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1、QUANTILE REGRESSIONAN INTRODUCTIONROGER KOENKER AND KEVIN F. HALLOCKUNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGNAbstract. Quantile regression as introduced in Koenker and Bassett #281978#29 maybe viewed as a natural extension of classical least squares estimation of conditionalmean models to the estim
2、ation of an ensemble of models for conditional quantilefunctions. The central special case is the median regression estimator that mini-mizes a sum of absolute errors. The remaining conditional quantile functions areestimated by minimizingan asymmetricallyweighted sum of absolute errors. Takentogeth
3、er the ensembleofestimated conditionalquantilefunctions o#0Bers amuchmorecomplete view of the e#0Bect of covariates on the location, scale and shape of the dis-tribution of the response variable.This essay provides a brief tutorial introduction to quantile regression methods,illustrating their appli
4、cation in several settings. Practical aspects of computation,inference, and interpretation are discussed and suggestions for further reading areprovided.1. IntroductionIn the classical mythology of least-squares regression the conditional mean function,the function that describes how the mean of y c
5、hanges with the vector of covariatesx, is #28almost#29 all weneedtoknow about the relationship between y and x. Inthe resilient terminology of Frisch #281934#29 and Koopmans #281937#29 it is #5Cthe systematiccomponent or true value, around which y #0Ductuates due to an #5Cerratic componentor #5Cacci
6、dental error. The crucial, and convenient, thing about this view is that theerror is assumed to have preciselythe same distribution whatever values maybetakenby the components of the vector x.We refer to this as a pure location shift modelsince it assumes that x a#0Bects only the location of the con
7、ditional distribution ofy, not its scale, or any other aspect of its distributional shape. If this is the case,we can be fully satis#0Ced with an estimated model of the conditional mean function,supplemented perhaps by an estimate of the conditional dispersion of y around itsmean.Version: December 2
8、8, 2000. Wewould like to thank Gib Bassett, John DiNardo, Olga Geling,and StevePortnoy for helpful comments on earlier drafts. This paper has been prepared for theJournal of Economic Perspectives #5CSymposium on Econometric Tools. The researchwas partiallysupported by NSF grant SBR-9911184.12 Quanti
9、le RegressionWhen we add the further requirement that the errors are Gaussian, least-squaresmethods deliver the maximum likelihood estimates of the conditional mean functionand achieveawell-publicized optimality. Indeed, Gauss seems to have #5Cdiscoveredthe Gaussian densityasanex post rationalizatio
10、n for the optimality of least-squaresmethods. But we will argue that there is more to econometric life than is dreamtof in this philosophy of the location shift model. Covariates may in#0Duence the con-ditional distribution of the response in myriad other ways: expanding its dispersionas in traditio
11、nal models of heteroscedasticity, stretching one tail of the distribution,compressing the other tail, and even inducing multimodality. Explicit investigation ofthese e#0Bects via quantile regression can provide a more nuanced view of the stochasticrelationship between variables, and therefore a more
12、 informative empirical analysis.The remainder of the paper is organized as follows. Section 2 brie#0Dy explains howthe ordinary quantiles, and consequently the regression quantiles, may be de#0Cnedas the solution to a simple minimization of a weighted sum of absolute residuals.We then illustrate the
13、 technique in the classical bivariate setting of Ernst Engels#281857#29 original food expenditure study. In Section 3 wesketch the outline of a moreambitious application to the analysis of infant birthweights in the U.S. In Section4we o#0Ber a brief review of recent empirical applications of quantil
14、e regression ineconomics. Section 5 contains some practical guidance on matters of computation,inference, and software. Section 6 o#0Bers several thumbnail sketches of #5Cwhat can gowrong and some possible remedies, and Section 7 concludes.2. Whatisit?Wesay that a student scores at the #1Cth quantil
15、e of a standardized exam if heperforms better than the proportion #1C, of the reference group of students, and worsethan the proportion #281 ,#1C#29. Thus, half of students perform better than the medianstudent, and half perform worse. Similarly, the quartiles divide the population intofour segments
16、with equal proportions of thereferencepopulation in eachsegment. Thequintiles divide the population into 5 parts; the deciles into 10 parts. The quantiles,or percentiles, or occasionally fractiles, refer to the general case. Quantile regressionseeks to extendthese ideas to the estimationof conditional quantile functions #7Bmodelsin whichquantilesof the conditional distributionof theresponse variable are expressedas functions of observed covariates. To accomplish this task weneeda
