计量经济学5ppt课件

.,1,Chapter5STATISTICALINFERENCE:ESTIMATIONANDHYPOTHESESTESTING,Statisticalinferencedrawsconclusionsaboutapopulationi.e.,probabilitydensityfunction(PDF)fromarandomsamplethathassupposedlybeendrawnfromthatpopulation.,.,2,5.1THEMEANINGOFSTATISTICALINFERENCE,Statisticalinference:thestudyoftherelationshipbetweenapopulationandasampledrawnforthatpopulation.Theprocessofgeneralizingfromthesamplevalue()tothepopulationvalueE(X)istheessenceofstatisticalinference.,.,3,5.2ESTIMATIONANDHYPOTHESISTESTING:TWINBRANCHESOFSTATISTICALINFERENCE,1.EstimationEstimation:thefirststepinstatisticalinference.:anestimator/statisticofthepopulationparameterE(X),estimate:theparticularnumericalvalueoftheestimatorsamplingvariation/samplingerror:thevariationinestimationfromsampletosample.2.HypothesistestingInhypothesistestingwemayhaveapriorjudgmentorexpectationaboutwhatvalueaparticularparametermayassume.,.,4,Theusualprocedureofestimation:toassumethatwehavearandomsampleofsizenfromtheknownprobabilitydistributionandusethesampletoestimatetheunknownparameters,thatis,usethesamplemeanasanestimateofthepopulationmean(orexpectedvalue)andthesamplevarianceasanestimateofthepopulationvariance.1.PointestimateApointestimator,orastatistic,isanr.v.,itsvaluewillvaryfromsampletosample.Howcanwerelyonjustoneestimateofthetruepopulationmean.,5.3ESTIMATIONOFPARAMETERS,.,5,2.Intervalestimate,Althoughisthesingle“best”guessofthetruepopulationmean,theinterval,say,from8to14,mostlikelyincludesthetrue?Thisisintervalestimation.Samplingorprobabilitydistribution:P(-n-1tn-1)=1criticaltvalues:n-1confidenceinterval:(lowerlimit-upperlimit)confidencecoefficient:1levelofsignificance/theprob.ofcommittingtypeIerror:,.,6,FIGURE5-1Thetdistribution,.,7,Note:Theintervalisrandom,andnottheparameterx.,Theconfidenceinterval:arandominterval,becauseitisbasedonandwhichwillvaryfromsampletosample.Thepopulationmean:althoughunknown,issomefixednumberanditisnotrandom.Youshouldnotsay:theprobabilityis0.95(1)thatxliesinthisinterval.Youshouldsay:theprobabilityis0.95thattherandominterval,containsthetruex.,.,8,Intervalestimation,incontrasttopointestimation,providesarangeofvaluesthatwillincludethetruevaluewithacertaindegreeofconfidenceorprobability(suchas0.95).P(LxU)=1-013,one-sidedalternativehypothesisH1:x13,one-sidedalternativehypothesisH1:x13,two-sidedalternativehypothesis,.,12,1.TheConfidenceIntervalApproachtoHypothesisTestingInhypothesistesting,the95%confidenceintervaliscalledtheacceptanceregionandtheareaoutsidetheacceptanceregioniscalledthecriticalregion/theregionofrejection,ofthenullhypothesis.Theboundariesoftheacceptanceregionarecalledcriticalvalues.Thenullhypothesisisrejectedifthevalueoftheparameterunderthenullhypothesiseitherexceedstheuppercriticalvalueorislessthanthelowercriticalvalueoftheacceptanceregion.,.,13,2.TypeIandTypeIIErrors:ADigression,TypeIerror:theerrorofrejectingahypothesiswhenitistrue.TypeIIerror:theerrorofacceptingafalsehypothesis.TypeIerror=prob.(rejectingH0|H0istrue)TypeIIerror=prob.(acceptingH0|H0isfalse)TheclassicalapproachtodealwithtypeI,typeIIproblems:ToassumeatypeIerrorismoreseriousthanatypeIIerror,trytokeeptheprob.ofcommittingatypeIerroratafairlylowlevel,andthenminimizeatypeIIerrorasmuchaspossible.Thatis,simplyspecifiesthevalueofwithoutworryingtoomuchabout.Thedecisiontoacceptorrejectanullhypothesisdependscriticallyonboththed.f.andtheprobabilityofcommittingatypeIerror.A95%confidencecoefficient/a5%levelofsignificance/a95%levelordegreeofconfidence:wearepreparedtoacceptatthemosta5percentprobabilityofcommittingatypeIerror.,.,14,3.TheTestofSignificanceApproachtoHypothesisTesting,Ifthedifferencebetweenandxissmall(inabsoluteterms),thenthe|t|valuewillalsobesmall.If=x,twillbezero,thenwecanacceptthenullhypothesis.Asthe|t|valuedeviatesfromzero,increasinglywewilltendtorejectthenullhypothesis.Ifthecomputedtvalueliesineitheroftherejectionregions,wecanrejectthenullhypothesis.Whenwerejectthenullhypothesis,wesaythat:ourfindingisstatisticallysignificant.whenwedonotrejectthenullhypothesis,wesaythat:ourfindingisnotstatisticallysignificant.,.,15,4.AWordonChoosingtheLevelofSignificance,，andthepValue,pvalue:theexactsignificancelevel,oftheteststatistic,thelowestsignificancelevelatwhichanullhypothesiscanberejected.Thesmallerthepvalue,thestrongertheevidenceagainstthenullhypothesis.5.Thetestofsignificance.(1)Theconfidenceintervalapproach:establisha(1-)%confidenceintervalforthetruebutunknown2usingthedistribution.(2)Hypothesistestingapproach:Justcomputevalueandtestitssignificanceagainstthecriticalvalue.6.TheFtestofsignificance.,.,16,Conclusion:,Summarizingthestepsinvolvedintestingastatisticalhypothesis:Step1:StatethenullhypothesisH0andthealternativehypothesisH1e.g.,H0:X=13andH1:X13.Step2:Selecttheteststatistic(e.g.,)Step3:Determinetheprobabilitydistributionoftheteststatistic(e.g.,.Step4:Choosethelevelofsignificance,thatis,theprobabilityofcommittingatypeIerror.(Keepinmindourdiscussionaboutthepvalue.)Step5:Choosetheconfidenceintervalorthetestofsignificanceapproach.,.,17,Step6:Acceptorrefusethenullhypothesis?,(1)Theconfidenceintervalapproach:Usingtheprobabilitydistributionoftheteststatistic,establisha100(1-)%confidenceinterval.Ifthisinterval(theacceptanceregion)includesthenull-hypothesizedvalue,donotrejectthenullhypothesis.Ifthisintervaldoesnotincludeit,rejectthenullhypothesis.(2)Thetestofsignificanceapproach:Obtainingtherelevantteststatistic(e.g.,thetstatistic)underthenullhypothesisandfindouttheprobabilityofobtainingaspecificvalueoftheteststatisticfromtheappropriateprobabilitydistributionTheprobabilityislessthantheprechosenvalueof,rejectt