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1、Appendix CReviewofStatisticalInferencePreparedbyVeraTabakova,EastCarolinaUniversityAppendix C:Review of Statistical InferenceC.1ASampleofDataC.2AnEconometricModelC.3EstimatingtheMeanofaPopulationC.4EstimatingthePopulationVarianceandOtherMomentsC.5IntervalEstimationSlide C-Slide C-22Principles of Eco

2、nometrics,3rd EditionPrinciples of Econometrics,3rd EditionAppendix C:Review of Statistical InferenceC.6HypothesisTestsAboutaPopulationMeanC.7SomeOtherUsefulTestsC.8IntroductiontoMaximumLikelihoodEstimationC.9AlgebraicSupplementsSlide C-Slide C-33Principles of Econometrics,3rd EditionPrinciples of E

3、conometrics,3rd EditionC.1 A Sample of DataSlide C-Slide C-44Principles of Econometrics,3rd EditionPrinciples of Econometrics,3rd EditionC.1 A Sample of DataFigureC.1HistogramofHipSizesSlide C-Slide C-55Principles of Econometrics,3rd EditionPrinciples of Econometrics,3rd EditionC.2 An Econometric Mo

4、delSlide C-Slide C-66Principles of Econometrics,3rd EditionPrinciples of Econometrics,3rd Edition(C.1)(C.2)C.3 Estimating the Mean of a PopulationSlide C-Slide C-77Principles of Econometrics,3rd EditionPrinciples of Econometrics,3rd Edition(C.3)(C.4)C.3 Estimating the Mean of a PopulationSlide C-Sli

5、de C-88Principles of Econometrics,3rd EditionPrinciples of Econometrics,3rd Edition(C.3)(C.4)C.3 Estimating the Mean of a PopulationSlide C-Slide C-99Principles of Econometrics,3rd EditionPrinciples of Econometrics,3rd EditionC.3.1 The Expected Value ofSlide C-Slide C-1010Principles of Econometrics,

6、3rd EditionPrinciples of Econometrics,3rd Edition(C.5)C.3.2 The Variance ofSlide C-Slide C-1111Principles of Econometrics,3rd EditionPrinciples of Econometrics,3rd Edition(C.6)C.3.3 The Sampling Distribution ofFigureC.2IncreasingSampleSizeandSamplingDistributionofSlide C-Slide C-1212Principles of Ec

7、onometrics,3rd EditionPrinciples of Econometrics,3rd EditionC.3.4 The Central Limit TheoremSlide C-Slide C-1313Principles of Econometrics,3rd EditionPrinciples of Econometrics,3rd EditionCentralLimitTheorem:IfY1,YNareindependentandidenticallydistributed（i.i.d.）randomvariableswithmeanandvariance2,and

8、,thenhasaprobabilitydistributionthatconvergestothestandardnormalN(0,1)asN.C.3.4 The Central Limit TheoremFigureC.3CentralLimitTheoremSlide C-Slide C-1414Principles of Econometrics,3rd EditionPrinciples of Econometrics,3rd EditionC.3.5 Best Linear Unbiased EstimationSlide C-Slide C-1515Principles of

9、Econometrics,3rd EditionPrinciples of Econometrics,3rd EditionApowerfulfindingabouttheestimatorofthepopulationmeanisthatitisthebestofallpossibleestimatorsthatarebothlinearandunbiased(線性不偏).AlinearestimatorissimplyonethatisaweightedaverageoftheYis,suchas,wheretheaiareconstants.“Best”meansthatitisthel

10、inearunbiasedestimatorwiththesmallestpossiblevariance.C.4 Estimating the Population Variance and Other MomentsSlide C-Slide C-1616Principles of Econometrics,3rd EditionPrinciples of Econometrics,3rd EditionC.4.1 Estimating the population varianceSlide C-Slide C-1717Principles of Econometrics,3rd Edi

11、tionPrinciples of Econometrics,3rd Edition(C.7)C.4.1 Estimating the population varianceSlide C-Slide C-1818Principles of Econometrics,3rd EditionPrinciples of Econometrics,3rd Edition(C.9)(C.8)C.4.2 Estimating higher momentsInstatisticstheLawofLargeNumbers(大數法則大數法則)saysthatsamplemeansconvergetopopul

12、ationaverages(expectedvalues)asthesamplesizeN.Slide C-Slide C-1919Principles of Econometrics,3rd EditionPrinciples of Econometrics,3rd EditionC.4.2 Estimating higher momentsSlide C-Slide C-2020Principles of Econometrics,3rd EditionPrinciples of Econometrics,3rd EditionC.5 Interval EstimationC.5.1Int

