Quantile Regression: Theory and Applications

290 Pages·2013·3.435 MB·English

by  Cristina Davino, Marilena Furno, Domenico Vistocco

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Quantile Regression
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WILEYSERIESINPROBABILITYANDSTATISTICS
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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
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Thiseditionfirstpublished2014
©2014JohnWiley&Sons,Ltd
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Davino,Cristina.
Quantileregression:theoryandapplications/CristinaDavino,MarilenaFurno,DomenicoVistocco.
pagescm–(Wileyseriesinprobabilityandstatistics)
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ISBN978-1-119-97528-1(hardback)
1. Quantileregression. 2. Regressionanalysis. I. Furno,Marilena,1957–. II. Vistocco,Domenico.
III. Title.
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1 2014
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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
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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
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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
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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
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