中級(jí)計(jì)量濟(jì)學(xué)2015s 12

ChapterPropertiesChapterPropertiesofOLSwithSeriallyCorrelatederrors誤TestingforSerialCorrectingforSerialCorrelationwithStrictlyExogenousRegressors當(dāng)自變量為嚴(yán)格外生時(shí)校正序列相DifferencingandSerial2LectureWhatisserialLectureWhatisserialBasicintroductionoftimeseriesTestPropertiesofOLSwithSeriallyCorrelated誤差序列相關(guān)時(shí)OLSTestingforSerial3BasicRegressionAnalysisBasicRegressionAnalysiswithTimeSeriesWefocusondiscussingtheGauss-Markovassumptionsfortimeseriesapplications.TheNatureofTimeSeriesAtimeseriesdatasetisasequenceofrandomvariablesindexedbytime.Timeseriesdatasetcomeswithatemporal4BasicRegressionAnalysiswithBasicRegressionAnalysiswithTimeSeriesExample:astaticyt01ztut,tyt01ztut,tAdynamicmodelytyt1xt5TimeSeriesData:FiniteSamplePropertiesofOLSUnderClassicalAssumptionsTimeSeriesData:FiniteSamplePropertiesofOLSUnderClassicalAssumptionsUnbiasednessofOLSAssumptionLinearin模型對(duì)于參數(shù)呈線性關(guān)AssumptionZeroconditionalTS.2:零條件AssumptionNoperfect假定TS.3：沒有6TS.1-unconditionallyasTS.1-unconditionallyas(OLS的無偏性):在假定TS.1-3下OLS估計(jì)量條件于X是無偏的，因此也是無條7Theassumption假定WeneedtodiscussmoreaboutTS2.ItassumesthatE(uTheassumption假定WeneedtodiscussmoreaboutTS2.ItassumesthatE(ut|X)=0,t=1,…,n,whereXdenotesalltheindependentvariablesinallthetimeperiods.我們需要更多的討論關(guān)于TS2。它假定了E(ut|X)=0t=1,…,n,Thisassumptionimpliesthatutisuncorrelatedwithanyofthe這個(gè)假定可推出ut與任何xkjWhenTS2holdswesaytheexplanatoryvariablesarestrictly當(dāng)TS28TimeSeriesData:FiniteTimeSeriesData:FiniteSamplePropertiesofOLSUnderClassicalAssumptionsTheVariancesoftheOLSestimatorsandtheGauss-MarkovTheoremsOLS估計(jì)量的方差和高斯-馬爾可AssumptionTS.4:HomoskedasticityinerrorAssumptionTS.5:Noserialcorrelationbetweenerror9TimeSeriesData:FiniteSamplePropertiesofOLSUnderClassicalAssumptionsTheorem10.2(OLSTimeSeriesData:FiniteSamplePropertiesofOLSUnderClassicalAssumptionsTheorem10.2(OLSsamplingvariances):UnderthetimeGauss-MarkovassumptionsTS.1-5,thevarianceof?,conditionaljX,定理10.2(OLS抽樣方差)：在時(shí)間序列的高斯—馬爾可夫假定5jV(?|X)2/ 12jjjwhereSSTjisthetotalsumofsquaresandRj2istheR-fromtheregressionofxjontheotherindependent其中SSTj是xij的總平方和，而Rj2是xj對(duì)其它自變量回歸得到的RTimeSeriesData:FiniteTimeSeriesData:FiniteSamplePropertiesofOLSUnderClassicalAssumptionsUnbiasedEstimationofσ2σ2的無偏估計(jì)Theorem10.3:UnderassumptionsTS.1–5,theunbiasedestimatorofσ2isσ2的無偏估計(jì)量定理10.3：在假定TS.1-5下TimeSeriesData:FiniteSamplePropertiesofOLSUnderClassicalAssumptionsTimeSeriesData:FiniteSamplePropertiesofOLSUnderClassicalAssumptionsGauss-Markov(Gauss-TS.1-bestlinearunbiasedestimatorson定理10.4(高斯-馬爾可夫定理)：在假定TS.1-5X而言，OLS估計(jì)量是最優(yōu)線性無偏估計(jì)量TimeSeriesData:TimeSeriesData:FiniteSamplePropertiesofOLSUnderClassicalAssumptionsNormalSampling TS.6:Theerrortermsarei.i.dnormallydistributed.TS.6:誤差項(xiàng)是i.i.d.