An Information Theoretic Approach to Econometrics Paperback

260 Pages·2011·2.253 MB·English

by  George G. Judge

ANINFORMATIONTHEORETICAPPROACHTOECONOMETRICS
Thisbookisintendedtoprovidethereaderwithafirmconceptualandempirical
understandingofbasicinformationtheoreticeconometricmodelsandmeth-
ods.Becausemostdataareobservational,practitionersworkwithindirectnoisy
observationsandill-posedeconometricmodelsintheformofstochasticinverse
problems. Consequently, traditional econometric methods in many cases are
notapplicableforansweringmanyofthequantitativequestionsthatanalysts
wish to ask. After initial chapters deal with parametric and semiparametric
linear probability models, the focus turns to solving nonparametric stochas-
tic inverse problems. In succeeding chapters, a family of power divergence
measure–likelihoodfunctionsisintroducedforarangeoftraditionalandnon-
traditional econometric-model problems. Finally, within either an empirical
maximumlikelihoodorlosscontext,GeorgeG.JudgeandRonC.Mittelham-
mersuggestabasisforchoosingamemberofthedivergencefamily.
GeorgeG.JudgeisaProfessorattheUniversityofCalifornia,Berkeley.Professor
JudgehasalsoservedonthefacultyoftheUniversityofIllinois,Universityof
Connecticut,andOklahomaStateUniversityandhasbeenavisitingprofessorat
severalU.S.andEuropeanuniversities.Heisthecoauthororeditorof15books
in econometrics and related fields and author or coauthor of more than 150
articlesinrefereedjournals.Hisresearchexploresspecificationandevaluation
ofstatisticaldecisionrules,improvedinferencemethods,andparametricand
semiparametric estimation and information recovery in the case of ill-posed
inverseproblemswithnoise.JudgeisaFellowoftheEconometricSociety,the
JournalofEconometrics,andtheAmericanAgriculturalEconomicsAssociation.
RonC.MittelhammerisRegentsProfessorofEconomicSciencesandStatistics
at Washington State University. He is the author of Mathematical Statistics
for Economics and Business (1996); lead coauthor with George G. Judge and
Douglas J. Miller of Econometric Foundations (Cambridge University Press,
2000); and the author of numerous book chapters and articles in refereed
economics, statistics, and econometrics journals. Professor Mittelhammer’s
current research focuses on econometric theory for applications in a range
of economics fields. With more than two decades of graduate-level teaching
experience,hisskillasateacherofstatisticsandeconometricsisdocumented
byteachingevaluationsandawards.HeservedaspresidentoftheAgricultural
andAppliedEconomicsAssociationin2009–10.
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Cambridge Books Online © Cambridge University Press, 2012
An Information Theoretic Approach
to Econometrics
GEORGE G. JUDGE
UniversityofCalifornia,Berkeley
RON C. MITTELHAMMER
WashingtonStateUniversity
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Thispublicationisincopyright.Subjecttostatutoryexception
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permissionofCambridgeUniversityPress.
Firstpublished2012
PrintedintheUnitedStatesofAmerica
AcatalogrecordforthispublicationisavailablefromtheBritishLibrary.
LibraryofCongressCataloginginPublicationData
Judge,GeorgeG.
Aninformationtheoreticapproachtoeconometrics/GeorgeG.Judge,RonC.Mittelhammer.
p. cm.
Includesbibliographicalreferencesandindex.
ISBN978-0-521-86959-1(hardback)–ISBN978-0-521-68973-1(paperback)
1.Econometrics. I.Mittelhammer,Ron. II.Title.
HB139.J795 2011
330.01′5195–dc23 2011018358
ISBN978-0-521-86959-1Hardback
ISBN978-0-521-68973-1Paperback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyofURLsfor
externalorthird-partyInternetWebsitesreferredtointhispublicationanddoesnotguarantee
thatanycontentonsuchWebsitesis,orwillremain,accurateorappropriate.
