中級(jí)計(jì)量經(jīng)濟(jì)學(xué)課件

第五講性響（QualitativeResponseBinaryResponseModels:LinearProbabilityModel(LPM)，LogitandProbitMultinomialResponseModels:MultinomialLogitandMultinomialProbitOrderedLogitandOrderedProbit第一節(jié)二值響應(yīng)模型（BinaryResponse一、二值響應(yīng)被解釋變量（BinaryResponseExplained一個(gè)家庭擁有住房（Y=1）或不擁有住房（Y成年人勞動(dòng)參與決定，參與（Y1）或不參與（Y夫妻雙方都參加工作（Y1）或只一人參加工作（YY=1ifownshome,0X e,1000Y=1ifownshome,0X e,1000108211071要建立模型，研究收入（X）會(huì)如何影響擁有住房（Y=1）發(fā)生的概1452530455678二、線性概率模型（LinearProbabilityModel,模y?β??β?x??β?x???β?x?????1,成功（失敗 失?。ǔ晒?y|X??y1p0E?y??var?y??E?y?E?y?????1?p??p??0?p???1?p??p?1?E?y|X??Xβ?p，0p?1var?y|X??Xβ?1?Xβ??p?1?p?例??0.9457? 001?0.10211個(gè)單位收入（$1000，家庭擁有住房的概率增加1x12（$12000）E(y|x12)0.94570.1021*12LPM模型存在的主要問(wèn)題干繞項(xiàng)的非正態(tài)性，導(dǎo)致在小樣本情況下推斷y0ε 當(dāng)y=1 1-β0- 當(dāng)y=0 -β0- var(ε)p(1-p)（由貝努里分布得到又因?yàn)閜E(yi|xiβ0+β1xi，所以ε的方差最終依賴于x，即具有異方差性。0E（yi|xi）1x值情況下都成立。比如上例中，預(yù)測(cè)yhat情況，有一些小于0，有一些大于1。110三、IndexModelsforBinaryResponseProbitandIndexp?y|X?? ResponseprobabilitydependsonX.Index:Xβ?β??β?x??ThefunctionGmapstheindexintotheresponseInmostapplication,Gisacumulativedistributionfunction(cdf)whosespecificformcansometimesbederivedfromanunderlyingeconomicmodel.IndexmodelswhereGisaCDFcanbederivedmoregenerallyfromanunderlyinglatentvariablemodel.y??Xβ? y?1?y???1???istheindicatory*rarelyhasawell-definedunitofmeasurement.Forexample,y*mightbemeasuredinutilityunit.p?y?1|X??p?y???p?Xβ?e??p?e??? 0 0ProbitandLogitModelse:standard ??p?y?1|X??G?Xβ?? ??v?dv?e?e:standardLogisticp?y?1|X??G?Xβ??Λ?Xβ??1?exp??

?1?expCumulativeDistributionFunctionF?x;μ,s? 1?e??Inthisequation,xistherandomvariable,μisthemean,andsisaparameterproportionaltothestandarddeviationProbabilityDistributionFunction e??fx;μ,

?s?1?e??

1?

1?

1?exp??

ln ??1?：oddsratio（機(jī)會(huì)比率ln???：logofoddsratio（機(jī)會(huì)比率對(duì)數(shù)）orPartialIfxjis

p??p?g?Xβ?β,where Therefore,thepartialeffectofxjonpdependsonXthroughg(Xβ).IfG(·)isastrictlyincreasingcdf,asintheprobitandlogitcase,g(z)>0forallz.thesignoftheeffectisgivenbythesignofthemagnitudesofthecoefficientsareNOTdirectlycomparableacrossmodels,althoughtheratiosofcoefficientsonthe(roughly)continuousexplanatoryvariablesare.therelativeeffectsdonotdependonx:??????? Ifxkisabinaryexplanatoryvariable,thenthepartialeffectfromchangingxkfromzerotoone,holdingallothervariablesfixed,issimplyG?β??β?x???β??x???β???G?β??β?x???umLikelihoodEstimationofBinaryResponseIndexY=0一般用最大似然法估計(jì)法估計(jì)參數(shù)。因?yàn)閅服從貝努里分布，我們有Pr(Yi=1)=piPr(Yi=0)=1-pi 于是觀測(cè)到nY值的聯(lián)合分布概率（ f(Y,Y,...,Y)f(Y)pYi(1p

function對(duì)數(shù)，我們有對(duì)數(shù)似然函數(shù)（loglikelihoodfunction）為nlnf(Y1,Y2,...,Yn)[Yilnpi(1Yi)ln(1pin[YilnpiYiln(1pi)ln(1pin Yiln ln(1pi pi ) ...11

1 2 k 1e01x1,i2x2,i...kxk,ilnf(Y,Y,...,Y)

Y( ... n

1 2 kln(1e01x1,i2x2,i...kxk,i以得到β的參數(shù)估計(jì)。法得到β的估計(jì)值以使似然函數(shù)最大。lnf?Y?,Y?…,Y???∑??Y?lnp???1?Y??ln?1?? X?? ?? X?? ???∑???Y?ln???e