正態(tài)分布Theorem10.5:UnderAssumptionsTS.1-6,theOLSestimatorsarenormallydistributed,conditionalonX.Eachtstatistichasatdistribution,andeachFstatistichasanFdistribution.Theusualconstructionofconfidenceintervalsisalsovalid.TimeSeriesData:TrendsandLineartimetrendTimeSeriesData:TrendsandLineartimetrend01tExponentialtrendmodel:Quadratictimetrendlog(yt)01tt 2yt012tTimeSeriesData:TimeSeriesData:TrendsandIfatimeseriesisobservedatmonthlyorquarterlyintervals,itmayexhibitseasonality.Whenweworkwithseasonallyunadjusteddata,wecanincludeasetofseasonaldummyvariablestoaccountforseasonalityinthedependentvariable,theindependentvariables,orboth.GrangerCausalityTest:GrangerCausalityTest:ChickensEggsWhichonecomesFirst?先有雞還是先有蛋？ChickensEggsWhichonecomesChickensEggsWhichonecomesFirst?Thishasbeenaquestiondebatedbyscientistsofdifferentfieldsforages.這是不同領(lǐng)域的科學(xué)家爭(zhēng)論了很久CNNreportedonMay26,2006that“Nowateammadeupofageneticist,philosopherandchickenfarmerclaimtohavefoundananswer.”2006年5月26日However,thesameconclusionhasbeendrawnbyeconomistsusingGrangerCausalitytest18yearsHowcanwefindoutHowcanwefindoutwhichonecomes我們?nèi)绾文軌虻贸稣l(shuí)在先Weoftensay“inChinese,thisisexactlythesimpleideaGrangerthe2003NobelPrizewinnerusedtoconstructtheso-calledGrangercausalitytest.在中文中我們常說“前因后果”，In1988，ThurmanandFisherusesthisideatotestwhetherchickensoreggscomefirst1988年，Thurman和Fisher利用該檢驗(yàn)TheycollectU.S.timeseriesfrom1930to1983ofeggproductionandchickenpopulation.Theyhavebeenverycarefultodefinethechickenpopulationasallthechicksthatarecapableofcausingeggs(excludechickensraisedonlyformeat).他們收集Review:timeseriesisaseriesofdatapointsobservedoverTheIdeaofTheIdeaoftheGrangerCausalityLet’sconsidertheeggproduction.Whatcanbeusedtopredictthenumberofeggsinyeart現(xiàn)在讓我們來考慮一Thenumberofeggslastperiod（上年的雞蛋數(shù)Thenumberofchickenslastperiod（上年的雞的數(shù)目Someotherunobservedfactors（一下無法觀測(cè)的因素Henceonecanexpressthetotalproductionofeggsatteggst1eggst11chickenst1Similarlythechickenproductionisexpressedas（雞的產(chǎn)量 2chickenst12eggst1TheIdeaoftheGrangerTheIdeaoftheGrangerCausalityIfthelaggedvaluesofchickenscanhelppredictcurrentvaluesofeggschickensaresaidtoGrangercauseeggsThisdoesnotcontradictwithchickenscomefirst.如果上期雞的數(shù)量有助于預(yù)測(cè)這Ifthelaggedvaluesofeggscanhelppredictcurrentvaluesofchickens,eggsaresaidtoGrangercausechickens.Thisdoesnotcontradictwitheggscomefirst反之如果上期雞蛋的數(shù)量有助于Ifwefindcausalityfromonedirectionbutnosignificantcausalityfromtheotherdirection,wecanfindoutwhichonecomesfirst.Otherwisequestionsremainunanswered.如果我們發(fā)現(xiàn)的證據(jù)只支TheIdeaoftheGrangerCausalityIneconometricterms,totestTheIdeaoftheGrangerCausalityIneconometricterms,totestwhetherchickenscomefirst,wecantest在計(jì)量中，要建議是否有雞在先的證據(jù)，就是要檢eggst1eggst11chickenst1H0:1Totestwhethereggscomefirst,wecantest要檢驗(yàn)蛋在先，chickenstchickenst1eggst1H0:2Ifwerejectthenullhypothesis21cannot,wefindevidenceseggsfirst.