Table Of Content

(cid:2) (cid:2) “9781119975281fm” — 2013/9/27 — 15:47 — page i — #1 (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) “9781119975281fm” — 2013/10/10 — 17:21 — page i — #1 (cid:2) (cid:2) Quantile Regression (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) “9781119975281fm” — 2013/10/10 — 17:21 — page ii — #2 (cid:2) (cid:2) WILEYSERIESINPROBABILITYANDSTATISTICS EstablishedbyWalterA.ShewhartandSamuelS.Wilks Editors DavidJ.Balding,NoelA.C.Cressie,GarrettM.Fitzmaurice,HarveyGoldstein, IainM.Johnstone,GeertMolenberghs,DavidW.Scott,AdrianF.M.Smith,RueyS. Tsay,SanfordWeisberg EditorsEmeriti VicBarnett,J.StuartHunter,JosephB.Kadane,JozefL.Teugels Acompletelistofthetitlesinthisseriesappearsattheendofthisvolume. (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) “9781119975281fm” — 2013/10/10 — 17:21 — page iii — #3 (cid:2) (cid:2) Quantile Regression Theory and Applications Cristina Davino Departmentof Political Science,Communication and International Relations, UniversityofMacerata,Italy Marilena Furno DepartmentofAgriculture UniversityofNaplesFedericoII, Italy Domenico Vistocco DepartmentofEconomicsandLaw Universityof Cassino,Italy (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) “9781119975281fm” — 2013/10/10 — 17:21 — page iv — #4 (cid:2) (cid:2) Thiseditionfirstpublished2014 ©2014JohnWiley&Sons,Ltd RegisteredOffice JohnWiley&Sons,Ltd,TheAtrium,SouthernGate,Chichester,WestSussex,PO198SQ,UnitedKingdom Fordetailsofourglobaleditorialoffices,forcustomerservicesandforinformationabouthowtoapplyfor permissiontoreusethecopyrightmaterialinthisbookpleaseseeourwebsiteatwww.wiley.com. Therightoftheauthortobeidentifiedastheauthorofthisworkhasbeenassertedinaccordancewiththe Copyright,DesignsandPatentsAct1988. Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmitted,in anyformorbyanymeans,electronic,mechanical,photocopying,recordingorotherwise,exceptaspermitted bytheUKCopyright,DesignsandPatentsAct1988,withoutthepriorpermissionofthepublisher. Wileyalsopublishesitsbooksinavarietyofelectronicformats.Somecontentthatappearsinprintmaynotbe availableinelectronicbooks. Designationsusedbycompaniestodistinguishtheirproductsareoftenclaimedastrademarks.Allbrandnames andproductnamesusedinthisbookaretradenames,servicemarks,trademarksorregisteredtrademarksof theirrespectiveowners.Thepublisherisnotassociatedwithanyproductorvendormentionedinthisbook. LimitofLiability/DisclaimerofWarranty:Whilethepublisherandauthorhaveusedtheirbesteffortsin preparingthisbook,theymakenorepresentationsorwarrantieswithrespecttotheaccuracyorcompletenessof thecontentsofthisbookandspecificallydisclaimanyimpliedwarrantiesofmerchantabilityorfitnessfora particularpurpose.Itissoldontheunderstandingthatthepublisherisnotengagedinrenderingprofessional servicesandneitherthepublishernortheauthorshallbeliablefordamagesarisingherefrom.Ifprofessional adviceorotherexpertassistanceisrequired,theservicesofacompetentprofessionalshouldbesought. LibraryofCongressCataloging-in-PublicationData Davino,Cristina. Quantileregression:theoryandapplications/CristinaDavino,MarilenaFurno,DomenicoVistocco. pagescm–(Wileyseriesinprobabilityandstatistics) Includesbibliographicalreferencesandindex. ISBN978-1-119-97528-1(hardback) 1. Quantileregression. 