?dv???1?Y??ln?1???思考線性模型的最大似然估計(jì)（MLE）？β?分布？:Asymptoticnormal(1)擁有住房與收入之logit/probit模STATA：e=x$14,000擁有住房，收入<$15,000不擁有住房，所以軟件無(wú)法匯報(bào)Logit/Probit的估計(jì)值。Adepartmentstorewantstodevelopadiscriminantruletodeterminewhetherlocalcollegestudentsshouldbegivencreditforfutureforfuturepurchases.DuringtheprecedingtwoyearsthestorecollectedinformationfromstudentsthatweregivenLogit.logitNRISKNSEXDum_MAJOR1Dum_MAJOR2Dum_MAJOR3GPTAGELogitNumberof LR Prob> Loglikelihood=-Pseudo NRISK Coef.Std. [95%Conf.+Dum_MAJOR1 -Dum_MAJOR2 Dum_MAJOR1 -Dum_MAJOR2 Dum_MAJOR3 GPT|----AGE|-.--.HRS|-.---_cons note:4failuresand0successesmpley

??log

??33.22?0.62NSEX?18um???m??5.01DumM??????3.94GPT?0.55AGE?eX???????偏效應(yīng)（partialNSEX=Male， ?ComparedwithBusinessmajor，Major= ?Major=Social ?????Major= ????? ????? ??????butnotsignificant ??????擬合優(yōu)度的測(cè)度pseudo-.tabulateRISKPredicted_Risk, |rowpercentage RISK GOOD BAD 8 9.20 GOOD 78 93.98 Total 86 50.59 pseudo-R2：最常用的是McFadden（1974）pseudo-McFaddenpseudo?R??1? 表示只有截距項(xiàng)的模型的對(duì)數(shù)似然為何可以用pseudo-R2刻劃擬合優(yōu)度nlnfY1Y2,.,Yn[Yilnpi1Yiln(1piY=0, lnf0，lnLlnL0如果解釋變量均無(wú)解釋能力，那么|lnL||lnL0|pseudo-R2通常情況下，|lnL|<|lnL0|,因此pseudo- <0ProbitprobitNRISKNSEXDum_MAJOR1Dum_MAJOR2Dum_MAJOR3GPTAGEProbit Numberof LR Prob> Loglikelihood=- Pseudo NRISK +Std. [95%NSEX|NRISK +Std. [95%NSEX|-Dum_MAJOR1 .- --Dum_MAJOR2Dum_MAJOR3..GPT|- --3.869698-AGE|- --.HRS|- ---_cons|

??risk?bad??Φ?19.07?0.33NSEX?。??e?? |rowpercentage RISK GOOD BAD 8 9.20 GOOD 78 93.98 Total 86 50.59 dataLogit模型與Probit模型的比42 1logistic分布具有稍微平坦的尾部（fattails42 10logitprobitCDF0研究者選用logit模型。ThreeAsymptoticallyEquivalenceTests(allbasedonumlikelihoodestimation）LikelihoodRatio(LR)LR?2?lnL??????????lnL???????Underthenullhypothesiswithqexclusion ?R?ar??R??R?r?T?Rar???R???R?H?:?β??2β???5β??3β?? ??β???? ?5 2?5?3?

?????

0 ?ov????? ar??v??,? ?5 ?????? a??? ?52?? ?5?3?H?:β?β??g?β??β?β?10 β? gβ? Tg???Tar????

g???????????

??

a?β? ?,???

?

?T?

????

?ov???? ar??????

???2????ov???,????

Reference:AllanW.GregoryandMichaelR.Veall,1985.FormulatingWaldTestsofNonlinearRestrictions.Econometrica,Vol.53,No.6(Nov.,1985),pp.1465-1468.MultinomialResponseUnorderedresponse,sometimescalledanominalHealthplanchoice:1/2/3;MultinomialLogitandMultinomialMultinomial

P?y?j|X?,j?0,1,2,…Forj?1,2,…,J,P?y?j|X? exp11??

expForj? P?y?0|X? PartialeffectsforthismodelareForcontinuousxk,wecan

1?1?

exp?P?y ?P?y?

