伍4ccer計(jì)量wooldrige第八講

IntermediateEconometrics,YanShen1MultipleRegressionAnalysis

多元回歸分析之序列相關(guān)

y=b0+b1xt1+b2xt2+...bkxtk+uSerialCorrelation 序列相關(guān)IntermediateEconometrics,YanShen2ChapterOutline

本章大綱PropertiesofOLSwithSeriallyCorrelatederrors誤差序列相關(guān)時(shí)OLS的性質(zhì)TestingforSerialCorrelation

檢驗(yàn)序列相關(guān)CorrectingforSerialCorrelationwithStrictlyExogenousRegressors

當(dāng)自變量為嚴(yán)格外生時(shí)校正序列相關(guān)DifferencingandSerialCorrelation

差分和序列相關(guān)HeteroskedasticityinTimeSeriesRegression

時(shí)間序列回歸中的異方差性IntermediateEconometrics,YanShen3LectureOutline

講義大綱Whatisserialcorrelation

什么是序列相關(guān)Basicintroductionoftimeseriesanalysis

時(shí)間序列分析的基本介紹PropertiesofOLSwithSeriallyCorrelatedErrors

誤差序列相關(guān)時(shí)OLS的性質(zhì)TestingforSerialCorrelation

檢驗(yàn)序列相關(guān)IntermediateEconometrics,YanShen4Whatisserialcorrelation

什么是序列相關(guān)Serialcorrelationhappenswhenthecovariancesoftheerrortermsarenotzero,thatis,forsomeindividualsiandm, 當(dāng)誤差項(xiàng)協(xié)方差不為零時(shí)，序列相關(guān)就出現(xiàn)了。即，對(duì)某些觀察值i和m,

cov(ui,um)?=0.Eventhoughtheproblemofserialcorrelationcanalsohappentocross-sectiondatawhenthedataareorderedinaspecificway,itisafrequentonewhenusingtimeseriesduetoinertiainthesystem.

分析截面數(shù)據(jù)時(shí)，如果我們把數(shù)據(jù)按特定方式排序，序列相關(guān)的問題也可能發(fā)生，然而由于系統(tǒng)產(chǎn)存在時(shí)間上的惰性，它在時(shí)間序列分析中更為常見。Forthisreasonitisoftencalledautocorrelation.因?yàn)檫@個(gè)原因，它常常被稱作是自相關(guān)。IntermediateEconometrics,YanShen5BasicRegressionAnalysiswithTimeSeriesData

時(shí)間序列數(shù)據(jù)的基本回歸分析

WefocusondiscussingtheGauss-Markovassumptionsfortimeseriesapplications.

我們集中討論時(shí)間序列版的高斯-馬爾可夫假定。TheNatureofTimeSeriesData

時(shí)間序列數(shù)據(jù)的本質(zhì)Atimeseriesdatasetisasequenceofrandomvariablesindexedbytime.

時(shí)間序列數(shù)據(jù)是以時(shí)間為指標(biāo)的一個(gè)隨機(jī)變量序列。Timeseriesdatasetcomeswithatemporalordering.

時(shí)間序列數(shù)據(jù)集伴隨著一個(gè)時(shí)間上的排序。IntermediateEconometrics,YanShen6BasicRegressionAnalysiswithTimeSeriesData

時(shí)間序列數(shù)據(jù)的基本回歸分析

Example:astaticmodel

例：一個(gè)靜態(tài)模型Adynamicmodel一個(gè)動(dòng)態(tài)模型IntermediateEconometrics,YanShen7TimeSeriesData:FiniteSamplePropertiesofOLSUnderClassicalAssumptions

時(shí)間序列數(shù)據(jù)：在經(jīng)典假定下OLS的有限樣本性質(zhì)UnbiasednessofOLSOLS的無(wú)偏性AssumptionTS.1:

Linearinparameters

假定TS.1:模型對(duì)于參數(shù)呈線性關(guān)系A(chǔ)ssumptionTS.2:

Zeroconditionalmean假定TS.2:零條件期望AssumptionTS.3:

Noperfectcollinearity假定TS.3：沒有完全共線性Theorem10.1(UnbiasednessofOLS):

UnderAssumptionsTS.1-3,theOLSestimatorsareunbiasedconditionalonX,andthereforeunconditionallyaswell:

定理10.1(OLS的無(wú)偏性):在假定TS.1-3下，OLS估計(jì)量條件于X是無(wú)偏的，因此也是無(wú)條件無(wú)偏。IntermediateEconometrics,YanShen8TheassumptionTS2

假定TS2WeneedtodiscussmoreaboutTS2.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)=0,t=1,…,n,

其中X表示所有時(shí)期的所有自變量。Thisassumptionimpliesthatut

isuncorrelatedwithanyofthexkj.

