Quantile Regression-An Introduction

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F.HACAbstract.uantileregression,asintroducedbyoenkerandBassett(1978),maybeviewedasanextensionofclassicalleastsquaresestimationofconditionalmeanmodelstotheestimationofanensembleofmodelsforseveralconditionalquan-tilefunctions.Thecentrals

2、pecialcaseisthemedianregressionestimatorwhichminimizesasumofabsoluteerrors.therconditionalquantilefunctionsareesti-matedbyminimizinganasymmetricallyweightedsumofabsoluteerrors.uantileregressionmethodsareillustratedwithapplicationstomodelsforCEpay,fodexpenditure,andinfantbirthweight.Wesayt

3、hatastudentscoresatthethquantileofastandardizedexamifheperformsbetterthantheproportion,ofthereferencegroupofstudents,andworsethantheproportion(1).Thus,halfofstudentsperformbetterthanthemedianstudent,andhalfperformworse.Similarly,thequartilesdividethepopulationintofoursegmentswithequalpr

4、oportionsofthereferencepopulationineachsegment.Thequintilesdividethepopulationinto5parts;thedecilesinto10parts.Thequantiles,orpercentiles,oroccasionallyfractiles,refertothegeneralcase.uantileregressionasintroducedbyoenkerandBassett(1978)seekstoextendtheseideastotheestimationofconditionalqu

5、antilefunctions{modelsinwhichquantilesoftheconditionaldistributionoftheresponsevariableareexpressedasfunctionsofobservedcovariates.
nFigure1weillustrateoneapproachtothistaskbasedonTukey'sboxplot.An-nualCEcompensationisplottedasafunctionofrm'smarketvalueofequity.Asampleof1660rmswassplitin

6、to10groupsofequalsizeaccordingtotheirmarketcapitalization.Foreachgroupof166rmswecomputethethreequartilesofCEcompensation:salary,bonus,andothercompensationincludingstockoptionsval-uedbyBlack-Scholesatthetimeofthegrant.Foreachgroupthebowtie-likeboxrepresentsthemiddlehalfofthesalarydistribut

7、ionlyingbetweentherstandthirdquartiles.Thehorizontallinenearthemiddleofeachboxrepresentsthemediancom-pensationforeachgroupofCEs,andthenotchesrepresentanestimatedcondenceintervalforeachmedianestimate.Thefullrangeoftheobservedsalariesineachgroupisrepres