Since Quetelet's work in the nineteenth century, social science has iconified the average man, that hypothetical man without qualities who is comfortable with his head in the oven and his feet in a bucket of ice. Conventional statistical methods since Quetelet have sought to estimate the effects of policy treatments for this average man. However, such effects are often quite heterogeneous: Medical treatments may improve life expectancy but also impose serious short-term risks; reducing class sizes may improve the performance of good students but not help weaker ones, or vice versa. Quantile regression methods can help to explore these heterogeneous effects. Some recent developments in quantile regression methods are surveyed in this review.


Article metrics loading...

Loading full text...

Full text loading...


Literature Cited

  1. Abadie A, Angrist J, Imbens G. 2002. Instrumental variables estimates of subsidized training on the quantile of trainee earnings. Econometrica 70:91–117 [Google Scholar]
  2. Arellano M, Bonhomme S. 2017a. Nonlinear panel data estimation via quantile regression. Econom. J. 19:C61–94 [Google Scholar]
  3. Arellano M, Bonhomme S. 2017b. Quantile selection models with an application to understanding changes in wage inequality. Econometrica 851–28 [Google Scholar]
  4. Arrow K, Hoffenberg M. 1959. A Time Series Analysis of Interindustry Demands Amsterdam: North-Holland [Google Scholar]
  5. Bassett GWJ, Tam MYS, Knight K. 2002. Quantile models and estimators for data analysis. Metrika 55:17–26 [Google Scholar]
  6. Belloni A, Chernozhukov V. 2011. 1-penalized quantile regression in high-dimensional sparse models. Ann. Stat. 39:82–130 [Google Scholar]
  7. Belloni A, Chernozhukov V, Kato K. 2015. Uniform post-selection inference for least absolute deviation regression and other Z-estimation problems. Biometrika 102:77–94 [Google Scholar]
  8. Belloni A, Chernozhukov V, Kato K. 2016. Valid post-selection inference in high-dimensional approximately sparse quantile regression models. arXiv: 1312.7186v4 [math.ST]
  9. Buchinsky M. 2001. Quantile regression with sample selection: estimating women's return to education in the U.S. Empir. Econ. 26:87–113 [Google Scholar]
  10. Carlier G, Chernozhukov V, Galichon A. 2016. Vector quantile regression: an optimal transport approach. Ann. Stat. 44:1165–92 [Google Scholar]
  11. Chamberlain G. 1984. Panel data. Handbook of Econometrics 2 Z Griliches, MD Intriligator 1247–318 Amsterdam: Elsevier [Google Scholar]
  12. Chamberlain G. 1994. Quantile regression, censoring, and the structure of wages. Advances in Econometrics C Sims 171–208 Cambridge, UK: Cambridge Univ. Press [Google Scholar]
  13. Chaudhuri P. 1991. Nonparametric estimates of regression quantiles and their local Bahadur representation. Ann. Stat. 19:760–77 [Google Scholar]
  14. Chen X. 2007. Large sample sieve estimation of semi-nonparametric models. Handbook of Econometrics 6 JJ Heckman, EE Leamer 5549–632 Amsterdam: Elsevier [Google Scholar]
  15. Chen X, Koenker R, Xiao Z. 2009. Copula-based nonlinear quantile autoregression. Econom. J. 12:S50–67 [Google Scholar]
  16. Chernozhukov V, Fernández-Val I, Galichon A. 2010. Quantile and probability curves without crossing. Econometrica 78:1093–125 [Google Scholar]
  17. Chernozhukov V, Hansen C. 2005a. The effects of 401(k) participation on the wealth distribution: an instrumental quantile regression analysis. Rev. Econ. Stat. 73:735–51 [Google Scholar]
  18. Chernozhukov V, Hansen C. 2005b. An IV model of quantile treatment effects. Econometrica 73:245–62 [Google Scholar]
  19. Chernozhukov V, Hansen C. 2006. Instrumental quantile regression inference for structural and treatment effect models. J. Econom. 132:491–525 [Google Scholar]
  20. Chernozhukov V, Hansen C. 2008. Instrumental variable quantile regression: a robust inference approach. J. Econom. 142:379–98 [Google Scholar]
  21. Chernozhukov V, Hansen C. 2013. Quantile models with endogeneity. Annu. Rev. Econ. 5:57–81 [Google Scholar]
  22. Chesher A. 2003. Identification in nonseparable models. Econometrica 71:1405–41 [Google Scholar]
