zbMATH — the first resource for mathematics

Smoothed GMM for quantile models. (English) Zbl 1456.62281
Summary: We consider estimation of finite-dimensional parameters identified by general conditional quantile restrictions, including instrumental variables quantile regression. Within a generalized method of moments framework, moment functions are smoothed to aid both computation and precision. Consistency and asymptotic normality are established under weaker assumptions than previously seen in the literature, allowing dependent data and nonlinear structural models. Simulations illustrate the finite-sample properties. An in-depth empirical application estimates the consumption Euler equation derived from quantile utility maximization. Advantages of quantile Euler equations include robustness to fat tails, decoupling risk attitude from the elasticity of intertemporal substitution, and error-free log-linearization.

62P20 Applications of statistics to economics
62G05 Nonparametric estimation
62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
GenSA; pracma; R
Full Text: DOI
[1] Amemiya, T., Two stage least absolute deviations estimators, Econometrica, 50, 3, 689-711, (1982), URL https://www.jstor.org/stable/1912608 · Zbl 0493.62098
[2] Andrews, D. W.K., Consistency in nonlinear econometric models: A generic uniform law of large numbers, Econometrica, 55, 6, 1465-1471, (1987), URL http://www.jstor.org/stable/1913568 · Zbl 0646.62101
[3] Andrews, D. W.K., Laws of large numbers for dependent non-identically distributed random variables, Econometric Theory, 4, 3, 458-467, (1988), URL http://www.jstor.org/stable/3532335
[4] Andrews, D. W.K., An empirical process central limit theorem for dependent non-identically distributed random variables, J. Multivariate Anal., 38, 2, 187-203, (1991), URL https://doi.org/10.1016/0047-259X(91)90039-5 · Zbl 0732.60026
[5] Andrews, D. W.K., Heteroskedasticity and autocorrelation consistent covariance matrix estimation, Econometrica, 59, 3, 817-858, (1991), URL http://www.jstor.org/stable/2938229 · Zbl 0732.62052
[6] Andrews, D. W.K., Generic uniform convergence, Econometric Theory, 8, 2, 241-257, (1992), URL http://www.jstor.org/stable/3532442
[7] Angrist, J.; Chernozhukov, V.; Fernández-Val, I., Quantile regression under misspecification, with an application to the U.S. wage structure, Econometrica, 74, 2, 539-563, (2006), URL https://doi.org/10.1111/j.1468-0262.2006.00671.x · Zbl 1145.62399
[8] Borchers, H.W., 2015. pracma: Practical numerical math functions. R package version 1.8.3. URL http://CRAN.R-project.org/package=pracma.
[9] Buchinsky, M., Recent advances in quantile regression models: A practical guideline for empirical research, J. Hum. Resour., 33, 1, 88-126, (1998), URL http://www.jstor.org/stable/146316
[10] Campbell, J. Y., Consumption-based asset pricing, (Constantinides, G. M.; Harris, M.; Stulz, R. M., Handbook of the Economics of Finance: Financial Markets and Asset Pricing, Vol. 1, Part B, (2003), North Holland), 803-887, URL https://doi.org/10.1016/S1574-0102(03)01022-7
[11] Campbell, J. Y.; Mankiw, N. G., Consumption, income, and interest rates: Reinterpreting the time series evidence, (Blanchard, O. J.; Fischer, S., NBER Macroeconomics Annual 1989, (1989), MIT Press: MIT Press Cambridge, MA), 185-216, URL http://www.nber.org/chapters/c10965
