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Conditional empirical likelihood estimation and inference for quantile regression models. (English) Zbl 1418.62165
Summary: This paper considers two empirical likelihood-based estimation, inference, and specification testing methods for quantile regression models. First, we apply the method of conditional empirical likelihood (CEL) by Y. Kitamura et al. [Econometrica 72, No. 6, 1667–1714 (2004; Zbl 1142.62331)] and J. Zhang and I. Gijbels [Scand. J. Stat. 30, No. 1, 1–24 (2003; Zbl 1034.62036)] to quantile regression models. Second, to avoid practical problems of the CEL method induced by the discontinuity in parameters of CEL, we propose a smoothed counterpart of CEL, called smoothed conditional empirical likelihood (SCEL). We derive asymptotic properties of the CEL and SCEL estimators, parameter hypothesis tests, and model specification tests. Important features are (i) the CEL and SCEL estimators are asymptotically efficient and do not require preliminary weight estimation; (ii) by inverting the CEL and SCEL ratio parameter hypothesis tests, asymptotically valid confidence intervals can be obtained without estimating the asymptotic variances of the estimators; and (iii) in contrast to CEL, the SCEL method can be implemented by some standard Newton-type optimization. Simulation results demonstrate that the SCEL method in particular compares favorably with existing alternatives.

##### MSC:
 62G08 Nonparametric regression and quantile regression 62G05 Nonparametric estimation 62G20 Asymptotic properties of nonparametric inference 62P20 Applications of statistics to economics
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##### References:
 [1] Ai, C.; Chen, X., Estimation of possibly misspecified semiparametric conditional moment restriction models with different conditioning variables, Journal of econometrics, 141, 5-43, (2007) · Zbl 1418.62405 [2] Amemiya, T., Two stage least absolute deviations estimators, Econometrica, 50, 689-712, (1982) · Zbl 0493.62098 [3] Andrews, D.W.K., Nonparametric kernel estimation for semiparametric models, Econometric theory, 11, 560-596, (1995) [4] Bierens, H.J.; Ginther, D.K., Integrated conditional moment testing of quadratic regression models, Empirical economics, 26, 307-324, (2001) [5] Buchinsky, M., Estimating the asymptotic covariance matrix for quantile regression models: a Monte Carlo study, Journal of econometrics, 68, 303-338, (1995) · Zbl 0825.62437 [6] Chen, S.X.; Hall, P., Smoothed empirical likelihood confidence intervals for quantiles, Annals of statistics, 21, 1166-1181, (1993) · Zbl 0786.62053 [7] Chernozhukov, V., Hansen, C., 2001. An IV model of quantile treatment effects. Working Paper. · Zbl 1152.91706 [8] Chernozhukov, V.; Hansen, C., Instrumental quantile regression inference for structural and treatment effect models, Journal of econometrics, 132, 491-525, (2006) · Zbl 1337.62353 [9] Chernozhukov, V.; Hong, H., An MCMC approach to classical estimation, Journal of econometrics, 115, 293-346, (2003) · Zbl 1043.62022 [10] Chernozhukov, V., Hansen, C., Jansson, M., 2006. Finite sample inference for quantile regression models. Working Paper. · Zbl 1431.62601 [11] Christoffersen, P.F., Hahn, J., Inoue, A., 2001. Testing, comparing, and combining value-at-risk measures. Working Paper. [12] Davidson, R.; MacKinnon, J.G., Implicit alternatives and the local power of test statistics, Econometrica, 55, 1305-1329, (1987) · Zbl 0644.62023 [13] Donald, S.G.; Imbens, G.; Newey, W.K., Empirical likelihood estimation and consistent tests with conditional moment restrictions, Journal of econometrics, 117, 55-93, (2003) · Zbl 1022.62046 [14] Hart, J.D., Nonparametric smoothing and lack-of-fit tests, (1997), Springer New York · Zbl 0886.62043 [15] Horowitz, J.L., Bootstrap methods for Median regression models, Econometrica, 66, 1327-1351, (1998) · Zbl 1056.62517 [16] Kemp, G.C.R., 2005. GEL estimation and inference with non-smooth moment indicators and dynamic data. Working Paper. [17] Kim, T.H.; White, H., Estimation, inference, and