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ToHeatherJudge-Price
ToLinda,andtomygoodfriendsandcolleagues
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Contents
Preface pagexv
1 EconometricInformationRecovery 1
1.1 BookObjectivesandProblemFormat 1
1.2 OrganizationoftheBook 3
1.3 SelectedReferences 4
PARTI TRADITIONALPARAMETRICANDSEMIPARAMETRICECONOMETRIC
MODELS:ESTIMATIONANDINFERENCE
2 FormulationandAnalysisofParametricandSemiparametric
LinearModels 7
2.1 DataSamplingProcesses(DSPs)andNotation 7
2.2 AParametricGeneralLinearModel 10
2.2.1 TheParametricModelandMaximumLikelihood
(ML)Estimationofβandσ2 12
2.2.2 TheParametricModelandInference 15
2.3 ASemiparametricGeneralLinearModel 16
2.3.1 TheSquaredErrorMetricandtheLeastSquares
(LS)Principle 17
2.3.2 TheLSEstimator 18
2.3.3 FiniteSampleStatisticalPropertiesoftheLS
Estimator 19
2.3.4 ConsistencyandAsymptoticNormalityoftheLS
Estimator 19
2.3.5 LinearSemiparametricModelInference 20
2.3.6 InferentialAsymptotics 21
2.3.7 HypothesisTesting:LinearEqualityRestrictions
onβ 22
vii
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viii Contents
2.4 GeneralLinearModelwithStochasticX 24
2.4.1 LinearModelAssumptions 25
2.4.2 LSEstimatorProperties:FiniteSamples 26
2.4.3 LSEstimatorProperties:Asymptotics 26
2.4.4 MLEstimationofβandσ2underConditional
Normality 27
2.4.5 HypothesisTestingandConfidenceRegion
Estimation 28
2.4.5a SemiparametricCase 28
2.4.5b ParametricCase 28
2.4.6 Summary:StatisticalImplicationsof
StochasticX 30
2.5 Extremum(E)EstimationandInference 30
2.5.1 MLandLSEstimatorsExpressedinEEstimator
Form 31
2.5.2 AsymptoticPropertiesofEEstimators 32
2.5.3 InferenceBasedonEEstimation 33
2.5.4 SummaryandForward:EEstimators 34
2.6 SelectedReferences 35
3 MethodofMoments,GeneralizedMethodofMoments,and
EstimatingEquations 36
3.1 Introduction 36
3.1.1 AJust-DeterminedMomentSystem
withRandomSamplingofScalars 37
3.2 Just-DeterminedMomentSystems,RandomSampling,
andMethodofMoments(MOM) 39
3.2.1 GeneralAsymptoticProperties 40
3.2.2 LinearModelSemiparametricEstimation
throughMomentEquations 41
3.2.3 MOMConclusions 42
3.3 GeneralizedMethodofMoments(GMM) 43
3.3.1 GMMFramework 43
3.3.2 GMMLinearModelEstimation 44
3.3.2a OptimalGMMWeightMatrix 45
3.3.2b SamplingPropertiesofEstimatedOptimal
GMM(EOGMM)Estimator 46
3.3.2c HypothesisTestingandConfidenceRegions 47
3.3.2d AdditionalPropertiesoftheGMM
Approach 48
3.3.2e SummaryandForward:TheGMM
Approach 49
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Contents ix
3.4 EstimatingEquations 50
3.4.1 DualitybetweenEstimatingEquations(EEs)andE
Estimators 51
3.4.2 LinearEstimatingFunctions(EFs) 52
3.4.3 OptimalUnbiasedEFs 54
3.4.3a Unbiasedness 54
3.4.3b OptimalEstimatingFunctions(OptEFs):
TheScalarCase 56
3.4.3c OptEFs:TheMultivariateCase 57
3.4.4 InferenceintheContextofEEEstimation 59
3.4.4a Wald(W)andZTestsandConfidence
Regions 59
3.4.4b GeneralizedScore(Lagrange
Multiplier-Type)TestsandConfidence
Regions 60
3.4.4.c Pseudo-LikelihoodRatioTestsand
ConfidenceRegions 61
3.5 EEstimationwithInstrumentalVariables 62
3.6 SummaryandForward 63
3.7 SelectedReferences 64
PARTII FORMULATIONANDSOLUTIONOFSTOCHASTICINVERSEPROBLEMS
4 AStochastic-EmpiricalLikelihoodInverseProblem:
FormulationandEstimation 69
4.1 Introduction 69
4.2 AStochasticLinearInverseProblem 71
4.2.1 AddressingtheIndeterminacyofUnknowns 73
4.3 NonparametricMLSolutionstoInverseProblems 74
