Introductory Econometrics

637 Pages·2017·7.412 MB·English

by  Phoebus Dhrymes (auth.)

Phoebus Dhrymes
Introductory
Econometrics
With Contributions by
John Guerard
Introductory Econometrics
Phoebus Dhrymes
Introductory Econometrics
With Contributions by John Guerard
PhoebusDhrymes(deceased)
ISBN978-3-319-65914-5 ISBN978-3-319-65916-9 (eBook)
DOI10.1007/978-3-319-65916-9
LibraryofCongressControlNumber:2017950078
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Contents
1 TheGeneralLinearModelI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Generalities. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 1
1.2 ModelsandTheirUses. . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 ModelSpecificationandEstimation. . . . . . . . . . . . . . . . . . . . . . . 3
2.1 Notation,BasicConcepts,andAssumptions. . . . . . . . . . . . 3
2.2 TheGeneralLinearModelDefined. . . . . . . . . . . . . . . . . . 7
2.3 EstimationofParameters:Assumptions. . . . . . . . . . . . . . . 8
2.4 PropertiesoftheOLSEstimatorofβ. . . . . . . . . . . . . . . . . 10
2.5 Estimationofσ2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 GoodnessofFit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.1 PropertiesoftheVectorofResiduals;the
CoefficientofDeterminationofMultipleRegression. . . . . . 17
3.2 TheGLMWithoutaConstantTerm. . . . . . . . . . . . . . . . . . 23
Appendix:AGeometricInterpretationoftheGLM:TheMultiple
CorrelationCoefficient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
TheGeometryoftheGLM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
AMeasureofCorrelationBetweenaScalarandaVector. . . . . . . 32
2 TheGeneralLinearModelII. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1 Generalities. . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 37
2 DistributionoftheEstimatorofβ. . . . . . . . . . . . . . . . . . . . . . . . . 38
2.1 EquivalenceofOLSandMLProcedures. . . . . . . . . . . . . . 38
2.2 DistributionoftheMLEstimatorofβ. . . . . . . . . . . . . . . . 40
2.3 DistributionofQuadraticForms
inNormalVariables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.4 TestsofSignificanceintheGLM
withNormalErrors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.5 FormalTestsofGoodnessofFit.. . . . . . .. . . . . . . .. . . . . 54
v
vi Contents
3 GeneralLinearRestriction:EstimationandTests. . . . . . . . . . . . . 56
4 TheInformationContentinResidualsandOutliers?. . . . . . . . . . . 70
5 MixedEstimatorsandtheBayesianApproach. . . . . . . . . . . . . . . . 73
Appendix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
NoncentralChiSquare. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
NoncentralF-Distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
MultipleComparisonTests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
GeometricPreliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
MultipleComparisonTests—TheS-Method. . . . . . . . . . . . . . . . . 104
3 TheGeneralLinearModelIII. . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
1 Generalities. . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 115
2 ViolationofStandardErrorProcessAssumptions. . . . . . . . . . . . . 116
2.1 NonzeroMean. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
2.2 NonscalarCovarianceMatrix. . . . . . . . . . . . . . . . . . . . . . . 118
2.3 Heteroskedasticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
2.4 AutocorrelatedErrors. . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
2.5 TestsforFirst-OrderAutoregression:
Durbin–WatsonTheory. . . . . . . . . . . . . . . . . . . . . . . . . . . 151
2.6 SystemsofGLM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
Appendix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
Durbin–WatsonTheory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
GapsinData. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
3 TablesforTestingHypothesesonthe
AutoregressiveStructureoftheErrorsinaGLM. . . . . . . . . . . . . . 220
4 TheGeneralLinearModelIV. . . . . . .. . . . . . . .. . . . . . . .. . . . . . 229
1 Multicollinearity:FailureoftheRankCondition. . . . . . . . . . . . . . 230
1.1 DefinitionoftheProblem. . . . . . . . . . . . . . . . . . . . . . . . . . 230
1.2 RecognitionofMulticollinearity
andProposedRemedies. . . . . . . . . . . . . . . . . . . . . . . . . . . 235
2 AnalysisofVariance:CategoricalExplanatoryVariables. . . . . . . . 255
3 AnalysisofCovariance:SomeCategorical
andSomeContinuousExplanatoryVariables. . . . . . . . . . . . . . . . 265