這個(gè)假定可推出ut

與任何xkj都不相關(guān)。WhenTS2holdswesaytheexplanatoryvariablesarestrictlyexogenous.

當(dāng)TS2成立時(shí)我們說(shuō)解釋變量是嚴(yán)格外生的。IntermediateEconometrics,YanShen9TimeSeriesData:FiniteSamplePropertiesofOLSUnderClassicalAssumptions

時(shí)間序列數(shù)據(jù)：在經(jīng)典假定下OLS的有限樣本性質(zhì)TheVariancesoftheOLSestimatorsandtheGauss-MarkovTheorems

OLS估計(jì)量的方差和高斯-馬爾可夫定理AssumptionTS.4:

Homoskedasticityinerrorterms

假定TS.4:誤差具有同方差性AssumptionTS.5:Noserialcorrelationbetweenerrorterms假定TS.5:誤差項(xiàng)之間沒有序列相關(guān)IntermediateEconometrics,YanShen10TimeSeriesData:FiniteSamplePropertiesofOLSUnderClassicalAssumptions

時(shí)間序列數(shù)據(jù)：在經(jīng)典假定下OLS的有限樣本性質(zhì)Theorem10.2(OLSsamplingvariances):UnderthetimeseriesGauss-MarkovassumptionsTS.1-5,thevarianceof,conditionalonX,is定理10.2(OLS抽樣方差)：在時(shí)間序列的高斯—馬爾可夫假定TS.1-5下，的方差，條件于X，為

whereSSTjisthetotalsumofsquaresofxijandRj2istheR-squaredfromtheregressionofxjontheotherindependentvariables.

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R方。IntermediateEconometrics,YanShen11TimeSeriesData:FiniteSamplePropertiesofOLSUnderClassicalAssumptions

時(shí)間序列數(shù)據(jù)：在經(jīng)典假定下OLS的有限樣本性質(zhì)UnbiasedEstimationofσ2

σ2

的無(wú)偏估計(jì)Theorem10.3:UnderassumptionsTS.1–5,theunbiasedestimatorofσ2is

定理10.3：在假定TS.1-5下，σ2的無(wú)偏估計(jì)量為IntermediateEconometrics,YanShen12TimeSeriesData:FiniteSamplePropertiesofOLSUnderClassicalAssumptions

時(shí)間序列數(shù)據(jù)：在經(jīng)典假定下OLS的有限樣本性質(zhì)Gauss-MarkovTheorem

高斯-馬爾可夫定理Theorem10.4(Gauss-MarkovTheorem):

UnderAssumptionsTS.1-5,theOLSestimatorsarethebestlinearunbiasedestimatorsonX.

定理10.4(高斯-馬爾可夫定理)：在假定TS.1-5下，對(duì)X而言，OLS估計(jì)量是最優(yōu)線性無(wú)偏估計(jì)量。IntermediateEconometrics,YanShen13TimeSeriesData:FiniteSamplePropertiesofOLSUnderClassicalAssumptions

時(shí)間序列數(shù)據(jù)：在經(jīng)典假定下OLS的有限樣本性質(zhì)NormalSamplingDistributions

正態(tài)抽樣分布TS.6:Theerrortermsarei.i.d.normallydistributed.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.

定理10.5:在假定TS.1-6下，OLS估計(jì)量是條件于X的正態(tài)分布。此時(shí)t統(tǒng)計(jì)量服從t分布，而F統(tǒng)計(jì)量服從F分布。據(jù)此建立的置信區(qū)間也是合適的。IntermediateEconometrics,YanShen14TimeSeriesData:TrendsandSeasonality

時(shí)間序列數(shù)據(jù)：趨勢(shì)和季節(jié)性Lineartimetrendmodel:

線性時(shí)間趨勢(shì)模型Exponentialtrendmodel:

指數(shù)趨勢(shì)模型：Quadratictimetrendmodel:

二次時(shí)間趨勢(shì)模型：IntermediateEconometrics,YanShen15TimeSeriesData:TrendsandSeasonality

時(shí)間序列數(shù)據(jù)：趨勢(shì)和季節(jié)性Ifatimeseriesisobservedatmonthlyorquarterlyintervals,itmayexhibitseasonality.Whenweworkwithseasonallyunadjusteddata,wecanincludeasetofseasonaldummyvariablestoaccountforseasonalityinthedependentvariable,theindependentvariables,orboth.