[12] Campbell, J. Y.; Viceira, L. M., Consumption and portfolio decisions when expected returns are time varying, Q. J. Econ., 114, 2, 433-495, (1999), URL https://doi.org/10.1162/003355399556043 · Zbl 0933.91021
[13] Chambers, C. P., An axiomatization of quantiles on the domain of distribution functions, Math. Finance, 19, 2, 335-342, (2009), URL https://doi.org/10.1111/j.1467-9965.2009.00369.x · Zbl 1168.91460
[14] Chen, X.; Chernozhukov, V.; Lee, S.; Newey, W. K., Local identification of nonparametric and semiparametric models, Econometrica, 82, 2, 785-809, (2014), URL https://doi.org/10.3982/ECTA9988 · Zbl 1410.62198
[15] Chen, L.-Y.; Lee, S., Exact computation of GMM estimators for instrumental variable quantile regression models, J. Appl. Econometrics, 33, 4, 553-567, (2018), URL https://doi.org/10.1002/jae.2619
[16] Chen, X.; Liao, Z., Sieve semiparametric two-step GMM under weak dependence, J. Econometrics, 189, 1, 163-186, (2015), URL https://doi.org/10.1016/j.jeconom.2015.07.001 · Zbl 1337.62249
[17] Chen, X.; Linton, O.; van Keilegom, I., Estimation of semiparametric models when the criterion function is not smooth, Econometrica, 71, 5, 1591-1608, (2003), URL https://doi.org/10.1111/1468-0262.00461 · Zbl 1154.62325
[18] Chen, X.; Pouzo, D., Efficient estimation of semiparametric conditional moment models with possibly nonsmooth residuals, J. Econometrics, 152, 1, 46-60, (2009), URL https://doi.org/10.1016/j.jeconom.2009.02.002 · Zbl 1431.62111
[19] Chen, X.; Pouzo, D., Estimation of nonparametric conditional moment models with possibly nonsmooth moments, Econometrica, 80, 1, 277-322, (2012), URL https://doi.org/10.3982/ECTA7888 · Zbl 1274.62232
[20] Chernozhukov, V.; Hansen, C., An IV model of quantile treatment effects, Econometrica, 73, 1, 245-261, (2005), URL http://www.jstor.org/stable/3598944 · Zbl 1152.91706
[21] Chernozhukov, V.; Hansen, C., Instrumental quantile regression inference for structural and treatment effect models, J. Econometrics, 132, 2, 491-525, (2006), URL https://doi.org/10.1016/j.jeconom.2005.02.009 · Zbl 1337.62353
[22] Chernozhukov, V.; Hansen, C., Instrumental variable quantile regression: A robust inference approach, J. Econometrics, 142, 1, 379-398, (2008), URL https://doi.org/10.1016/j.jeconom.2007.06.005 · Zbl 1418.62154
[23] Chernozhukov, V.; Hansen, C.; Wüthrich, K., Instrumental variable quantile regression, (Koenker, R.; Chernozhukov, V.; He, X.; Peng, L., Handbook of Quantile Regression, (2017), CRC/Chapman-Hall), 119-144, URL https://www.routledgehandbooks.com/doi/10.1201/9781315120256
[24] Chernozhukov, V.; Hong, H., An MCMC approach to classical estimation, J. Econometrics, 115, 2, 293-346, (2003), URL https://doi.org/10.1016/S0304-4076(03)00100-3 · Zbl 1043.62022
[25] Cochrane, J. H., Asset Pricing, (2005), Princeton University Press: Princeton University Press Princeton, NJ
[26] de Castro, L. I.; Galvao, A. F., Dynamic Quantile Models of Rational Behavior, (2019), University of Arizona: University of Arizona mimeo, Available at SSRN: https://dx.doi.org/10.2139/ssrn.2932382
[27] Fernandes, M., Guerre, E., Horta, E., 2017. Smoothing quantile regressions. Mimeo, Available at http://bibliotecadigital.fgv.br/dspace/handle/10438/18390.