4 CaseStudyofReturns,Risk,PortfolioSelection
andEvaluation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
4.1 ExpectedReturnsModelingandStockSelection
Models:RecentEvidence.. . . . . . . . .. . . . . . . . . .. . . . . . 268
4.2 EvaluationofPortfolioPerformance:Origins. . . . . . . . . . . 283
4.3 PortfolioSimulationResultswiththeUSER
andGLERModelsandaUSERModelUpdate. . . . . . . . . . 285
5 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
Appendix:ModernRegressionAnalysis,theCase
ofLeastAngleRegression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
Contents vii
5 MisspecificationAnalysisandErrorsinVariables. . . . . . . . . . . . . 293
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
2 MisspecificationAnalysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
2.1 GeneralTheoryintheContextoftheGLM. . . . . . . . . . . . . 296
2.2 ProxyVariablesandTheirUse. . . . . . . . . . . . . . . . . . . . . . 305
2.3 NearCollinearity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
3 ErrorsinVariables(EIV):BivariateModel. . . . . . . . . . . . . . . . . . 321
3.1 InconsistencyoftheOLSEstimator. . . . . . . . . . . . . . . . . . 321
3.2 WaldandMLEstimators. . . . . . . . . . . . . . . . . . . . . . . . . . 325
4 ErrorsinVariables(EIV):GeneralModel. . . . . . . . . . . . . . . . . . . 331
4.1 DerivationoftheEstimator. . . . . . . . . . . . . . . . . . . . . . . . 331
4.2 AsymptoticProperties. . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
5 MisspecificationErrorAnalysisforEIVModels. . . . . . . . . . . . . . 343
5.1 TheGeneralCase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
6 SystemsofSimultaneousEquations. . . . . . . . . . . . . . . . . . . . . . . . 353
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
2 TheSimultaneousEquationsModel(SEM):Definitions,
Conventions,andNotation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
2.1 TheNatureoftheProblem. . . . . . . . . . . . . . . . . . . . . . . . . 354
2.2 DefinitionofGLSEM. . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
2.3 BasicConditionsUnderWhichtheGLSEM
IsEstimated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
3 TheIdentificationProblem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
4 EstimationoftheGLSEM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
4.1 FailureofOLSMethods. . . . . . . . . . . . . . . . . . . . . . . . . . 373
4.2 TwoStageLeastSquares(2SLS). . . . . . . . . . . . . . . . . . . . 376
4.3 ThreeStageLeastSquares(3SLS). . . . . . . . . . . . . . . . . . . 380
4.4 AsymptoticPropertiesof2SLSand3SLS. . . . . . . . . . . . . . 383
5 PredictionfromtheGLSEM. . . . . . . . . . . . . . . . . . . . . . . . . . . . 390
6 TheGLSEMandUndersizedSamples. . . . . . . . . . . . . . . . . . . . . 401
7 MaximumLikelihood(ML)Estimators. . . . . .. . . . .. . . . . .. . . . 406
7 TimeSeriesModeling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
1 BasicStatisticalPropertiesofEconomicSeries. . . . . . . . . . . . . . . 416
1.1 StationarityofEconomicSeries. . . . . . . . . . . . . . . . . . . . . 417
2 ARMAModelIdentificationinPractice. . . . . . . . . . . . . . . . . . . . 426
3 TimeSeriesModelingofRealGDPAnalysis,
1947–2015Q2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
4 AutomaticTimeSeriesModelSelection. . . . . . . . . . . . . . . . . . . . 436
5 TransferFunctionsusingLeadingEconomic
Indicators(LEI)toForecastRealGDPAnalysis,
1959-2015Q2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440
6 TestingforCausality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
7 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
Appendix:AdvancedTimeSeriesModeling. . . . . . . . . . . . . . . . . . . . 473
viii Contents
8 Forecasting:AccuracyandEvaluation. . . . . . . . . . . . . . . . . . . . . . 477
1 MeasuringForecastAccuracyandEstablishing
Benchmarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478
2 ForecastRationality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
2.1 AbsoluteandRelativeForecastAccuracy. . . . . . . . . . . . . 486
3 AnEqually-WeightedForecast. . . . . . . . . . . . . . . . . . . . . . . . . . 488
4 GNPForecasts:IsEqual-WeightingofForecasts
OptimalandCanTimeSeriesModelsBeatExperts?. . . . . . . . . . 493
5 LeadingEconomicIndicators(LEI)andReal
GDPAnalysis:ReviewingtheStatisticalEvidence,
1970–2002,andanUpdate,2003–6/2016. . . . . . . . . . . . . . . . . . 497
6 ForecastingUnemployment. . . . . . . . . . . . . . . . . . . . . . . . . . . . 500
7 Forecasts,ForecastRevisions,andtheApplicability
