistics. Springer-Verlag.
Heyde, C.C. (1997) Quasi-Likelihood and Its Application: A General Approach to Optimal Parameter Estimation. Springer Series in Statistics. Springer.
Horowitz, J.L. (1992) A smoothed maximum score estimator for the binary response model. Econometrica 60, 505531.
Horowitz, J.L. (1998) Bootstrap methods for median regression models. Econometrica 66, 13271351.
Horowitz, J.L. (2002) Bootstrap critical values for tests based on the smoothed maximum score estimator. Journal of Econometrics 111, 141167.
Huber, P.J. (1964) Robust estimation of a location parameter. The Annals of Mathematical Statistics 35, 73101.
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.
Jun, S.J. (2008) Weak identification robust tests in an instrumental quantile model. Journal of Econometrics 144, 118138.
Kinal, T.W. (1980) The existence of moments of k-class estimators. Econometrica 48, 241249.
Koenker, R. & Bassett, G. Jr. (1978) Regression quantiles. Econometrica 46, 3350.
Kwak, D.W. (2010) Implementation of instrumental variable quantile regression (IVQR) methods Working paper, Michigan State University.
Liang, K.-Y. & Zeger, S. (1986) Longitudinal data analysis using generalized linear models. Biometrika 73, 1322.
MaCurdy, T. & Hong, H. (1999) Smoothed quantile regression in generalized method of moments. Working paper, Stanford University.
Müller, H.-G. (1984) Smooth optimum kernel estimators of densities, regression curves and modes. The Annals of Statistics 12, 766774.
Nagar, A.L. (1959) The bias and moment matrix of the general k-class estimators of the parameters in simultaneous equations. Econometrica 27, 573595.
Newey, W.K. (1990) Efficient instrumental variables estimation of nonlinear models. Econometrica 58, 809837.
Newey, W.K. (2004) Efficient semiparametric estimation via moment restrictions. Econometrica 72, 18771897.
Newey, W.K. & Powell, J.L. (1990) Efficient estimation of linear and type I censored regression models under conditional quantile restrictions. Econometric Theory 6, 295317.
Otsu, T. (2008) Conditional empirical likelihood estimation and inference for quantile regression models. Journal of Econometrics 142, 508538.
Phillips, P.C.B. (1982) Small sample distribution theory in econometric models of simultaneous equations. Cowles Foundation Discussion Paper 617, Yale University.
Ruppert, D. & Carroll, R.J. (1980) Trimmed least squares estimation in the linear model. Journal of the American Statistical Association 75, 828838.
van der Vaart, A.W. (1998) Asymptotic Statistics. Cambridge University Press.
Whang, Y.-J. (2006) Smoothed empirical likelihood methods for quantile regression models. Econometric Theory 22, 173205.
Zhou, Y., Wan, A.T.K., & Yuan, Y. (2011) Combining least-squares and quantile regressions. Journal of Statistical Planning and Inference 141, 38143828.


  • David M. Kaplan (a1) and Yixiao Sun (a2)


Altmetric attention score

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.