  23. Chesher A. 2005. Nonparametric identification under discrete variation. Econometrica 73:1525–50 [Google Scholar]
  24. Chowdhury J, Chowdhury P. 2017. Nonparametric quantile regression for Banach-valued response. Handbook of Quantile Regression R Koenker, V Chernozhukov, X He, L Peng Boca Raton, FL: CRC Press. In press [Google Scholar]
  25. Cole TJ, Green P. 1992. Smoothing reference centile curves: the LMS method and penalized likelihood. Stat. Med. 11:1305–19 [Google Scholar]
  26. Cox DR. 1984. Interaction. Int. Stat. Rev. 52:1–24 [Google Scholar]
  27. Diaz I. 2016. Efficient estimation of quantiles in missing data models. arXiv: 1512.08110 [stat.ME]
  28. Donoho D, Chen S, Saunders M. 1998. Atomic decomposition by basis pursuit. SIAM J. Sci. Comput. 20:33–61 [Google Scholar]
  29. Durbin J. 1954. Errors in variables. Rev. Int. Stat. Inst. 22:23–32 [Google Scholar]
  30. Edgeworth F. 1888a. A mathematical theory of banking. J. R. Stat. Soc. 51:113–27 [Google Scholar]
  31. Edgeworth F. 1888b. On a new method of reducing observations relating to several quantities. Philos. Mag. 25:184–91 [Google Scholar]
  32. Efron B. 1967. The two sample problem with censored data. Proc. Berkeley Symp. Math. Statist. Prob., 5th, Berkeley, CA, 4 831–53 New York: Prentice-Hall [Google Scholar]
  33. Firpo S. 2007. Efficient semiparametric estimation of quantile treatment effects. Econometrica 75:259–76 [Google Scholar]
  34. Fitzenberger B, Wilke R, Zhang X. 2009. Implementing Box-Cox quantile regression. Econom. Rev. 29:158–81 [Google Scholar]
  35. Fox M, Rubin H. 1964. Admissibility of quantile estimates of a single location parameter. Ann. Math. Stat. 35:1019–30 [Google Scholar]
  36. Freedman DA. 2008. On regression adjustments in experiments with several treatments. Ann. Appl. Stat. 2:176–96 [Google Scholar]
  37. Galichon A. 2016. Optimal Transport Methods in Economics Princeton, NJ: Princeton Univ. Press [Google Scholar]
  38. Galvao AF. 2011. Quantile regression for dynamic panel data with fixed effects. J. Econom. 164:142–57 [Google Scholar]
  39. Gronau R. 1974. Wage comparisons—a selectivity bias. J. Polit. Econ. 82:1119–43 [Google Scholar]
  40. Hagemann A. 2011. Robust spectral analysis. arXiv: 1111.1965 [math.ST]
  41. Hagemann A. 2017. Cluster-robust bootstrap inference in quantile regression models. J. Am. Stat. Assoc. 112446–56 [Google Scholar]
  42. Hallin M, Lu Z, Paindaveine D, Šiman M. 2015. Local bilinear multiple-output quantile/depth regression. Bernoulli 21:1435–66 [Google Scholar]
  43. Hallin M, Paindaveine D, Šiman M. 2010. Multivariate quantiles and multiple-output regression quantiles: from l1 optimization to halfspace depth. Ann. Stat. 38:635–69 [Google Scholar]
  44. He X. 2017. Resampling methods. Handbook of Quantile Regression R Koenker, V Chernozhukov, X He, L Peng Boca Raton, FL: CRC Press In press [Google Scholar]
  45. He X, Liang H. 2000. Quantile regression estimates for a class of linear and partially linear errors-in-variables models. Stat. Sin. 10:129–40 [Google Scholar]
  46. Heckman JJ. 1974. Shadow prices, market wages, and labor supply. Econometrica 42:679–94 [Google Scholar]
  47. Heckman JJ. 1979. Sample selection bias as a specification error. Econometrica 47:153–61 [Google Scholar]
  48. Heckman JJ, Smith J, Clements N. 1997. Making the most out of programme evaluations and social experiments: accounting for heterogeneity in programme impacts. Rev. Econ. Stud. 64:487–535 [Google Scholar]
  49. Hotelling H. 1939. Tubes and spheres in n-spaces, and a class of statistical problems. Am. J. Math. 61:440–60 [Google Scholar]
  50. Imbens G, Angrist J. 1994. Identification and estimation of local average treatment effects. Econometrica 62:467–75 [Google Scholar]
  51. Kadane JB, Seidenfeld T. 1996. Statistical issues in the analysis of data gathered in the new designs. Bayesian Methods and Ethics in a Clinical Trial Design JB Kadane 115–25 Hoboken, NJ: Wiley [Google Scholar]
  52. Kato K. 2012. Estimation in functional linear quantile regression. Ann. Stat. 40:3108–36 [Google Scholar]