4.3.1 NonparametricML 74
4.3.2 EmpiricalLikelihood(EL)Functionforθ 76
4.3.3 ComparingtheUseofEstimatingFunctions
inEEandELContexts 78
4.3.4 TheFunctionalFormoftheELFunction 80
4.3.5 SummaryoftheELConcept 81
4.3.6 MaximumEmpiricalLikelihood(MEL)Estimation
ofaPopulationMean 82
4.3.7 MELLinearModelEstimationforStochasticX 85
4.4 Epilogue 86
4.5 SelectedReferences 87
Appendix4. ANumericalExample:ComputingMEL
Estimates 87
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x Contents
5 AStochasticEmpiricalLikelihoodInverseProblem:Estimation
andInference 90
5.1 Introduction 90
5.2 MELInference:iidCase 90
5.2.1 MELEfficiencyProperty 91
5.3 EmpiricalExampleofMELEstimationBasedonTwo
Moments 94
5.4 HypothesisTestsandConfidenceRegions:iidCase 95
5.4.1 EmpiricalLikelihoodRatioTestsandConfidence
Regionsforc(θ) 95
5.4.2 WaldTestsandConfidenceRegionsforc(θ) 96
5.4.3 LagrangeMultiplierTestsandConfidence
Regionsforc(θ) 97
5.4.4 Z-TestofInequalityHypothesesfortheValueofc(θ) 97
5.4.5 TestingtheValidityofMomentEquations 98
5.4.6 MELTestingandConfidenceIntervalsfor
PopulationMean 99
5.4.7 IllustrativeMELConfidenceIntervalExample 100
5.5 ConcludingComments 101
5.6 SelectedReferences 103
6 Kullback-LeiblerInformationandtheMaximumEmpirical
ExponentialLikelihood 104
6.1 Introduction 104
6.1.1 SolutionstoSystemsofEstimatingEquationsand
Kullback-LeiblerInformation 104
6.2 Kullback-LeiblerInformationCriterion(KLIC) 106
6.2.1 RelationshipbetweenMaximumEmpirical
ExponentialLikelihood(MEEL)andKL
Information 108
6.2.1a ObjectiveofMEEL 109
6.3 TheGeneralMEELAlternativeEmpirical
LikelihoodFormulation 111
6.3.1 TheMEELEstimatorandAlternativeEmpirical
Likelihood 111
6.3.2 MEELAsymptotics 112
6.3.3 MEELInference 114
6.3.3a TestingH :c(θ)=r 114
0
6.3.3b TestingH :c(θ)≤rorH :c(θ)≥r 115
0 0
6.3.3c TestingtheValidityofMomentEquations 115
6.3.3d ConfidenceRegions 116
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Contents xi
6.3.4 ContrastingtheUseofEstimatingFunctions
inEEandMEELContexts 116
6.4 CombiningEstimationEquations
underKullback-LeiblerLoss 117
6.4.1 CombatingModelUncertainty:General
CombiningFormulations 117
6.4.2 Example:ACombinedEstimator 119
6.4.2a FiniteSamplePerformance 120
6.4.2b Implications 121
6.5 AnInformativeReferenceDistribution 121
6.6 ConcludingRemarks 123
6.7 ReaderIdeaChecklist 124
6.8 SelectedReferences 125
Appendix6.A RelationshipbetweentheMaximumEmpirical
Likelihood(MEL)ObjectiveandKLInformation 126
Appendix6.B NumericalIllustrationofMEELandMEL
EstimationofaProbabilityDistribution 128
Appendix6.C Shannon’sEntropy–SomeHistoricalPerspective 130
PARTIII AFAMILYOFMINIMUMDISCREPANCYESTIMATORS
7 TheCressie-ReadFamilyofDivergenceMeasuresandEmpirical
MaximumLikelihoodFunctions 135
7.1 Introduction 135
7.1.1 FamilyofLikelihoodFunctions 136
7.2 TheCressie-Read(CR)PowerDivergenceFamily 137
7.3 ThreeMainVariantsofI(p,q,γ) 139
7.4 MinimumPowerDivergenceandEmpiricalMaximum
Likelihood(EML)Estimation 140
7.5 Inference 142
7.5.1 TestStatistics 142
7.5.1a MomentValidityTests 143
7.5.1b TestsofParameterRestrictions 144
7.6 ConcludingRemarks 145
7.7 SelectedReferences 146
Appendix7.A Propositions,Proofs,andDefinitions 148
Appendix7.B EntropyFamilies 152
8 Cressie-Read-MPD-TypeEstimatorsinPractice:MonteCarlo
EvidenceofEstimationandInferenceSamplingPerformance 153
8.1 Introduction 153
8.2 DesignofSamplingExperiments 154
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