Introduction to the Mathematical and Statistical Foundations of Econometrics

345 Pages·2004·3.106 MB·English

by  Herman J. Bierens

This page intentionally left blank
IntroductiontotheMathematicalandStatisticalFoundations
ofEconometrics
ThisbookisintendedforuseinarigorousintroductoryPh.D.-levelcourseinecono-
metricsorinafieldcourseineconometrictheory.Itcoversthemeasure–theoretical
foundationofprobabilitytheory,themultivariatenormaldistributionwithitsap-
plication to classical linear regression analysis, various laws of large numbers,
and central limit theorems and related results for independent random variables
aswellasforstationarytimeseries,withapplicationstoasymptoticinferenceof
M-estimatorsandmaximumlikelihoodtheory.Somechaptershavetheirownap-
pendixescontainingmoreadvancedtopicsand/ordifficultproofs.Moreover,there
arethreeappendixeswithmaterialthatissupposedtobeknown.AppendixIcon-
tainsacomprehensivereviewoflinearalgebra,includingalltheproofs.AppendixII
reviewsavarietyofmathematicaltopicsandconceptsthatareusedthroughoutthe
main text, and Appendix III reviews complex analysis. Therefore, this book is
uniquelyself-contained.
Herman J. Bierens is Professor of Economics at the Pennsylvania State Univer-
sity and part-time Professor of Econometrics at Tilburg University, The Nether-
lands. He is Associate Editor of the Journal of Econometrics and Econometric
Reviews, and has been an Associate Editor of Econometrica. Professor Bierens
has written two monographs, Robust Methods and Asymptotic Theory in Nonlin-
ear Econometrics and Topics in Advanced Econometrics (Cambridge University
Press 1994), as well as numerous journal articles. His current research interests
aremodel(mis)specificationanalysisineconometricsanditsapplicationinempir-
icalresearch,timeserieseconometrics,andtheeconometricanalysisofdynamic
stochasticgeneralequilibriummodels.

ThemesinModernEconometrics
Managingeditor
PETERC.B.PHILLIPS,YaleUniversity
Serieseditors
RICHARDJ.SMITH,UniversityofWarwick
ERICGHYSELS,UniversityofNorthCarolina,ChapelHill
ThemesinModernEconometricsisdesignedtoservicethelargeandgrow-
ingneedforexplicitteachingtoolsineconometrics.Itwillprovideanorga-
nizedsequenceoftextbooksineconometricsaimedsquarelyatthestudent
population and will be the first series in the discipline to have this as its
express aim. Written at a level accessible to students with an introduc-
tory course in econometrics behind them, each book will address topics
or themes that students and researchers encounter daily. Although each
book will be designed to stand alone as an authoritative survey in its own
right, the distinct emphasis throughout will be on pedagogic excellence.
Titlesintheseries
StatisticsandEconometricModels:Volumes1and2
CHRISTIANGOURIEROUXandALAINMONFORT
TranslatedbyQUANGVUONG
TimeSeriesandDynamicModels
CHRISTIANGOURIEROUXandALAINMONFORT
TranslatedandeditedbyGIAMPIEROGALLO
UnitRoots,Cointegration,andStructuralChange
G.S.MADDALAandIN-MOOKIM
GeneralizedMethodofMomentsEstimation
EditedbyLA´SZLO´ MA´TYA´S
NonparametricEconometrics
ADRIANPAGANandAMANULLAH
EconometricsofQualitativeDependentVariables
CHRISTIANGOURIEROUX
TranslatedbyPAULB.KLASSEN
TheEconometricAnalysisofSeasonalTimeSeries
ERICGHYSELSandDENISER.OSBORN
SemiparametricRegressionfortheAppliedEconometrician
ADONISYATCHEW

INTRODUCTION
TO THE MATHEMATICAL AND
STATISTICAL FOUNDATIONS
OF ECONOMETRICS
HERMAN J. BIERENS
PennsylvaniaStateUniversity
  
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge  , UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521834315
© Herman J. Bierens 2005
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Contents
Preface page xv
1 ProbabilityandMeasure 1
1.1 TheTexasLotto 1
1.1.1 Introduction 1
1.1.2 BinomialNumbers 2
1.1.3 SampleSpace 3
1.1.4 AlgebrasandSigma-AlgebrasofEvents 3
1.1.5 ProbabilityMeasure 4
1.2 QualityControl 6
1.2.1 SamplingwithoutReplacement 6
1.2.2 QualityControlinPractice 7
1.2.3 SamplingwithReplacement 8
1.2.4 LimitsoftheHypergeometricandBinomial
Probabilities 8
1.3 WhyDoWeNeedSigma-AlgebrasofEvents? 10
1.4 PropertiesofAlgebrasandSigma-Algebras 11
1.4.1 GeneralProperties 11
1.4.2 BorelSets 14
1.5 PropertiesofProbabilityMeasures 15
1.6 TheUniformProbabilityMeasure 16
1.6.1 Introduction 16
1.6.2 OuterMeasure 17
1.7 LebesgueMeasureandLebesgueIntegral 19
1.7.1 LebesgueMeasure 19
1.7.2 LebesgueIntegral 19
1.8 RandomVariablesandTheirDistributions 20
1.8.1 RandomVariablesandVectors 20
1.8.2 DistributionFunctions 23
1.9 DensityFunctions 25
vii
viii Contents
1.10 ConditionalProbability,Bayes’Rule,
andIndependence 27
1.10.1 ConditionalProbability 27
1.10.2 Bayes’Rule 27
1.10.3 Independence 28
1.11 Exercises 30
Appendix1.A – CommonStructureoftheProofsofTheorems
1.6and1.10 32
Appendix1.B – ExtensionofanOuterMeasuretoa
ProbabilityMeasure 32
2 BorelMeasurability,Integration,andMathematical
Expectations 37
2.1 Introduction 37
2.2 BorelMeasurability 38
2.3 IntegralsofBorel-MeasurableFunctionswithRespect
toaProbabilityMeasure 42
2.4 GeneralMeasurabilityandIntegralsofRandom
VariableswithRespecttoProbabilityMeasures 46
2.5 MathematicalExpectation 49
2.6 SomeUsefulInequalitiesInvolvingMathematical
Expectations 50
2.6.1 Chebishev’sInequality 51
2.6.2 Holder’sInequality 51
2.6.3 Liapounov’sInequality 52
2.6.4 Minkowski’sInequality 52
2.6.5 Jensen’sInequality 52
2.7 ExpectationsofProductsofIndependentRandom
Variables 53
2.8 Moment-GeneratingFunctionsandCharacteristic
Functions 55
2.8.1 Moment-GeneratingFunctions 55
2.8.2 CharacteristicFunctions 58
2.9 Exercises 59
Appendix2.A – UniquenessofCharacteristicFunctions 61
3 ConditionalExpectations 66
3.1 Introduction 66
3.2 PropertiesofConditionalExpectations 72
3.3 ConditionalProbabilityMeasuresandConditional
Independence 79
3.4 ConditioningonIncreasingSigma-Algebras 80