  53. Kato K, Galvao AF, Montes-Rojas GV. 2012. Asymptotics for panel quantile regression models with individual effects. J. Econom. 170:76–91 [Google Scholar]
  54. Khmaladze EV. 1981. Martingale approach in the theory of goodness-of-fit tests. Theory Probab. Appl. 26:240–57 [Google Scholar]
  55. Kley T, Volgushev S, Dette H, Hallin M. 2016. Quantile spectral processes: asymptotic analysis and inference. Bernoulli 22:1770–807 [Google Scholar]
  56. Knight K, Bassett G. 2007. Second order improvements of sample quantiles using subsamples Work. Pap., Dep. Stat., Univ. Toronto, Toronto, Can. [Google Scholar]
  57. Koenker R. 2004. Quantile regression for longitudinal data. J. Multivar. Anal. 91:74–89 [Google Scholar]
  58. Koenker R. 2005. Quantile Regression Cambridge, UK: Cambridge Univ. Press [Google Scholar]
  59. Koenker R. 2011. Additive models for quantile regression: model selection and confidence bandaids. Braz. J. Probab. Stat. 25:239–62 [Google Scholar]
  60. Koenker R. 2016. Quantreg: quantile regression. R Package Software, Version 5.27 https://cran.r-project.org [Google Scholar]
  61. Koenker R. 2017. Handbook of Quantile Regression R Koenker, V Chernozhukov, X He, L Peng Boca Raton, FL: CRC Press In press [Google Scholar]
  62. Koenker R, Bassett G. 1978. Regression quantiles. Econometrica 46:33–50 [Google Scholar]
  63. Koenker R, d'Orey V. 1987. Computing regression quantiles. Appl. Stat. 36:383–93 [Google Scholar]
  64. Koenker R, Geling O. 2001. Reappraising medfly longevity: a quantile regression survival analysis. J. Am. Stat. Assoc. 96:458–68 [Google Scholar]
  65. Koenker R, Mizera I. 2004. Penalized triograms: total variation regularization for bivariate smoothing. J. R. Stat. Soc. B 66:145–63 [Google Scholar]
  66. Koenker R, Ng P, Portnoy S. 1994. Quantile smoothing splines. Biometrika 81:673–80 [Google Scholar]
  67. Koenker R, Park B. 1996. An interior point algorithm for nonlinear quantile regression. J. Econom. 71:265–83 [Google Scholar]
  68. Koenker R, Xiao Z. 2002. Inference on the quantile regression process. Econometrica 70:1583–612 [Google Scholar]
  69. Koenker R, Xiao Z. 2006. Quantile autoregression, with discussion and rejoinder. J. Am. Stat. Assoc. 101:980–1006 [Google Scholar]
  70. Kong L, Mizera I. 2012. Quantile tomography: using quantiles with multivariate data. Stat. Sin. 22:1589–610 [Google Scholar]
  71. Lamarche C. 2010. Robust penalized quantile regression estimation for panel data. J. Econom. 157:396–408 [Google Scholar]
  72. Lee S. 2003. Efficient semiparametric estimation of a partially linear quantile regression model. Econom. Theory 19:1–31 [Google Scholar]
  73. Lee YK, Mammen E, Park BU. 2010. Backfitting and smooth backfitting for additive quantile models. Ann. Stat. 38:2857–83 [Google Scholar]
  74. Lehmann E. 1974. Nonparametrics: Statistical Methods Based on Ranks San Francisco: Holden-Day [Google Scholar]
  75. Li R, Peng L. 2017. Survival analysis with competing risks and semi-competing risks data. Handbook of Quantile Regression R Koenker, V Chernozhukov, X He, L Peng Boca Raton, FL: CRC Press In press [Google Scholar]
  76. Li TH. 2008. Laplace periodogram for time series analysis. J. Am. Stat. Assoc. 103:757–68 [Google Scholar]
  77. Li TH. 2012. Quantile periodograms. J. Am. Stat. Assoc. 107:765–76 [Google Scholar]
  78. Lipsitz S, Fitzmaurice G, Molenberghs G, Zhao L. 1997. Quantile regression methods for longitudinal data with drop-outs: application to CD4 cell counts of patients infected with human immunodeficiency virus. Appl. Stat. 46:463–76 [Google Scholar]
  79. Ma L, Koenker R. 2006. Quantile regression methods for recursive structural equation models. J. Econom. 134:471–506 [Google Scholar]
  80. Machado J, Mata J. 2000. Box-Cox quantile regression and the distribution of firm sizes. J. Appl. Econom. 15:253–74 [Google Scholar]
  81. Manski C. 1975. Maximum score estimation of the stochastic utility model of choice. J. Econom. 3:205–28 [Google Scholar]
  82. Manski C. 1993. The selection problem in econometrics and statistics. Handbook of Statistics 11 GS Maddala, CR Rao, H Vinod 73–84 Amsterdam: Elsevier [Google Scholar]
  83. Manski CF. 2004. Statistical treatment rules for heterogeneous populations. Econometrica 72:1221–46 [Google Scholar]