如果時(shí)間序列以月或者季度的時(shí)間間隔被觀測(cè)，它便會(huì)顯示出季節(jié)性。當(dāng)我們處理未調(diào)整季節(jié)變動(dòng)的數(shù)據(jù)時(shí)，我們可以包含一個(gè)季節(jié)虛擬變量的集合去解釋自變量，因變量（或者同時(shí)兩者）所具有的季節(jié)性。IntermediateEconometrics,YanShen16Serialcorrelation:Howdoesitlooklike?

序列相關(guān)：它長(zhǎng)什么樣？IntermediateEconometrics,YanShen17Serialcorrelation:Howdoesitlooklike?

序列相關(guān)：它長(zhǎng)什么樣？IntermediateEconometrics,YanShen18IntermediateEconometrics,YanShen19IntermediateEconometrics,YanShen20IntermediateEconometrics,YanShen21IntermediateEconometrics,YanShen22Howdoesserialcorrelationlooklike?

序列相關(guān)：它長(zhǎng)什么樣？Theaboveslidesshowwhatdowemeanbystable,orstationary.

上面的幻燈片顯示了我們所指的穩(wěn)定或者平穩(wěn)性的含義。Toseehowserialcorrelationlookslike,weneedtoploterrortermsagainsttime,ifweknowtheerrorterms.

如果我們知道誤差項(xiàng)，為了看出序列相關(guān)的樣子，我們需要畫出它和時(shí)間的關(guān)系。Thisisplishedthroughsimulation.

這可以通過仿真來(lái)實(shí)現(xiàn)。IntermediateEconometrics,YanShen23Howdoesserialcorrelationlooklike?

序列相關(guān)：它看起來(lái)像什么？Stepsinstata: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。

gene=0.09*invnorm(uniform())ThenwegeneratetheAR(1)procedures,consideringρ=0.8,-0.8,1,-1,respectively. 我們生成一個(gè)AR(1)過程，令ρ分別為ρ=0.8,-0.8,1,-1, (ForsimplificationyoucandothisinExcel.)

簡(jiǎn)單起見，你可以在Excel里完成這些。Thenweplotthegenerateduseriestoseehowtheylooklike.Weuseregressiontocheckwhetherourgenerationsarefine.

那么我們可以畫出生成的序列u來(lái)看它們的樣子。我們用回歸來(lái)檢查我們是否生成了一個(gè)我們想要的序列。IntermediateEconometrics,YanShen24IntermediateEconometrics,YanShen25Checkingthesimulationthroughregression

通過回歸來(lái)檢查仿真First,generatethetimetrendandtellstatathatvariabletimegivesthetimetrend:

首先，生成時(shí)間趨勢(shì)并告訴stata變量time給出時(shí)間趨勢(shì)。 gentime=_n tssettime

Oncewetssetthedata,l.vargivesthefirstorderlag.

一旦我們tsset數(shù)據(jù)，1.var給出滯后一階的變量。regut8l.ut8

Thisregressiongivesanestimatedcoefficientsof0.7966,verycloseto0.8.

這個(gè)自回歸給出系數(shù)估計(jì)為0.7966，非常接近0.8。IntermediateEconometrics,YanShen26IntermediateEconometrics,YanShen27IntermediateEconometrics,YanShen28PropertiesofOLSwithSeriallyCorrelatedErrors

誤差序列相關(guān)時(shí)OLS的性質(zhì)Theorem10.1involvesonlyTS.1toTS.3,noassumptionabouttheserialcorrelationoftheerrorterm,hence,OLSisstillunbiasedwithserialcorrelation.

定理10.1僅僅涉及Ts.1到Ts.3，沒有任何關(guān)于誤差項(xiàng)是否序列相關(guān)的假定，因此序列相關(guān)時(shí)OLS仍然是無(wú)偏的。Thegoodnessoffitmeasuresarestillfinewithserialcorrelation,aslongasthedataarestationaryandweaklydependent(whichmeansthedependencebetweenxtandxt+hesweakerandweakerashgetslarger).