13、ervalEstimation:2KnownSlide C-Slide C-2121Principles of Econometrics,3rd EditionPrinciples of Econometrics,3rd Edition(C.10)C.5.1 Interval Estimation:2 KnownFigureC.4CriticalValuesfortheN(0,1)DistributionSlide C-Slide C-2222Principles of Econometrics,3rd EditionPrinciples of Econometrics,3rd Edition

14、C.5.1 Interval Estimation:2 KnownSlide C-Slide C-2323Principles of Econometrics,3rd EditionPrinciples of Econometrics,3rd Edition(C.11)C.5.1 Interval Estimation:2 KnownSlide C-Slide C-2424Principles of Econometrics,3rd EditionPrinciples of Econometrics,3rd Edition(C.13)(C.12)C.5.3 Interval Estimatio

15、n:2 UnknownWhen2isunknownitisnaturaltoreplaceitwithitsestimatorSlide C-Slide C-2525Principles of Econometrics,3rd EditionPrinciples of Econometrics,3rd Edition(C.14)C.5.3 Interval Estimation:2 UnknownSlide C-Slide C-2626Principles of Econometrics,3rd EditionPrinciples of Econometrics,3rd Edition(C.1

16、5)C.5.3 Interval Estimation:2 UnknownSlide C-Slide C-2727Principles of Econometrics,3rd EditionPrinciples of Econometrics,3rd EditionRemark:The confidence interval(C.15)is based upon theassumptionthatthepopulationisnormallydistributed,sothatisnormally distributed.If the population is not normal,then

17、 weinvokethecentrallimittheorem,andsaythatisapproximatelynormalin“large”samples,whichfromFigureC.3youcanseemightbeasfewas30observations.Inthiscasewecanuse(C.15),recognizing that there is an approximation error introduced insmallersamples.C.5.5 Interval estimation using the hip dataGivenarandomsample

18、ofsizeN=50weestimatedthemeanU.S.hipwidthtobe=17.158inches.Slide C-Slide C-2828Principles of Econometrics,3rd EditionPrinciples of Econometrics,3rd EditionC.6 Hypothesis Tests About A Population MeanSlide C-Slide C-2929Principles of Econometrics,3rd EditionPrinciples of Econometrics,3rd EditionCompon

19、entsofHypothesisTestsAnullhypothesis,H0(虛無假設)Analternative hypothesis,H1（對立假設）Ateststatistic（檢定統計量）Arejectionregion（拒絕域）Aconclusion（結論）C.6.1 Components of Hypothesis TestsTheNullHypothesis（虛無假設）（虛無假設）The“null”hypothesis,whichisdenotedH0(H-naught),specifiesavaluecforaparameter.Wewritethenullhypothesi

20、sasAnullhypothesisisthebeliefwewillmaintainuntilweareconvincedbythesampleevidencethatitisnottrue,inwhichcasewerejectthenullhypothesis.Slide C-Slide C-3030Principles of Econometrics,3rd EditionPrinciples of Econometrics,3rd EditionC.6.1 Components of Hypothesis TestsTheAlternativeHypothesis（對立假設）（對立假

21、設）H1:c Ifwerejectthenullhypothesisthat=c,weacceptthealternativethatisgreaterthanc.H1:)FigureC.5Therejectionregionfortheone-tailtestofH1:=c against H1:cSlide C-Slide C-3636Principles of Econometrics,3rd EditionPrinciples of Econometrics,3rd EditionC.6.3 One-tail Tests with Alternative“Less Than”()Fig

22、ureC.6Therejectionregionfortheone-tailtestofH1:=c against H1:1.68werejectthenullhypothesis.Thesampleinformationwehaveisincompatiblewiththehypothesisthat=16.5.Weacceptthealternativethatthepopulationmeanhipsizeisgreaterthan16.5inches,atthe=.05levelofsignificance.C.6.6 Example of a Two-tail Test Using

23、the Hip DataSlide C-Slide C-4141Principles of Econometrics,3rd EditionPrinciples of Econometrics,3rd EditionThenullhypothesisisThealternativehypothesisisTheteststatisticif the null hypothesis is true.Thelevelofsignificance=.05,thereforeC.6.6 Example of a Two-tail Test Using the Hip DataSlide C-Slide

24、 C-4242Principles of Econometrics,3rd EditionPrinciples of Econometrics,3rd EditionThevalueoftheteststatisticisConclusion:Since wedo not rejectthenullhypothesis.Thesampleinformationwehaveiscompatiblewiththehypothesisthatthepopulationmeanhipsize=17.C.6.7 The p-valueSlide C-Slide C-4343Principles of E