如果零假設(shè)拒能21，則我們發(fā)現(xiàn)了蛋在先的證據(jù)TheEmpiricalTestingTheEmpiricalTesting實(shí)證ThurmanandFisherdofindthat,thehypothesisthateggsdonotGrangercausechickensgetsaFstatisticof10.36(pvalue=0.0002),butchickensdonotGrangercauseeggsgetsaFstatisticof1.71(p=0.19).ThurmanandFisher發(fā)現(xiàn)，蛋不是雞之因的檢驗(yàn)得到F值為10.36,雞不是蛋之因的檢驗(yàn)得到的F值為1.71也就是說，Therefore,theyconcludethattheeggscame故TheCNNReportTheCNNReport18YearsAlsoreportthateggscomefirst.也說明Evidences:ThelivingorganisminsidetheeggshellwouldhavehadthesameDNAasthechickenitwoulddevelopinto,reportedbyProfessorJohnBrookfield,aspecialistinevolutionarygeneticsattheUniversityofNottingham證據(jù)：諾丁漢大學(xué)的進(jìn)化遺傳學(xué)專家JohnBrookfield教授指出，蛋殼中有和雞相同的所有DNAComment:WhatFurtherCanBeComment:WhatFurtherCanBeThisisoneexampleusingGrangercausalitybutineconomicsithasbeenusedtostudymuchmoresignificantrelationsbetweeneconomicvariablesliketherelationbetweenmoneyandincome.Itbestfitstotestthetime-seriesrelationsbetweentwovariables.Otherinteresting“testable”hypothesis:Hewholaughslastlaughsbest該檢驗(yàn)也可以用于檢驗(yàn)其他有趣的論斷，如“誰(shuí)笑到WhatisserialSerialWhatisserialSerialcorrelationhappenswhenthecovariancesoftheerrortermsarenotzero,thatis,forsomeindividualsiandm,Eventhoughtheproblemofserialcorrelationcanalsohappentocross-sectiondatawhenthedataareorderedinaspecificway,itisafrequentonewhenusingtimeseriesduetoinertiainthesystem.ForthisreasonitisoftencalledSerialcorrelation:HowdoesitlookSerialcorrelation:HowdoesitlookThemodelyt01x1t...kxktissaidtohaveseriallycorrelatederrors被稱作是具有序列相關(guān)的誤差1ut...qutqetwhereei.i.d.(0,2te issaidtobeautoregressiveoforderq 被稱作是q階自回歸Serialcorrelation:HowSerialcorrelation:HowdoesitlookMostoftenweconsiderAR(1)多數(shù)情況我們考慮AR(1)模型，utut1etwhereweassume|其中我們假定When||<1holds,wesay{ut}isastable當(dāng)||<1成立時(shí)，我們稱{ut}是一個(gè)穩(wěn)定的過程Whatisthepropertiesofastableprocess?一個(gè)穩(wěn)定的過程具有什么性質(zhì)？Consideratperiodt-1ut-1increasesby1unitandnothing考慮在時(shí)期t-1，ut-1增加1單位，而其它保持不變，elsechanges,whatisimpactonuover這對(duì)隨后的u會(huì)產(chǎn)生什么影響？114477cumulativeimpactofu_(t-1),6543210Netimpactofu_(t-1),10114477cumulativeimpactofu_(t-1),6543210Netimpactofu_(t-1),10Netimpactofu_(t-1),rou=-10cumulativeimpactofu_(t-1),rou=-10114477Netimpactofu_(t-1),rou=-10cumulativeimpactofu_(t-1),rou=-10114477114477cumulativeimpactofu_(t-1),50Netimpactofu_(t-1),10114477cumulativeimpactofu_(t-1),50Netimpactofu_(t-1),10114477cumulativeimpactofu_(t-1),rou=-10Netimpactofu_(t-1),rou=-10---114477cumulativeimpactofu_(t-1),rou=-10Netimpactofu_(t-1),rou=-10---HowdoesserialHowdoesserialcorrelationlookTheaboveslidesshowwhatdowemeanbystable,orToseehowserialcorrelationlookslike,weneedtoploterrortermsagainsttime,ifweknowtheerrorterms.ThisisaccomplishedthroughHowdoesserialcorrelationlookHowdoesserialcorrelationlookStepsinstata:wefirstsimulatetheiidv,letittobenormalwithzeromeanandvariance0.0081.