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xt和xt+h的相關(guān)性隨著h的增大而變得越來(lái)越弱)，擬合優(yōu)度仍然有效。

IntermediateEconometrics,YanShen29PropertiesofOLSwithSeriallyCorrelatedErrors

誤差序列相關(guān)時(shí)OLS的性質(zhì)SincetheGauss-MarkovTheoremrequiresbothHMKandseriallyuncorrelatederrors,inthepresenceofserialcorrelation,OLSisnolongerBLUE,andtheusualOLSstandarderrorsandteststatisticsarenotvalidanymore.

因?yàn)楦咚?馬爾可夫定理同時(shí)要求HMK和誤差序列不相關(guān)，那么存在序列相關(guān)時(shí)，OLS不再是BLUE，而且通常的OLS標(biāo)準(zhǔn)誤和檢驗(yàn)統(tǒng)計(jì)量不再是有效的。Toseethis,weassumethattheerrortermisanAR(1),wherethestartingpointisu0

andetisani.i.d.sequencewithzeromeanandvariance

σe2

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et是具有零期望值和方差σe2的一個(gè)iid序列。IntermediateEconometrics,YanShen30PropertiesofOLSwithSeriallyCorrelatedErrors

誤差序列相關(guān)時(shí)OLS的性質(zhì)IntermediateEconometrics,YanShen31PropertiesofOLSwithSeriallyCorrelatedErrors

誤差序列相關(guān)時(shí)OLS的性質(zhì)IntermediateEconometrics,YanShen32PropertiesofOLSwithSeriallyCorrelatedErrors

誤差序列相關(guān)時(shí)OLS的性質(zhì)IntermediateEconometrics,YanShen33PropertiesofOLSwithSeriallyCorrelatedErrors

誤差序列相關(guān)時(shí)OLS的性質(zhì)Whenρ>0,andtheindependentvariablesarepositivelycorrelatedovertime,thesecondtermispositive,sotheusualOLSvarianceunderestimatesthetruevarianceoftheestimator.

當(dāng)ρ>0，而且自變量在時(shí)間上是正相關(guān)時(shí)，第二項(xiàng)為正，因此通常的OLS方差低估了估計(jì)量真實(shí)的方差。Insuchcase,theusualOLSstandarderrorisinvalid.Thetstatisticsisalsoinvalid,andtendtobetoolargeinthecaseofρ>0.

在這種情況下，通常的OLS標(biāo)準(zhǔn)差不再正確。t統(tǒng)計(jì)量也不再正確。在第二項(xiàng)為正的情況下，t會(huì)變大。TheFandLMstatisticsformultiplehypothesisarealsoinvalid.

聯(lián)合假設(shè)的F和LM統(tǒng)計(jì)量也不再正確。IntermediateEconometrics,YanShen34TestingforAR(1)SerialCorrelation

檢驗(yàn)AR(1)的序列相關(guān)

Wanttobeabletotestforwhethertheerrorsareseriallycorrelatedornot

想要檢驗(yàn)是否誤差是序列相關(guān)的Wanttotestthenullthatr=0inut=rut-1+et,t=2,…,n,whereutisthemodelerrortermandetisiid

想要檢驗(yàn)零假設(shè)：在ut=rut-1+et,t=2,…,n中r=0，其中ut

是模型的誤差項(xiàng)而et是iid的。Withstrictlyexogenousregressors,thetestisverystraightforward–simplyregresstheresidualsonlaggedresidualsanduseat-test

當(dāng)自變量為嚴(yán)格外生時(shí)，檢驗(yàn)很直接。只要把殘差對(duì)滯后的殘差作回歸并使用t檢驗(yàn)即可。IntermediateEconometrics,YanShen35Example:TherelationofSOCBloanandSOEoutput

例：SOCB的貸款和SOE的產(chǎn)出的關(guān)系CantheSOEoutputexplainmuchvariationinSOCBloan?

是否國(guó)有企業(yè)產(chǎn)出解釋了國(guó)有商業(yè)銀行貸款中大部分的變動(dòng)？IsittruethatmostoftheincreaseinSOCBloansaresupportingthedevelopmentofSOEs?

是否國(guó)有商業(yè)銀行增加的貸款大部分用于支持國(guó)有企業(yè)的發(fā)展？Usetheyearlydatafrom1978–2002toillustrateit.