[28] Galvao, A. F.; Kato, K., Smoothed quantile regression for panel data, J. Econometrics, 193, 1, 92-112, (2016), URL https://doi.org/10.1016/j.jeconom.2016.01.008 · Zbl 1420.62483
[29] Giovannetti, B. C., Asset pricing under quantile utility maximization, Rev. Financ. Econ., 22, 4, 169-179, (2013), URL https://doi.org/10.1016/j.rfe.2013.05.008
[30] Hall, R. E., Intertemporal substitution in consumption, J. Political Econ., 96, 2, 339-357, (1988), URL https://doi.org/10.1086/261539
[31] Hansen, L. P., Large sample properties of generalized method of moments estimators, Econometrica, 50, 4, 1029-1054, (1982), URL http://www.jstor.org/stable/1912775 · Zbl 0502.62098
[32] Hansen, L. P.; Singleton, K. J., Stochastic consumption, risk aversion, and the temporal behavior of asset returns, J. Political Econ., 92, 2, 249-265, (1983), URL http://www.jstor.org/stable/1832056
[33] Horowitz, J. L., Bootstrap methods for median regression models, Econometrica, 66, 6, 1327-1351, (1998), URL http://www.jstor.org/stable/2999619 · Zbl 1056.62517
[34] Hwang, J.; Sun, Y., Should we go one step further? an accurate comparison of one-step and two-step procedures in a generalized method of moments framework, J. Econometrics, (2019), (in press)
[35] de Jong, R. M., Weak laws of large numbers for dependent random variables, Ann. Écon. Stat., 51, 209-225, (1998), URL http://www.jstor.org/stable/20076144
[36] de Jong, R. M.; Davidson, J., Consistency of kernel estimators of heteroscedastic and autocorrelated covariance matrices, Econometrica, 68, 2, 407-423, (2000), URL http://www.jstor.org/stable/2999433 · Zbl 1016.62030
[37] Kaplan, D. M.; Sun, Y., Smoothed estimating equations for instrumental variables quantile regression, Econometric Theory, 33, 1, 105-157, (2017), URL https://doi.org/10.1017/S0266466615000407 · Zbl 1441.62768
[38] Kato, K., Asymptotic normality of Powell’s kernel estimator, Ann. Inst. Statist. Math., 64, 2, 255-273, (2012), URL https://doi.org/10.1007/s10463-010-0310-9 · Zbl 1440.62123
[39] Kinal, T. W., The existence of moments of k-class estimators, Econometrica, 48, 1, 241-249, (1980), URL http://www.jstor.org/stable/1912027 · Zbl 0418.62047
[40] Koenker, R.; Bassett Jr., G., Regression quantiles, Econometrica, 46, 1, 33-50, (1978), URL http://www.jstor.org/stable/1913643 · Zbl 0373.62038
[41] Lancaster, T.; Jun, S. J., Bayesian quantile regression methods, J. Appl. Econometrics, 25, 2, 287-307, (2010), URL https://doi.org/10.1002/jae.1069
[42] Ljungqvist, L.; Sargent, T. J., Recursive Macroeconomic Theory, (2012), MIT Press: MIT Press Cambridge, Massachusetts
[43] Lucas Jr., R. E., Asset prices in an exchange economy, Econometrica, 46, 6, 1429-1446, (1978), URL https://www.jstor.org/stable/1913837 · Zbl 0398.90016
[44] MaCurdy, T., A practitioner’s approach to estimating intertemporal relationships using longitudinal data: Lessons from applications in wage dynamics, (Heckman, J. J.; Leamer, E. E., Handbook of Econometrics, Vol. 6A, (2007), Elsevier), 4057-4167, URL https://doi.org/10.1016/S1573-4412(07)06062-X
[45] MaCurdy, T., Hong, H., 1999. Smoothed quantile regression in generalized method of moments, mimeo.
[46] MaCurdy, T., Timmins, C., 2001. Bounding the influence of attrition on intertemporal wage variation in the NLSY. mimeo. Available at http://public.econ.duke.edu/ timmins/bounds.pdf.
[47] Manski, C. F., Ordinal utility models of decision making under uncertainty, Theory and Decision, 25, 1, 79-104, (1988), URL https://doi.org/10.1007/BF00129169
[48] Newey, W. K.; McFadden, D., Large sample estimation and hypothesis testing, (Engle, R. F.; McFadden, D. L., Handbook of Econometrics, Vol. 4, (1994), Elsevier), 2111-2245, URL https://doi.org/10.1016/S1573-4412(05)80005-4
[49] Newey, W. K.; West, K. D., A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix, Econometrica, 55, 3, 703-708, (1987), URL http://www.jstor.org/stable/1913610 · Zbl 0658.62139
[50] Oberhofer, W.; Haupt, H., Asymptotic theory for nonlinear quantile regression under weak dependence, Econometric Theory, 32, 3, 686-713, (2016), URL https://doi.org/10.1017/S0266466615000031 · Zbl 1441.62822