在stata中的步驟：我們首先模擬一個(gè)iid的v，讓它服從期望為零，方差為0.0081的正態(tài)分布。genThenwegeneratetheAR(1)procedures,consideringρ=0.8,-0.8,1,-1,我們生成一個(gè)AR(1)過程，令ρ分別為ρ=0.80.811,(ForsimplificationyoucandothisinExcel.)Thenweplotthegenerateduseriestoseehowtheylooklike.Weuseregressiontocheckwhetherourgenerationsarefine.--thei.i.d.e--thei.i.d.eover CheckingthesimulationCheckingthesimulationthroughFirst,generatethetimetrendandtellstatathatvariabletimegivesthetimetrend:gentime=_ntssettimeOncewetssetthedata,l.vargivesthefirstorder一旦我們tsset數(shù)據(jù)，1.var給出滯后一階的變量regut8Thisregressiongivesanestimatedcoefficientsof0.7966,verycloseto0.8.這個(gè)自回歸給出系數(shù)估計(jì)為0.7966，非常接近0.8theAR(1)ut,00AR(1tu,0北京大學(xué)中國(guó)經(jīng)濟(jì)tim究中心沈0----theAR(1)ut,00AR(1tu,0北京大學(xué)中國(guó)經(jīng)濟(jì)tim究中心沈0----thenonstationaryut,0nonstatite0100學(xué)中thenonstationaryut,0nonstatite0100學(xué)中國(guó)經(jīng)濟(jì)研究中00--1--01PropertiesofOLSwithSeriallyPropertiesofOLSwithSeriallyCorrelatedTheorem10.1involvesonlyTS.1toTS.3,noassumptionabouttheserialcorrelationoftheerrorterm,hence,OLSisstillunbiasedwithserialcorrelation.Thegoodnessoffitmeasuresarestillfinewithserialcorrelation,aslongasthedataarestationaryandweaklydependent(whichmeansthedependencebetweenxtandxt+hbecomesweakerandweakerashgetslarger).存在序列相關(guān)時(shí)，只要數(shù)據(jù)是平穩(wěn)的而且是弱相關(guān)的(意味著xtPropertiesofOLSPropertiesofOLSwithSeriallyCorrelatedSincetheGauss-MarkovTheoremrequiresbothHMKandseriallyuncorrelatederrors,inthepresenceofserialcorrelation,OLSisnolongerBLUE,andtheusualOLSstandarderrorsandteststatisticsarenotvalidanymore.Toseethis,weassumethattheerrortermisanAR(1),wherethestartingpointisu0andetisani.i.d.sequencewithzeromeanandvarianceσe2為了看清這一點(diǎn)，我們假定誤差項(xiàng)滿足AR(1)，其中起點(diǎn)為u0而是具有零期望值和方差σe2的一個(gè)iid序列PropertiesofOLSwithSeriallyCorrelatedConsiderthevarianceoftheOLSslopeestimatorinyt01xtut,assumingx0forsimplicity.在yt0PropertiesofOLSwithSeriallyCorrelatedConsiderthevarianceoftheOLSslopeestimatorinyt01xtut,assumingx0forsimplicity.在yt01xtut中考慮OLS系數(shù)估計(jì)量的方差，簡(jiǎn)單起見假定x0，那么我們有：NowwehavenxuV(?)nni i21xVx 11t nii1tnxninix x2V(u)SinceE(u t t tttjtweneedtogetV(ut)andE(ututj我們需要得到V(ut)E(ututjPropertiesofOLSwithSeriallyCorrelatedByHMKassumptionwehaveassumedthatV(ut)tobePropertiesofOLSwithSeriallyCorrelatedByHMKassumptionwehaveassumedthatV(ut)tobedenoteitasV(ut)2.AlsonoticethatE(ututj)cov(ut,utj通過HMK假定我們假設(shè)V(ut)為常數(shù)，把它記作V(utE(ututj)cov(ut,utj).2utjutj1et(utjjuj...ettjttUsingetj,j1isuncorrelatedwithutweetjj1與ut不相關(guān) )E(u )ju2ju...eut tt1tjt ttE(u)j2j2tPropertiesofOLSwithSeriallyCorrelated1因此，?的方差11?)PropertiesofOLSwithSeriallyCorrelated1因此，?