用1978-2002的年度數(shù)據(jù)討論這個(gè)問題。IntermediateEconometrics,YanShen36Example:PlottinglnrsocbloanagainstLnrsoeoutput例：畫出lnrsocbloan和lnrsoeoutput的關(guān)系IntermediateEconometrics,YanShen37IntermediateEconometrics,YanShen38Example:Plottingtheresidualsagainsttime例：畫出殘差和時(shí)間的關(guān)系IntermediateEconometrics,YanShen39.reguhatl.uhat

Source|SSdfMSNumberofobs=24-------------+------------------------------F(1,22)=15.93Model|.2140086711.214008671Prob>F=0.0006Residual|.29563897822.013438135R-squared=0.4199-------------+------------------------------AdjR-squared=0.3935Total|.50964764923.022158593RootMSE=.11592

------------------------------------------------------------------------------uhat|Coef.Std.Err.tP>|t|[95%Conf.Interval]-------------+----------------------------------------------------------------uhat|L1|.6456036.16177813.990.001.3100963.9811108_cons|.0031941.02366310.130.894-.0458801.0522682------------------------------------------------------------------------------Example:Regressingtheresidualsonitslag例:將殘差對(duì)它的滯后變量進(jìn)行回歸IntermediateEconometrics,YanShen40Example:theHSKtestbeforedrawingconclusions

例：做結(jié)論前的HSK檢驗(yàn)NoticethatthetstatisticisappropriateundertheHMKassumption.WethereforetestforHSKusinghettestinstataafterregressinguhatonitslag.

注意到t統(tǒng)計(jì)量在HMK假定下是正確的。我們因此可以用stata中的hettest在uhat對(duì)它的滯后變量作回歸后檢驗(yàn)HSKTheteststatiticis0.04withapvalueofover0.8,henceHMKisnotrejected.t統(tǒng)計(jì)量為0.04，其p值超過0.8，所以HMK沒有被拒絕。Thereexistpositiveserialcorrelationamongtheresiduals.

殘差之間存在正的序列相關(guān)。IntermediateEconometrics,YanShen41TheDWtestofAR(1)serialcorrelation

AR(1)序列相關(guān)的DW檢驗(yàn)IntermediateEconometrics,YanShen42TheDWtestofAR(1)serialcorrelation

AR(1)序列相關(guān)的DW檢驗(yàn)IntermediateEconometrics,YanShen43TheDWtestofAR(1)serialcorrelation

AR(1)序列相關(guān)的DW檢驗(yàn)dL=1.2244-dL=2.776dU=1.5534-dU=2.4472r=0r<0r>0undeterminedregion（不能決定的區(qū)域）undeterminedregion（不能決定的區(qū)域）IntermediateEconometrics,YanShen44TheDWtestofAR(1)serialcorrelation

AR(1)序列相關(guān)的DW檢驗(yàn)TheDWstatisticfortheaboveexampleisgetbytyping

上面例子中的DW統(tǒng)計(jì)量可在回歸命令后通過輸入dwstata來(lái)得到：

reglnrsocbloanlnrsoeoutput

dwstatTheteststatisticis0.598,indicatingstrongpositiveserialcorrelation.

統(tǒng)計(jì)量為0.598，說(shuō)明存在很強(qiáng)的正的序列相關(guān)性。IntermediateEconometrics,YanShen45TestingforAR(1)SerialCorrelation

檢驗(yàn)AR(1)序列相關(guān)Noticethatthecriticalvaluesarecalculatedunderthenormalityassumption.

注意到DW檢驗(yàn)的臨界值是在正態(tài)性假定下計(jì)算得到的。Furthermore,thecriticalvaluesaredifficulttocalculate,makingthettesteasiertoworkwith.

另外，臨界值的計(jì)算較困難，而t檢驗(yàn)則更容易進(jìn)行。However,weneedtoknowwhatisDWtestsinceitisverypopular.