25、conometrics,3rd EditionPrinciples of Econometrics,3rd Editionp-valuerule:Rejectthenullhypothesiswhenthep-valueislessthan,orequalto,thelevelofsignificance.Thatis,ifpthenrejectH0.IfpthendonotrejectH0C.6.7 The p-valueHowthep-valueiscomputeddependsonthealternative.Iftisthecalculatedvaluenotthecriticalva

26、luetcofthet-statisticwithN1degreesoffreedom,then:ifH1:c,p=probabilitytotherightoftifH1:c,p=probabilitytotheleftoftifH1:c,p=sumofprobabilitiestotherightof|t|andtotheleftof|t|Slide C-Slide C-4444Principles of Econometrics,3rd EditionPrinciples of Econometrics,3rd EditionC.6.7 The p-valueFigureC.8Thep-

27、valueforaright-tailtestSlide C-Slide C-4545Principles of Econometrics,3rd EditionPrinciples of Econometrics,3rd EditionC.6.7 The p-valueFigureC.9Thep-valueforatwo-tailedtestSlide C-Slide C-4646Principles of Econometrics,3rd EditionPrinciples of Econometrics,3rd EditionC.6.9 Type I and Type II errors

28、Slide C-Slide C-4747Principles of Econometrics,3rd EditionPrinciples of Econometrics,3rd EditionCorrectDecisionsThenullhypothesisisfalseandwedecidetoreject it.Thenullhypothesisistrueandwedecidenottorejectit.IncorrectDecisionsThenullhypothesisistrueandwedecidetoreject it(aTypeIerror)Thenullhypothesis

29、isfalseandwedecidenottorejectit(aTypeIIerror)C.6.10 A Relationship Between Hypothesis Testing and Confidence IntervalsIfwefailtorejectthenullhypothesisatthelevelofsignificance,thenthevaluecwillfallwithina100(1)%confidenceintervalestimateof.Ifwerejectthenullhypothesis,thenc willfalloutsidethe100(1)%c

30、onfidenceintervalestimateof.Slide C-Slide C-4848Principles of Econometrics,3rd EditionPrinciples of Econometrics,3rd EditionC.6.10 A Relationship Between Hypothesis Testing and Confidence IntervalsWefailtorejectthenullhypothesiswhenorwhenSlide C-Slide C-4949Principles of Econometrics,3rd EditionPrin

31、ciples of Econometrics,3rd EditionC.7 Some Useful TestsC.7.1TestingthepopulationvarianceSlide C-Slide C-5050Principles of Econometrics,3rd EditionPrinciples of Econometrics,3rd EditionC.7.1 Testing the Population VarianceSlide C-Slide C-5151Principles of Econometrics,3rd EditionPrinciples of Econome

32、trics,3rd EditionC.7.2 Testing the Equality of two Population MeansCase1:PopulationvariancesareequalSlide C-Slide C-5252Principles of Econometrics,3rd EditionPrinciples of Econometrics,3rd EditionC.7.2 Testing the Equality of two Population MeansCase2:PopulationvariancesareunequalSlide C-Slide C-535

33、3Principles of Econometrics,3rd EditionPrinciples of Econometrics,3rd EditionC.7.3 Testing the ratio of two population variancesSlide C-Slide C-5454Principles of Econometrics,3rd EditionPrinciples of Econometrics,3rd EditionC.7.4 Testing the normality of a populationThenormaldistributionissymmetric,

34、andhasabell-shapewithapeakednessandtail-thicknessleadingtoakurtosisof3.Wecantestfordeparturesfromnormalitybycheckingtheskewness（偏態）andkurtosis（峰態）fromasampleofdata.Slide C-Slide C-5555Principles of Econometrics,3rd EditionPrinciples of Econometrics,3rd EditionKeywordsSlide C-56Principles of Economet

35、rics,3rd EditionalternativehypothesisasymptoticdistributionBLUEcentrallimittheoremcentralmomentsestimateestimatorexperimentaldesigninformationmeasureintervalestimateLagrangemultipliertestLawoflargenumberslevelofsignificancelikelihoodfunctionlikelihoodratiotestlinearestimatorloglikelihoodfunctionmaxi

36、mumlikelihoodestimationnullhypothesispointestimatepopulationparameterp-valuerandomsamplerejectionregionsamplemeansamplevariancesamplingdistributionsamplingvariationstandarderrorstandarderrorofthemeanstandarderroroftheestimatestatisticalinferenceteststatistictwo-tailtestsTypeIerrorTypeIIerrorunbiasedestimatorsWaldtest