的方差11?)nini2VxV2x1t 2jx. tninix2(tjtttThefirsttermisthevarianceof?when1Sincethesecondtermisnotzeroingeneral,thevarianceestimatorignoringserialcorrelationwillbebiased.不等于零，那么忽視序列相關(guān)的方差估計(jì)量將會(huì)是有PropertiesofOLSPropertiesofOLSwithSeriallyCorrelatedWhenρ>0,andtheindependentvariablesarepositivelycorrelatedovertime,thesecondtermispositive,sotheusualOLSvarianceunderestimatesthetruevarianceoftheestimator.Insuchcase,theusualOLSstandarderrorisinvalid.Thetstatisticsisalsoinvalid,andtendtobetoolargeinthecaseof在第二項(xiàng)為正的情況下，t會(huì)變大。TheFandLMstatisticsformultiplehypothesisarealso聯(lián)合假設(shè)的F和LM統(tǒng)計(jì)TestingforAR(1)SerialTestingforAR(1)SerialWanttobeabletotestforwhethertheerrorsareseriallycorrelatedornotWanttotestthenullthat=0inut=ut-1+et,t=2,…,n,utisthemodelerrortermandetis想要檢驗(yàn)零假設(shè)：在utut-1ett=2,…n中0，其中ut是Withstrictlyexogenousregressors,thetestisverystraightforward–simplyregresstheresidualsonlaggedresidualsanduseat-testExample:TherelationExample:TherelationofSOCBloanandSOECantheSOEoutputexplainmuchvariationinSOCBloan?IsittruethatmostoftheincreaseinSOCBloansaresupportingthedevelopmentofSOEs?Usetheyearlydatafrom1978–2002toillustrate用1978-2002的年度數(shù)據(jù)討論這個(gè)問題Example:Plottinglnrsocbloanagainst例：畫出lnrsocbloan和lnrsoeoutput345Example:Plottinglnrsocbloanagainst例：畫出lnrsocbloan和lnrsoeoutput345 +======+1Root+======+1Root||+t||Example:Plottingtheresidualsagainst--Example:Plottingtheresidualsagainst-- Example:Regressingtheresidualsonits.reguhatSource======-------------+-----------------------------Example:Regressingtheresidualsonits.reguhatSource======-------------+-----------------------------Model|1ProbFResidualAdjR-squaredRootMSE-------------+-----------------------------Total-----------------------------------------------------------------------------|Std.t[95%Conf.-------------+---------------------------------------------------------------|L1|------------------------------------------------------------------------------Example:theHSKtestExample:theHSKtestbeforedrawingNoticethatthetstatisticisappropriateundertheHMKassumption.WethereforetestforHSKusinghettestinstataafterregressinguhatonitslag.Theteststatiticis0.04withapvalueofover0.8,henceHMKisnotrejected.t統(tǒng)計(jì)量為0.04，其p值超過0.8，所以HMKThereexistpositiveserialcorrelationamongtheTheDWtestofAR(1)serialThealternativetestforserialcorrelationisTheDWtestofAR(1)serialThealternativetestforserialcorrelationisbyDurbinandWaston(1950,1951),hencecalledDWu?tTheteststatisticis.2tu?t。檢驗(yàn)統(tǒng)計(jì)量為2tTheDWtestofAR(1)serialWhentislarge,TheDWtestofAR(1)serialWhentislarge,u?2,therefore,dis2t-t ).當(dāng)t22? d t-t-tcov(?t?t-cov(?t?t-cov(??t-cov(??t-ifcov(?t?t-10thend2.cov(??t-Empirically,theregionsofcriticalvaluescanbeshownbythefigurebelow,fork=2,n=26. TheDWtestofAR(1)serialTheDWtestofAR(1)serialundetermined（不能決定的區(qū)域=><24- 4-