An Introduction to Quantile Regression and the QUANTREG Proc

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Paper213-30

AnIntroductiontoQuantileRegressionandtheQUANTREGProcedure

Colin(Lin)Chen,SASInstituteInc.,Cary,NC

ABSTRACT

Ordinaryleast-squaresregressionmodelstherelationshipbetweenoneormorecovariatesXandthecon-ditionalmeanofaresponsevariableYgivenX=x.Incontrast,quantileregressionmodelstherelationshipbetweenXandtheconditionalquantilesofYgivenX=x,soitisespeciallyusefulinapplicationswhereextremesareimportant,suchasenvironmentalstudieswhereupperquantilesofpollutionlevelsarecriticalfromapublichealthperspective.Quantileregressionalsoprovidesamorecompletepictureofthecondi-tionaldistributionofYgivenX=xwhenbothlowerandupperorallquantilesareofinterest,asintheanalysisofbodymassindexwherebothlower(underweight)andupper(overweight)quantilesarecloselywatchedhealthstandards.ThispaperdescribesthenewQUANTREGprocedureinSAS9.1,whichcom-putesestimatesandrelatedquantitiesforquantileregressionbysolvingamodi cationoftheleast-squarescriterion.

INTRODUCTION

ThispaperintroducestheQUANTREGprocedure,whichcomputesestimatesandrelatedquantitiesforquantileregression.ForSAS9.1,http://doc.guandang.net.

Ordinaryleast-squaresregressionmodelstherelationshipbetweenoneormorecovariatesXandtheconditionalmeanoftheresponsevariableYgivenX=x.Quantileregression,whichwasintroducedbyKoenkerandBassett(1978),extendstheregressionmodeltoconditionalquantilesoftheresponsevariable,suchasthe90thpercentile.Quantileregressionisparticularlyusefulwhentherateofchangeintheconditionalquantile,expressedbytheregressioncoef cients,dependsonthequantile.

Asanexampleofdatawiththisstructure,considerthescatterplotinFigure1ofbodymassindex(BMI)againstagefor8,250menfromafour-year(1999–2002)surveybytheNationalCenterforHealthStatistics.MoredetailsaboutthedatacanbefoundinChen(2004).Bodymassindex,de nedastheratioofweight(kg)tosquaredheight(m2),isameasureofoverweightorunderweight.ThepercentilesofBMIforspeci edagesareofparticularinterest.Asageincreases,thesepercentilesprovidegrowthpatternsofBMInotonlyforthemajorityofthepopulation,butalsoforunderweightoroverweightextremesofthepopulation.Inaddition,thepercentilesofBMIforaspeci edageprovideareferenceforinpidualsatthatagewithrespecttothepopulation.

ThecurvesinFigure1represent ttedconditionalquantilesofBMI,includingthemedian,computedwiththeQUANTREGprocedureforapolynomialregressionmodelinage.Duringthequickgrowthperiod(ages2to20),thedispersionofBMIincreasesdramatically;itbecomesstableduringmiddleage,andthenitcontractsafterage60.Thispatternsuggeststhataneffectivewaytocontroloverweightinapopulationistostartinchildhood.

Notethatordinaryleast-squaresregressioncanbeusedtoestimateconditionalpercentilesbymakingadistributionalassumptionsuchasnormalityfortheerrorterminthemodel.However,itwouldnotbeappropriateheresincethedifferencebetweeneach ttedpercentilecurveandthemeancurvewouldbeconstantwithage.Least-squaresregressionassumesthatthecovariatesaffectonlythelocationoftheconditionaldistributionoftheresponse,andnotitsscaleoranyotheraspectofitsdistributionalshape.Themainadvantageofquantileregressionoverleast-squaresregressionisits exibilityformodelingdatawithheterogeneousconditionaldistributions.Dataofthistypeoccurinmany elds,includingeconomet-rics,survivalanalysis,andecology;refertoKoenkerandHallock(2001).Quantileregressionprovidesa

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completepictureofthecovariateeffectwhenasetofpercentilesismodeled,anditmakesnodistributional

assumptionabouttheerrorterminthe

model.

Figure1.BMIwithGrowthPercentileCurves

Thenextsectionprovidesamoreformalde nitionofquantileregression,followedbyacloserlookattheuseoftheQUANTREGprocedureintheBMIexample.Asecondexampleintroducesnonparametricquantileregression.Subsequentsectionsdiscussvariousaspectsofquantileregression,includingalgorithmsforestimatingregressioncoef cients,con denceintervals,statisticaltests,detectionofleveragepointsandoutliers,andquantileprocessplots.Theseaspectsareillustratedwithathirdexampleusingeconomicgrowthdata.ThelastsectiondiscussesthescalabilityoftheQUANTREGprocedure.

QUANTILEREGRESSION

Quantileregressiongeneralizestheconceptofaunivariatequantiletoaconditionalquantilegivenoneormorecovariates.

ForarandomvariableYwithprobabilitydistributionfunction

F(y)=Prob(Y≤y)

theτthquantileofY isde nedastheinversefunction

Q(τ)=inf{y:F(y)≥τ}

thatastudent’sscoreonatestisattheτthquantileifhis(orher)gradeisbetterthan100τ%ofthestudentswhotookthe

test.Thescoreisalsosaidtobeatthe100τthpercentile.

Recall

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where 0<τ<1.Inparticular,themedianisQ(1/2).

Forarandomsample{y1,...,yn}ofY,itiswellknownthatthesamplemedianistheminimizerofthesumofabsolutedeviations

min

n

ξ∈Ri=1

|yi ξ|

Likewise,thegeneralτthsamplequantileξ(τ),whichistheanalogueofQ(τ),maybeformulatedasthe

solutionoftheoptimizationproblem

n

ξ∈Ri=1

minρτ(yi ξ)

whereρτ(z)=z(τ I(z<0)),0<τ<1.HereI(·)denotestheindicatorfunction.Justasthesamplemean,whichminimizesthesumofsquaredresiduals

n i=1

µ =argminµ∈R

(yi µ)2

canbeextendedtothelinearconditionalmeanfunctionE(Y|X=x)=x βbysolving

=argminpββ∈R

n i=1

2

(yi x iβ)

thelinearconditionalquantilefunction,Q(τ|X=x)=x β(τ),canbeestimatedbysolving

(τ)=argminpββ∈R

n i=1

ρτ(yi x iβ)

(τ)iscalledtheτthregressionquantile.Thecaseτ=1/2,foranyquantileτ∈(0,1).Thequantityβ

whichminimizesthesumofabsoluteresiduals,correspondstomedianregression,whichisalsoknownasL1regression.

USINGTHEQUANTREGPROCEDURE

TheQUANTREGprocedurecomputesthequantilefunctionQ(τ|X=x)andconductsstatisticalinferences

(τ).ThissectionintroducestheQUANTREGprocedurebyrevisitingtheontheestimatedparametersβ

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