  84. Matzkin RL. 2015. Estimation of nonparametric models with simultaneity. Econometrica 83:1–66 [Google Scholar]
  85. Meyer M, Woodroofe M. 2000. On the degrees of freedom in shape-restricted regression. Ann. Stat. 28:1083–104 [Google Scholar]
  86. Mu Y, He X. 2007. Power transformation toward a linear regression quantile. J. Am. Stat. Assoc. 102:269–79 [Google Scholar]
  87. Neyman J, Scott EL. 1948. Consistent estimates based on partially consistent observations. Econometrica 16:1–32 [Google Scholar]
  88. Parikh N, Boyd S. 2014. Proximal algorithms. Found. Trends Optim. 1:123–231 [Google Scholar]
  89. Peng L. 2017. Quantile regression for survival analysis. Handbook of Quantile Regression R Koenker, V Chernozhukov, X He, L Peng Boca Raton, FL: CRC Press In press [Google Scholar]
  90. Peng L, Huang Y. 2008. Survival analysis with quantile regression models. J. Am. Stat. Assoc. 103:637–49 [Google Scholar]
  91. Portnoy S. 2003. Censored quantile regression. J. Am. Stat. Assoc. 98:1001–12 [Google Scholar]
  92. Portnoy S, Koenker R. 1997. The Gaussian hare and the Laplacian tortoise: computability of squared-error versus absolute-error estimators. Stat. Sci. 12:279–300 [Google Scholar]
  93. Powell JL. 1986. Censored regression quantiles. J. Econom. 32:143–55 [Google Scholar]
  94. R Development Core Team 2016. R: A Language and Environment for Statistical Computing Vienna: R Found. Stat. Comput. [Google Scholar]
  95. Ramsay J, Silverman B. 1997. Functional Data Analysis Berlin: Springer [Google Scholar]
  96. Rosenblatt M. 1952. Remarks on a multivariate transformation. Ann. Math. Stat. 23:470–72 [Google Scholar]
  97. Schennach SM. 2008. Quantile regression with mismeasured covariates. Econom. Theory 24:1010–43 [Google Scholar]
  98. Schmeidler D. 1986. Integral representation without additivity. Proc. Am. Math. Soc. 97:255–61 [Google Scholar]
  99. Stone CJ. 1994. The use of polynomial splines and their tensor products in multivariate function estimation. Ann. Stat. 22:118–71 [Google Scholar]
  100. Strotz R, Wold H. 1960. A triptych on causal systems. Econometrica 28:417–63 [Google Scholar]
  101. Student 1931. The Lanarkshire milk experiment. Biometrika 23:398–406 [Google Scholar]
  102. Tibshirani R. 1996. Regression shrinkage and selection via the lasso. J. R. Stat. Soc. B 58:267–88 [Google Scholar]
  103. Wald A. 1940. The fitting of straight lines if both variables are subject to error. Ann. Math. Stat. 11:284–300 [Google Scholar]
  104. Wang HJ, Li D, He X. 2012. Estimation of high conditional quantiles for heavy-tailed distributions. J. Am. Stat. Assoc. 107:1453–64 [Google Scholar]
  105. Wang HJ, Stefanski LA, Zhu Z. 2012. Corrected-loss estimation for quantile regression with covariate measurement errors. Biometrika 99:405–21 [Google Scholar]
  106. Wang L, Zhou Y, Song R, Sherwood B. 2016. Quantile optimal treatment regimes Work. Pap., Dep. Stat., Univ. Minn Minneapolis, MN: [Google Scholar]
  107. Wei Y. 2008. An approach to multivariate covariate-dependent quantile contours with application to bivariate conditional growth charts. J. Am. Stat. Assoc. 103:397–409 [Google Scholar]
  108. Wei Y. 2017. Quantile regression with measurement errors and missing data. Handbook of Quantile Regression R Koenker, V Chernozhukov, X He, L Peng Boca Raton, FL: CRC Press In press [Google Scholar]
  109. Wei Y, Carroll RJ. 2009. Quantile regression with measurement error. J. Am. Stat. Assoc. 104:1129–43 [Google Scholar]
  110. Wei Y, Pere A, Koenker R, He X. 2005. Quantile regression for reference growth charts. Stat. Med. 25:1369–82 [Google Scholar]
  111. Yang X, Narisetty N, He X. 2016. A new approach to censored quantile regression estimation Work. Pap., Dep. Stat., Univ. Mich., Ann Arbor, MI [Google Scholar]
  112. Ying Z, Sit T. 2017. Survival analysis: a quantile perspective. Handbook of Quantile Regression R Koenker, V Chernozhukov, X He, L Peng Boca Raton, FL: CRC Press In press [Google Scholar]

Data & Media loading...

  • Article Type: Review Article
This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error