specification testing for possibly misspecified quantile regressions, () [18] Kitamura, Y., 2003. A likelihood-based approach to the analysis of a class of nested and non-nested models. Working Paper. [19] Kitamura, Y.; Tripathi, G.; Ahn, H., Empirical likelihood-based inference in conditional moment restriction models, Econometrica, 72, 1667-1714, (2004) · Zbl 1142.62331 [20] Koenker, R., Confidence intervals for regression quantiles, () · Zbl 0908.62056 [21] Koenker, R., Quantile regression, (2005), Cambridge University Press Cambridge · Zbl 1111.62037 [22] Koenker, R.; Bassett, G., Regression quantiles, Econometrica, 46, 33-50, (1978) · Zbl 0373.62038 [23] Koenker, R.; Machado, J.A.F., Goodness of fit and related inference processes for quantile regression, Journal of the American statistical association, 94, 1296-1310, (1999) · Zbl 0998.62041 [24] Koenker, R.; Park, B.J., An interior point algorithm for nonlinear quantile regression, Journal of econometrics, 71, 265-283, (1996) · Zbl 0855.62030 [25] Komunjer, I., Quasi-maximum likelihood estimation for conditional quantiles, Journal of econometrics, 128, 137-164, (2005) · Zbl 1337.62235 [26] Komunjer, I., Vuong, Q., 2006a. Efficient conditional quantile estimation: the time series case. Working Paper. · Zbl 1431.62119 [27] Komunjer, I., Vuong, Q., 2006b. Semiparametric efficiency bound and M-estimation in time-series models for conditional quantiles. Working Paper. · Zbl 1185.62156 [28] LeBlanc, M.; Crowley, J., Semiparametric regression functionals, Journal of the American statistical association, 90, 95-105, (1995) · Zbl 0818.62040 [29] Nelder, J.A.; Mead, R., A simplex algorithm for function minimization, Computer journal, 7, 308-313, (1965) · Zbl 0229.65053 [30] Newey, W.K.; Powell, J.L., Efficient estimation of linear and type I censored regression models under conditional quantile restrictions, Econometric theory, 6, 295-317, (1990) [31] Otsu, T., 2006. RESET for quantile regression. Working Paper. [32] Owen, A., Empirical likelihood, (2001), Chapman & Hall London · Zbl 0989.62019 [33] Parente, P.M.D.C., Smith, R.J., 2005. GEL methods for non-smooth moment indicators. Working Paper. [34] Parzen, M.I.; Wei, L.; Ying, Z., A resampling method based on pivotal estimating functions, Biometrika, 81, 341-350, (1994) · Zbl 0807.62038 [35] Powell, J.L., Least absolute deviation estimator for the censored regression model, Journal of econometrics, 25, 303-325, (1984) · Zbl 0571.62100 [36] Powell, J.L., Censored regression quantiles, Journal of econometrics, 32, 143-155, (1986) · Zbl 0605.62139 [37] Stein, C., 1956. Efficient nonparametric testing and estimation. In: Proceedings of the 3rd Berkeley Symposium in Mathematical Statistics and Probability, vol. 1. University of California Press, Berkeley, pp. 187-196. · Zbl 0074.34801 [38] Stone, C.J., Optimal rates of convergence for nonparametric estimators. annals of statistics, 8, 1348-1360, (1980) · Zbl 0451.62033 [39] Stone, C.J., Optimal global rates of convergence for nonparametric regression, Annals of statistics, 10, 1040-1053, (1982) · Zbl 0511.62048 [40] Tripathi, G.; Kitamura, Y., Testing conditional moment restrictions, Annals of statistics, 31, 2059-2095, (2003) · Zbl 1044.62049 [41] van der Vaart, A.W.; Wellner, J.A., Weak convergence and empirical process, (1996), Springer New York · Zbl 0862.60002 [42] Whang, Y.-J., Smoothed empirical likelihood methods for quantile regression models, Econometric theory, 22, 173-205, (2006) · Zbl 1138.62017 [43] Zhang, J.; Gijbels, I., Sieve empirical likelihood and extensions of the generalized least squares, Scandinavian journal of statistics, 30, 1-24, (2003) · Zbl 1034.62036 [44] Zhao, Q., Asymptotically efficient Median regression in the presence of heteroskedasticity of unknown form, Econometric theory, 17, 765-784, (2001) · Zbl 1109.62327 [45] Zheng, J.X., A consistent nonparametric test of parametric regression models under conditioning quantile restrictions, Econometric theory, 14, 123-138, (1998)
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