ofAnalystsForecastsinFinancialMarkets. . . . . . . . . . . . . . . . . 502
8 ForecastingRecessions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508
9 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514
AppendixA:ExponentialSmoothing. . . . . . . . . . . . . . . . . . . . . . . . . 514
AppendixB:TheThetaModel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517
AppendixC:AutomaticModelingoftheUnemploymentRate. . . . . . 519
9 DiscreteChoiceModels:LogitandProbitAnalysis. . . . . . . . . . . . 527
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527
2 TheNatureofDiscreteChoiceModels. . . . . . . . . . . . . . . . . . . . . 528
3 FormulationofDichotomousChoiceModels. . . . . . . . . . . . . . . . 529
4 ABehavioralJustificationfortheDichotomous
ChoiceModel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533
5 InapplicabilityofOLSProcedures. . . . . . . . . . . . . . . . . . . . . . . . 535
6 MaximumLikelihoodEstimation. . . . . . . . . . . . . . . . . . . . . . . . . 538
7 InferenceforDiscreteChoiceModels. . . . . . . . . . . . . . . . . . . . . . 540
8 PolytomousChoiceModels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543
8.1 GeneralDiscussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543
8.2 Estimation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544
Appendix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545
ARandomChoiceMotivationfortheLogisticModel. . . . . . . . . . 545
ConvexityoftheLikelihoodFunction. . .. . . . . . .. . . . . . .. . . . . 550
ConvexityoftheLikelihoodFunction
inthePolytomousLogisticCase. . . . . . . . . . . . . . . . . . . . . . . . . 554
10 StatisticalandProbabilisticBackground. . . . . . . . . . . . . . . . . . . . 561
1 MultivariateDensityandDistributionFunctions. . . . . . . . . . . . . . 561
1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561
1.2 MultivariateDistributions. . . . . . . . . . . . . . . . . . . . . . . . . . 561
1.3 ExpectationandCovarianceOperators. . . . . . . . . . . . . . . . . 564
1.4 AMathematicalDigression. . . . . . . . . . . . . . . . . . . . . . . . . 568
Contents ix
2 TheMultivariateNormalDistribution. . . . . . . . . . . . . . . . . . . . . . 570
2.1 Joint,Marginal,andConditionalDensity
Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570
2.2 TheMomentGeneratingFunction. . . . . . . . . . . . . . . . . . . . 579
3 PointEstimation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587
4 ElementsofBayesianInference. . . . . . . . . . . . . . . . . . . . . . . . . . 598
4.1 PriorandPosteriorDistributions. . . . . . . . . . . . . . . . . . . . . 598
4.2 InferenceinaBayesianContext. . . . . . . . . . . . . . . . . . . . . . 605
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613
A Tribute to Phoebus J. Dhrymes (1932–2016)
ArisSpanos,VirginiaTech,June2017
PhoebusJamesDhrymes wasborninKtima (Cyprus)
on October 1, 1932, and died on April 8, 2016, in
New York. His surname derives from the place of his
ancestry, “Dhrymou” (nowadays spelled “Drimou”), a
small village in the Paphos district, only a few miles
fromKtima.
His life journey from the humble beginnings of a
village boy to the Edwin W. Rickert Professor of Eco-
nomics at Columbia University is an amazing story of
diligenceandperseverance,sprinkledwithseeminglyinadvertentdecisions,suchas
hisvolunteeringtobedraftedintotheUSArmyin1952,butalwaysdrivenbythe
single-mindedness to succeed, and guided by the most honorable virtues of his
Greek heritage: fairness (δικαιoσύνη), temperance (σωφρoσύνη), fortitude
(ανδρεία), and wisdom (φρo´νηση): virtues that fostered an unquenched thirst for
knowledge,passionforteaching,andalife-timedevotiontohisfamily.
Education TherewasnothinginadvertentaboutanyofPhoebus’schoicesduring
hislife,includinghisvolunteeringtobedraftedintheUSarmy,onlyafewmonths
afterarrivingintheUSA.Havingnosourceofmoneywashisonlyreasonablepath
to a university education, thanks to the GI bill (The Servicemen’s Readjustment
Actof1944)!Indeed,tobeabletotakefulladvantageoftheGIbill’sallowanceof
$110 for 36 months after his two years of military service, he enrolled at the
Universityof Texas at Austinin 1954and completed his undergraduate degreein
Economics within 30 months by 1957. He could not afford to go to Berkeley or
NYC,eventhoughhehadoffers,becausehismonthlyallowance wouldnotcover
hislivingexpenses.
BasedonhisoutstandingperformanceasanundergraduateattheUniversityof
Texas(Austin),hemanagedtosecureaWoodrowWilsonFellowshiptostudyfor
xi