Quantile regression is gradually emerging as a unified statistical methodology for estimating models of conditional quantile functions. By complementing the exclusive focus of classical least squares regression on the conditional mean, quantile regression offers a systematic strategy for examining how covariates influence the location, scale and shape of the entire response distribution. This monograph is the first comprehensive treatment of the subject, encompassing models that are linear and nonlinear, parametric and nonparametric. The author has devoted more than 25 years of research to this topic. The methods in the analysis are illustrated with a variety of applications from economics, biology, ecology and finance. The treatment will find its core audiences in econometrics, statistics, and applied mathematics in addition to the disciplines cited above.

• First comprehensive study of quantile regression methods • Tutorial on associated statistical software in R • Illustrative applications from a broad variety of disciplines

### Contents

Part I. Introduction: 1. Means and ends; 2. The first regression: an historical prelude; 3. Quantiles, ranks, and optimization; 4. Preview of quantile regression; 5. Three examples; 6. Conclusion; Part II. Fundamentals of Quantile Regression: 7. Quantile treatment effects; 8. How does quantile regression work?; 9. Robustness; 10. Interpreting quantile regression models; 11. Caution: quantile crossing; 12. A random coefficient interpretation; 13. Inequality measures and their decomposition; 14. Expectiles and other variations; 15. Interpreting misspecified quantile regressions; 16. Problems; Part III. Inference for Quantile Regression: 17. The finite sample distribution of regression quantiles; 18. A heuristic introduction to quantile regression asymptotics; 19. Wald tests; 20. Estimation of asymptotic covariance matrices; 21. Rank based Inference for quantile regression; 22. Quantile likelihood ratio tests; 23. Inference on the quantile regression process; 24. Tests of the location/acale hypothesis; 25. Resampling methods and the bootstrap; 26. Monte-Carlo comparison of methods; 27. Problems; Part IV. Asymptotic Theory of Quantile Regression: 28. Consistency; 29. Rates of convergence; 30. Bahadur representation; 31. Nonlinear quantile regression; 32. The quantile regression rankscore process; 33. Quantile regression asymptotics under dependent conditions; 34. Extremal quantile regression; 35. The method of quantiles; 36. Model selection, penalties, and large-p asymptotics; 37. Asymptotics for inference; 38. Resampling schemes and the bootstrap; 39. Asymptotics for the quantile regression process; 40. Problems; Part V. L-Statistics and Weighted Quantile Regression: 41. L-Statistics for the linear model; 42. Kernel smoothing for quantile regression; 43. Weighted quantile regression; 44 Quantile regression for location-scale models; 45. Weighted sums of p-functions; 46. Problems; Part VI. Computational Aspects of Quantile Regression: 47. Introduction to linear programming; 48. Simplex methods for quantile regression; 49. Parametric programming for quantile regression; 50 Interior point methods for canonical LPs; 51. Preprocessing for quantile regression; 52. Nonlinear quantile regression; 53. Inequality constraints; 54. Weighted sums of p-functions; 55. Sparsity; 56. Conclusion; 57. Problems; Part VII. Nonparametric Quantile Regression: 58. Locally polynomial quantile regression; 59. Penalty methods for univariate smoothing; 60. Penalty methods for bivariate Smoothing; 61. Additive models and the Role of sparsity; Part VIII. Twilight Zone of Quantile Regression: 62. Quantile regression for survival data; 63. Discrete Response models; 64. Quantile autoregression; 65. Copula functions and nonlinear quantile regression; 66. High breakdown alternatives to quantile regression; 67. Multivariate quantiles; 68. Penalty methods for longitudinal data; 69. Causal effects and structural models; 70. Choquet utility, risk and pessimistic portfolios; Part IX. Conclusion: A. Quantile regression in R: a vignette; A.1. Introduction; A.2. What is a vignette?; A.3. Getting started; A.4. Object orientation; A.5. Formal Inference; A.6. More on testing; A.7. Inference on the quantile regression process; A.8. Nonlinear quantile regression; A.9. Nonparametric quantile regression; A.10. Conclusion; B. Asymptotic critical values.

### Reviews

'… well written and easy to read, with useful illustrations of important aspects of quantile regression. It is obvious that the author knows the subject inside out, giving an up-to-date, exhaustive account of the subject. … The book is a valuable contribution to the statistical literature, and a must have for every statistician or econometrician interested in quantile regression methods.' Journal of the Royal Statistical Society

'It is well written and easy to read, with useful ilustrations of important aspects of quantile regression … a valuable contribution to the statistical literature and is essential for every statistician or econometrician who is interested in quantile regression methods.' Andreas Karlsson, Uppsala University