[51] Ogaki, M.; Reinhart, C. M., Measuring intertemporal substitution: The role of durable goods, J. Political Econ., 106, 5, 1078-1098, (1998), URL https://www.jstor.org/stable/10.1086/250040
[52] Otsu, T., Conditional empirical likelihood estimation and inference for quantile regression models, J. Econometrics, 142, 1, 508-538, (2008), URL https://doi.org/10.1016/j.jeconom.2007.08.016 · Zbl 1418.62165
[53] Pötscher, B. M.; Prucha, I. R., Generic uniform convergence and equicontinuity concepts for random functions: An exploration of the basic structure, J. Econometrics, 60, 1, 23-63, (1994), URL https://doi.org/10.1016/0304-4076(94)90037-X · Zbl 0788.60042
[54] Powell, J. L., Least absolute deviations estimation for the censored regression model, J. Econometrics, 25, 3, 303-325, (1984), URL https://doi.org/10.1016/0304-4076(84)90004-6 · Zbl 0571.62100
[55] Powell, J. L., Estimation of monotonic regression models under quantile restrictions, (Barnett, W. A.; Powell, J.; Tauchen, G., Nonparametric and semiparametric methods in econometrics and statistics: proceedings of the Fifth International Symposium in Economic Theory and Econometrics, (1991), Cambridge University Press), 357-384 · Zbl 0754.62023
[56] Powell, J. L., Estimation of semiparametric models, (Engle, R. F.; McFadden, D. L., Handbook of Econometrics, Vol. 4, (1994), Elsevier), 2443-2521, URL https://doi.org/10.1016/S1573-4412(05)80010-8
[57] R Core Team, ., R Core Team, 2013. R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL http://www.R-project.org/.
[58] Rostek, M., Quantile maximization in decision theory, Rev. Econom. Stud., 77, 1, 339-371, (2010), URL https://doi.org/10.1111/j.1467-937X.2009.00564.x · Zbl 1189.91053
[59] Schennach, S. M., Bayesian exponentially tilted empirical likelihood, Biometrika, 92, 1, 31-46, (2005), URL http://www.jstor.org/stable/20441164 · Zbl 1068.62035
[60] Schennach, S. M., Point estimation with exponentially tilted empirical likelihood, Ann. Statist., 35, 2, 634-672, (2007), URL https://projecteuclid.org/euclid.aos/1183667287 · Zbl 1117.62024
[61] Su, L., Yang, Z., 2011. Instrumental variable quantile estimation of spatial autoregressive models. Working paper. Available at http://www.mysmu.edu/faculty/ljsu/Publications/ivqr_sar20110505.pdf.
[62] Toda, A. A.; Walsh, K., The double power law in consumption and implications for testing euler equations, J. Political Econ., 123, 5, 1177-1200, (2015), URL https://doi.org/10.1086/682729
[63] Toda, A. A.; Walsh, K. J., Fat tails and spurious estimation of consumption-based asset pricing models, J. Appl. Econometrics, 32, 6, 1156-1177, (2017), URL https://doi.org/10.1002/jae.2564
[64] van der Vaart, A. W., Asymptotic Statistics, (1998), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0910.62001
[65] Whang, Y.-J., Smoothed empirical likelihood methods for quantile regression models, Econometric Theory, 22, 2, 173-205, (2006), URL https://doi.org/10.1017/S0266466606060087 · Zbl 1138.62017
[66] Wooldridge, J. M., Asymptotic properties of econometric estimators, (1986), Department of Economics, University of California: Department of Economics, University of California San Diego, (Ph.D. thesis)
[67] Wüthrich, K., A comparison of two quantile models with endogeneity, J. Bus. Econom. Statist., (2019), (in press), URL https://doi.org/10.1080/07350015.2018.1514307
[68] Wüthrich, K., A closed-form estimator for quantile treatment effects with endogeneity, J. Econom., (2019), (in press), URL https://doi.org/10.1016/j.jeconom.2018.11.017 · Zbl 1452.62966
[69] Xiang, Y.; Gubian, S.; Suomela, B.; Hoeng, J., Generalized simulated annealing for global optimization: The GenSA package, R J., 5, 1, 13-28, (2013), URL https://journal.r-project.org/archive/2013/RJ-2013-002/index.html
[70] Yogo, M., Estimating the elasticity of intertemporal substitution when instruments are weak, Rev. Econ. Stat., 86, 3, 797-810, (2004), URL https://doi.org/10.1162/0034653041811770
[71] Zhu, Y., K-step correction for mixed integer linear programming: a new approach for instrumental variable quantile regressions and related problems, (2018), URL https://arxiv.org/abs/1805.06855
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.