Probability Theory And Statistical Inference: Empirical Modeling With Observational Data

788 Pages·2019·33.66 MB·english

by  Aris Spanos

Probability Theory and Statistical
Inference
Doubt over the trustworthiness of published empirical results is not unwarranted
and is often a result of statistical misspecification: invalid probabilistic assump-
tionsimposedondata.Nowinitssecondedition,thisbestsellingtextbookoffers
acomprehensivecourseinempiricalresearchmethods,teachingtheprobabilistic
andstatisticalfoundationsthatenablethespecificationandvalidationofstatistical
models, providing the basis for an informed implementation of statistical proce-
duretosecurethetrustworthinessofevidence.Eachchapterhasbeenthoroughly
updated,accountingfordevelopmentsinthefieldandtheauthor’sownresearch.
The comprehensive scope of the textbook has been expanded by the addition of
a new chapter on the Linear Regression and related statistical models. This new
edition is now more accessible to students of disciplines beyond economics and
includesmorepedagogicalfeatures,withanincreasednumberofexamplesaswell
asreviewquestionsandexercisesattheendofeachchapter.
ARIS SPANOS is Wilson E. Schmidt Professor of Economics at Virginia Poly-
technicInstitute andStateUniversity.HeistheauthorofStatisticalFoundations
of Econometric Modelling (Cambridge, 1986) and, with D. G. Mayo, Error and
Inference: Recent Exchanges on Experimental Reasoning, Reliability, and the
ObjectivityandRationalityofScience(Cambridge,2010).

Probability Theory
and Statistical
Inference
Empirical Modeling with
Observational Data
Second Edition
Aris Spanos
Virginia Tech
(Virginia Polytechnic Institute & State University)
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www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107185142
DOI:10.1017/9781316882825
(cid:2)c ArisSpanos2019
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished1999
Thirdprinting2007
Secondedition2019
PrintedintheUnitedKingdombyTJInternationalLtd.PadstowCornwall
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary.
LibraryofCongressCataloging-in-PublicationData
Names:Spanos,Aris,1952–author.
Title:Probabilitytheoryandstatisticalinference:empiricalmodelling
withobservationaldata/ArisSpanos(VirginiaCollegeofTechnology).
Description:Cambridge;NewYork,NY:CambridgeUniversityPress,2019.|
Includesbibliographicalreferencesandindex.
Identifiers:LCCN2019008498(print)|LCCN2019016182(ebook)|ISBN9781107185142|ISBN
Subjects:LCSH:Probabilities–Textbooks.|Mathematical
statistics–Textbooks.
Classification:LCCQA273(ebook)|LCCQA273.S68752019(print)|DDC519.5–dc23
LCrecordavailableathttps://lccn.loc.gov/2019008498
ISBN978-1-107-18514-2Hardback
ISBN978-1-316-63637-4Paperback
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wifeEviefortheirunconditionalloveandsupport

Contents
PrefacetotheSecondEdition pagexix
1 AnIntroductiontoEmpiricalModeling 1
1.1 Introduction 1
1.2 StochasticPhenomena:APreliminaryView 3
1.2.1 ChanceRegularityPatterns 3
1.2.2 FromChanceRegularitiestoProbabilities 7
1.2.3 ChanceRegularityPatternsandReal-WorldPhenomena 11
1.3 ChanceRegularitiesandStatisticalModels 12
1.4 ObservedDataandEmpiricalModeling 14
1.4.1 Experimentalvs.ObservationalData 14
1.4.2 ObservedDataandtheNatureofaStatisticalModel 15
1.4.3 MeasurementScalesandData 16
1.4.4 MeasurementScaleandStatisticalAnalysis 18
1.4.5 Cross-Sectionvs.TimeSeries,isthattheQuestion? 20
1.4.6 LimitationsofEconomicData 22
1.5 StatisticalAdequacy 23
∗
1.6 Statisticalvs.SubstantiveInformation 25
1.7 LookingAhead 27
1.8 QuestionsandExercises 28
2 ProbabilityTheoryasaModelingFramework 30
2.1 Introduction 30
2.1.1 PrimaryObjective 30
2.1.2 Descriptivevs.InferentialStatistics 30
2.2 SimpleStatisticalModel:APreliminaryView 32
2.2.1 TheBasicStructureofaSimpleStatisticalModel 33
2.2.2 TheNotionofaRandomVariable:ANaiveView 34
2.2.3 DensityFunctions 35
2.2.4 ARandomSample:APreliminaryView 36
2.3 ProbabilityTheory:AnIntroduction 40
2.3.1 OutliningtheEarlyMilestonesofProbabilityTheory 40
2.3.2 ProbabilityTheory:AModelingPerspective 42
2.4 ASimpleGenericStochasticMechanism 42
2.4.1 TheNotionofaRandomExperiment 42
2.4.2 ABird’s-EyeViewoftheUnfoldingStory 44
vii
viii Contents
2.5 FormalizingCondition[a]:TheOutcomesSet 45
2.5.1 TheConceptofaSetinSetTheory 45
2.5.2 TheOutcomesSet 45
2.5.3 SpecialTypesofSets 46
2.6 FormalizingCondition[b]:EventsandProbabilities 48
2.6.1 Set-TheoreticOperations 48
2.6.2 Eventsvs.Outcomes 51
2.6.3 EventSpace 51
2.6.4 ADigression:WhatisaFunction? 58
2.6.5 TheMathematicalNotionofProbability 59
2.6.6 ProbabilitySpace(S,(cid:4),P(.)) 63
2.6.7 MathematicalDeduction 64
2.7 ConditionalProbabilityandIndependence 65
2.7.1 ConditionalProbabilityanditsProperties 65
2.7.2 TheConceptofIndependenceAmongEvents 69
2.8 FormalizingCondition[c]:SamplingSpace 70
2.8.1 TheConceptofRandomTrials 70
2.8.2 TheConceptofaStatisticalSpace 72
2.8.3 TheUnfoldingStoryAhead 74
2.9 QuestionsandExercises 75
3 TheConceptofaProbabilityModel 78
3.1 Introduction 78
3.1.1 TheStorySoFarandWhatComesNext 78
3.2 TheConceptofaRandomVariable 79
3.2.1 TheCaseofaFiniteOutcomesSet:S={s ,s ,...,s } 80
1 2 n
3.2.2 KeyFeaturesofaRandomVariable 81
3.2.3 The Case of a Countable Outcomes Set:
S={s ,s ,...,s ,...} 85
1 2 n
3.3 TheGeneralConceptofaRandomVariable 86
3.3.1 TheCaseofanUncountableOutcomesSetS 86
3.4 CumulativeDistributionandDensityFunctions 89
3.4.1 TheConceptofaCumulativeDistributionFunction 89
3.4.2 TheConceptofaDensityFunction 91
3.5 FromaProbabilitySpacetoaProbabilityModel 95
3.5.1 ParametersandMoments 97
3.5.2 FunctionsofaRandomVariable 97
3.5.3 NumericalCharacteristicsofRandomVariables 99
3.5.4 HigherMoments 102
3.5.5 TheProblemofMoments* 110
3.5.6 OtherNumericalCharacteristics 112
3.6 Summary 118
3.7 QuestionsandExercises 119
Appendix3.A:UnivariateDistributions 121