2. Regressionanalysis. I. Furno,Marilena,1957–. II. Vistocco,Domenico. III. Title. QA278.2.D382013 519.5’36–dc23 2013023591 AcataloguerecordforthisbookisavailablefromtheBritishLibrary. Thecoverimagecontainsadetailofthecoverplateofthe‘TombofDiver’,reproducedbykindpermissionof theArchaeologicalMuseumofPaestum,Italy(grantn.19/2013MinisteroperiBenieleAttivitàCulturali, SoprintendenzaperiBeniArcheologicidiSalerno,Avellino,BeneventoeCaserta,Italy). ThedetailwasdrawnfromaphotooftheMuseumcollection(authors:FrancescoVallettaeGiovanniGrippo) ISBN:978-1-119-97528-1 Setin10/12ptTimesbySPiPublishersServices,Pondicherry,India 1 2014 (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) “9781119975281fm” — 2013/10/10 — 17:21 — page v — #5 (cid:2) (cid:2) Contents Preface ix Acknowledgments xi Introduction xii Nomenclature xv 1 Avisualintroductiontoquantileregression 1 Introduction 1 1.1 Theessentialtoolkit 1 1.1.1 Unconditionalmean,unconditionalquantilesandsurroundings 2 1.1.2 Technicalinsight:Quantilesassolutionsofaminimization problem 4 1.1.3 Conditionalmean,conditionalquantilesandsurroundings 6 1.2 ThesimplestQRmodel:Thecaseofthedummyregressor 8 1.3 AslightlymorecomplexQRmodel:Thecaseofanominalregressor 13 1.4 AtypicalQRmodel:Thecaseofaquantitativeregressor 15 1.5 Summaryofkeypoints 20 References 21 2 Quantileregression:Understandinghowandwhy 22 Introduction 22 2.1 Howandwhyquantileregressionworks 22 2.1.1 Thegenerallinearprogrammingproblem 23 2.1.2 ThelinearprogrammingformulationfortheQRproblem 26 2.1.3 Methodsforsolvingthelinearprogrammingproblem 31 2.2 Asetofillustrativeartificialdata 33 2.2.1 Homogeneouserrormodels 33 2.2.2 Heterogeneouserrormodels 35 2.2.3 Dependentdataerrormodels 36 2.3 HowandwhytoworkwithQR 38 2.3.1 QRforhomogeneousandheterogeneousmodels 38 2.3.2 QRpredictionintervals 42 2.3.3 Anoteonthequantileprocess 48 2.4 Summaryofkeypoints 60 References 62 (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) “9781119975281fm” — 2013/10/10 — 17:21 — page vi — #6 (cid:2) (cid:2) vi CONTENTS 3 Estimatedcoefficientsandinference 64 Introduction 64 3.1 Empiricaldistributionofthequantileregressionestimator 64 3.1.1 Thecaseofi.i.d.errors 66 3.1.2 Thecaseofi.ni.d.errors 71 3.1.3 Thecaseofdependenterrors 73 3.2 InferenceinQR,thei.i.d.case 76 3.3 Wald,Lagrangemultiplier,andlikelihoodratiotests 84 3.4 Summaryofkeypoints 92 References 93 4 Additionaltoolsfortheinterpretationandevaluationofthe quantileregressionmodel 94 Introduction 94 4.1 Datapre-processing 95 4.1.1 Explanatoryvariabletransformations 95 4.1.2 Dependentvariabletransformations 97 4.2 Responseconditionaldensityestimations 107 4.2.1 Thecaseofdifferentscenariosimulations 107 4.2.2 Thecaseoftheresponsevariablereconstruction 117 4.3 Validationofthemodel 117 4.3.1 Goodnessoffit 117 4.3.2 Resamplingmethods 120 4.4 Summaryofkeypoints 128 References 128 5 Modelswithdependentandwithnon-identicallydistributeddata 131 Introduction 131 5.1 