然而，因?yàn)镈W檢驗(yàn)被廣泛使用，我們需要知道它是什么。IntermediateEconometrics,YanShen46TestingforAR(1)SerialCorrelationwithoutstrictexogenousregressors

在沒有嚴(yán)格外生自變量時(shí)檢驗(yàn)AR(1)序列相關(guān)

Iftheregressorsarenotstrictlyexogenous,thenneitherthetorDWtestwillwork如果自變量不是嚴(yán)格外生的，那么t檢驗(yàn)或DW檢驗(yàn)都不再正確。Regresstheresidualonthelaggedresidualandallofthex’s

將殘差對(duì)滯后的殘差以及所有的x作回歸。Theinclusionofthex’sallowseachxtj

tobecorrelatedwithut-1,sodon’tneedassumptionofstrictexogeneity

將x包含進(jìn)來(lái)允許每一個(gè)xtj

和ut-1相關(guān)，因此不需要嚴(yán)格外生性的假定。IntermediateEconometrics,YanShen47ExampleContinued

國(guó)企產(chǎn)出和國(guó)有商業(yè)商業(yè)銀行貸款關(guān)系例（續(xù)）.reguhatlnrsoeoutputl.uhat

Source|SSdfMSNumberofobs=24-------------+------------------------------F(2,21)=7.80Model|.2172622262.108631113Prob>F=0.0029Residual|.29238542321.013923115R-squared=0.4263-------------+------------------------------AdjR-squared=0.3717Total|.50964764923.022158593RootMSE=.118

------------------------------------------------------------------------------uhat|Coef.Std.Err.tP>|t|[95%Conf.Interval]-------------+----------------------------------------------------------------lnrsoeoutput|-.0345891.0715531-0.480.634-.183392.1142138uhat|L1|.6512551.1650863.940.001.30794.9945702_cons|.1456964.29577130.490.627-.4693937.7607864------------------------------------------------------------------------------IntermediateEconometrics,YanShen48TestingforHigherOrderSerialCorrelation

檢驗(yàn)更高階的序列相關(guān)

CantestforAR(q)serialcorrelationinthesamebasicmannerasAR(1).

和檢驗(yàn)AR(1)同樣的方式可以用來(lái)檢驗(yàn)AR(q)序列相關(guān)。Step1:RuntheOLSregressionofyonallexplanatoryvariables,savetheresiduals.

步驟1：將y對(duì)所有解釋變量作OLS回歸，保存殘差。Step2:regresstheresidualsonalltheexplanatoryvariablesanditsownqorderlags.

步驟2:將殘差對(duì)所有解釋變量和它自身的1到q階滯后變量作回歸。Testthejointsignificanceofthecoefficientsr1

torq.

檢驗(yàn)r1

到

rq

的聯(lián)合顯著性。IntermediateEconometrics,YanShen49TestingforHigherOrderSerialCorrelation

檢驗(yàn)更高階的序列相關(guān)WecanuseFtestorLMtest,wheretheLMversioniscalledaBreusch-Godfreytestandis(n-q)Ru

2usingRu2fromresidualregression,

我們可以用F檢驗(yàn)或LM檢驗(yàn)，其中LM檢驗(yàn)被稱作是Breusch-Godfrey檢驗(yàn)而且為(n-q)Ru

2

，其中Ru2來(lái)自殘差回歸，TheLMstatisticfollowschisquaredistributionwithqdegreesoffreedomunderthenull.LM統(tǒng)計(jì)量在零假設(shè)下服從q個(gè)自由度的卡方分布。Whynowtimes(n-q)insteadn?

為什么乘上n-q而不是n？IntermediateEconometrics,YanShen50Examplecontinued

國(guó)企產(chǎn)出和國(guó)有商業(yè)商業(yè)銀行貸款關(guān)系例:檢驗(yàn)高階序列相關(guān).reguhatslnrsoeoutputl.uhatsl2.uhats

Source|SSdfMSNumberofobs=23-------------+------------------------------F(3,19)=7.08Model|.252363013.084121003Prob>F=0.0022Residual|.22565679619.011876673R-squared=0.5279-------------+------------------------------AdjR-squared=0.4534Total|.47801980622.021728173RootMSE=.10898

------------------------------------------------------------------------------uhats|Coef.Std.Err.tP>|t|[95%Conf.Interval]-------------+----------------------------------------------------------------lnrsoeoutput|-.0491576.070751-0.690.496-.1972411.0989258uhats|L1|.8731938.20088274.350.000.45274151.293646L2|-.3695032.2022297-1.830.083-.7927748.0537684_cons|.2114154.29409330.720.481-.4041291.8269598------------------------------------------------------------------------------IntermediateEconometrics,YanShen51Examplecontinued

范例（續(xù)）.testl.uhatsl2.uhats

(1)L.uhats=0(2)L2.uhats=0

F(2,19)=10.30Prob>F=0.0009IntermediateEconometrics,YanShen52CorrectingforSerialCorrelation

校正序列相關(guān)

Startwithcaseofstrictlyexogenousregressors,andmainta