Acloserlookatthescaleparameter,theindependentand identicallydistributedcase 131 5.1.1 Estimatingthevarianceofquantileregressions 131 5.1.2 Confidenceintervalsandhypothesistestingonthe estimatedcoefficients 134 5.1.3 Exampleforthei.i.d.case 134 5.2 Thenon-identicallydistributedcase 137 5.2.1 Exampleforthenon-identicallydistributedcase 141 5.2.2 Quickwaystotestequalityofcoefficientsacrossquantiles inStata 145 5.2.3 Thewageequationrevisited 147 5.3 Thedependentdatamodel 152 5.3.1 Examplewithdependentdata 155 5.4 Summaryofkeypoints 158 References 158 Appendix5.AHeteroskedasticitytestsandweightedquantile regression,StataandRcodes 159 (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) “9781119975281fm” — 2013/10/10 — 17:21 — page vii — #7 (cid:2) (cid:2) CONTENTS vii 5.A.1 KoenkerandBassettestforheteroskedasticitycomparing twoquantileregressions 159 5.A.2 KoenkerandBassettestforheteroskedasticitycomparing allquantileregressions 159 5.A.3 Quick tests for heteroskedasticity comparing quantile regressions 160 5.A.4 Computetheindividualroleofeachexplanatoryvariable tothedependentvariable 161 5.A.5 R-codesfortheKoenkerandBassettestfor heteroskedasticity 161 Appendix5.BDependentdata 162 6 Additionalmodels 163 Introduction 163 6.1 Nonparametricquantileregression 163 6.1.1 Localpolynomialregression 164 6.1.2 Quantilesmoothingsplines 169 6.2 Nonlinearquantileregression 172 6.3 Censoredquantileregression 175 6.4 Quantileregressionwithlongitudinaldata 183 6.5 Groupeffectsthroughquantileregression 187 6.6 Binaryquantileregression 195 6.7 Summaryofkeypoints 197 References 197 AppendixA QuantileregressionandsurroundingsusingR 201 Introduction 201 A.1 Loadingdata 202 A.1.1 Textdata 202 A.1.2 Spreadsheetdata 203 A.1.3 Filesfromotherstatisticalpackages 204 A.2 Exploringdata 205 A.2.1 Graphicaltools 205 A.2.2 Summarystatistics 209 A.3 Modelingdata 211 A.3.1 Ordinaryleastsquaresregressionanalysis 211 A.3.2 Quantileregressionanalysis 212 A.4 Exportingfiguresandtables 217 A.4.1 Exportingfigures 217 A.4.2 Exportingtables 218 References 218 AppendixB QuantileregressionandsurroundingsusingSAS 220 Introduction 220 B.1 Loadingdata 221 (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) “9781119975281fm” — 2013/10/10 — 17:21 — page viii — #8 (cid:2) (cid:2) viii CONTENTS B.1.1 Textdata 221 B.1.2 Spreadsheetdata 222 B.1.3 Filesfromotherstatisticalpackages 222 B.2 Exploringdata 223 B.2.1 Graphicaltools 223 B.2.2 Summarystatistics 227 B.3 Modelingdata 229 B.3.1 Ordinaryleastsquaresregressionanalysis 229 B.3.2 Quantileregressionanalysis 233 B.4 Exportingfiguresandtables 239 References 241 AppendixC QuantileregressionandsurroundingsusingStata 242 Introduction 242 C.1 Loadingdata 243 C.1.1 Textdata 243 C.1.2 Spreadsheetdata 244 C.1.3 Filesfromotherstatisticalpackages 245 C.2 Exploringdata 245 C.2.1 Graphicaltools 245 C.2.2 Summarystatistics 248 C.3 Modelingdata 249 C.3.1 Ordinaryleastsquaresregressionanalysis 249 C.3.2 Quantileregressionanalysis 251 C.4 Exportingfiguresandtables 255 C.4.1 Exportingfigures 255 C.4.2 Exportingtables 255 References 256 Index 257 (cid:2) (cid:2) (cid:2) (cid:2)