β??exp?? 1?∑??expEventhedirectionoftheeffectisnotdeterminedbyAsimplerinterpretationofβ?isgivenLogodds

?exp?Xβ??,forj?1,2,…,log?P??X,β???P??X, log?P??X,β???X?β?βP??X, umlikelihoodForeachitheconditionalloglikelihoodcanbewritten????????1??????log??????,?Asusual,weestimateβ izing

?MultinomialProbitUnderlyingutilityTheutilityfromchoosingalternativej??????????Letyidenotethechoiceofindividuali ize??,,…??,??????? Amoreflexibleassumptionisthat??hasamultivariatenormaldistributionwithcalledemultinomialprobitmodel.Theoretically,themultinomialprobitmodelisattractive,butishassomepracticalTheresponseprobabilitiesareverycomplicated,involvinga(J+1)-dimensionalintegral.Thiscomplexitynotonlymakesitdifficulttoobtainthepartialeffectsontheresponseprobabilities,italsomakes umlikelihoodestimation(MLE)infeasibleformorethanaboutfivealternatives.Example:SchoolandEmploymentDecisionsforYoungsch=home=work=.mlogitstatuseducexperexpersqblackify87==1,basecategory(0)Iteration0: loglikelihood=-1199.7182Iteration loglikelihood=-Iteration loglikelihood=-Iteration loglikelihood=-Iteration loglikelihood=-Iteration loglikelihood=-Multinomiallogistic Numberof LR Prob> Loglikelihood=- Pseudo status Coef.Std. [95%Conf.+ educ|educ|- ---exper|- --.expersq|- --black ._cons educ|- ---exper .expersq|- ---black -._cons estatus==0isthecomparison??home?log???

??10.28?0.67educ?0.11exper?0.01expersq????work? ??5.54?0.31educ?0.85exper?0.08expersq?P?P?home? P?

????home?black?1,log???school???Themagnitudesaredifficulttointerpret.Insteadwecaneithercomputepartialeffectsorcomputedifferencesinprobabilities.Forexample,considertwoblackmen,eachwithfiveyearsofexperience.Ablackmanwith16yearsofeducationhasemploymentprobabilitythatis0.042higherthanamanwith12yearsofeducation,andtheat-homeprobabilityis0.072lower.??Status exp?10.28?0.67educ?0.11exper?0.01expersq?1?exp?10.28?0.67educ?0.11exper?0.01expersq?0.81black??exp??Status 1?exp?10.28?0.67educ?0.11exper?0.01expersq?0.81black??exp??Status 1?exp?10.28?0.67educ?0.11exper?0.01expersq?0.81black??expwhere??5.54?0.31educ?0.85exper?0.08expersq?Greene,W.H.andD.A.Hensher,2010.ModelingOrderedChoices:ACambridgeUniversityWooldridge,J.M.,2010.EconometricysisofCrossSectionandPanelData,2ndedition.TheMITPress.OrderedCreditrating:Bondrating:Healthstatus:Productquality:個(gè)人程度（Happiness:0/1/2/3/4/5）悲慘（凄慘？）=0略感痛苦=無(wú)感覺（沒(méi)心沒(méi)肺？）=有點(diǎn)快樂(lè)=較快樂(lè)=極度快樂(lè)、癲狂狀態(tài)（convergetocase32)?沒(méi)心沒(méi)肺0<1<2<3<4<OrderedProbitandOrderedLogitUnderlyinglatenty??Xβ? y?:latentXdoesnotcontainaLetα??α???αJbeunknowncutpoints(orthresholdy?0ify??y?1ifα????y?2ifα?????α?y?Jify??Giventhestandardnormalassumptionfore,itisstraightforwardtoderivetheconditionaldistributionofygivenX.P?y?0|X??P?y??α?|X??P?Xβ?e?α?|X??Φ?α??P?y?1|X??P?α?????α???P?α???????P?α??Xβ???α??Xβ|X??Φ?α????α?|X??P?α??Xβ???α??Xβ|X??Φ?α??Xβ???P?y?J|X??P?y??αJ|X??P?e?αJ?Xβ?X??1?Φ?αJα α α2‐Xβ P?y?0|X??P?y?1|X???P?y?J|X??ReplacingΦwiththeLogitfunction,Λ,givestheorderedLogitumLikelihoodPartialIneithercase,wemustrememberthatβ,byitself,isoflimitedinterest.InmostcasewearenotinterestedinE(y*|X)=Xβ,asy*isan construct.Instead,weareinterestedintheresponseprobabilitiesP(y=j|X).?P???β??α? ?P??β

Xβ?

—Xβ??,0??? ??PJ?β??α— Forintermediate es1,2,…J-1,β?doesNOTalwaysdeterminethedirectionoftheeffectfortheintermediate Notethat exp???.oprobitrep77foreignlengthIteration loglikelihood=-Iteration loglikelihood=-Iteration loglikelihood=-Iteration loglikelihood=-Orderedprobit Numberof LR Prob> Loglikelihood=- Pseudo rep77 Coef.Std. [95%Conf. . . ..foreign|lengthmpg+_cut1 (Ancillary_cut2_cut3_cut4y?Poor,ify??y?Fair,if10.1589????y?Average,if11.21003????y?Good,if12.54561????y?Excellent,ify??13.98059??1.704861foreign? length? Aswithmultinomiallogit,fororderedresponseswecancomputethepresentcorrectlypredicted,foreach easwellasoverall:ourpredicationforyissimplythe ewiththehighestestimatedprobability.第六講時(shí)間序列模型(TimeSeries時(shí)間序列的平穩(wěn)性及其檢驗(yàn)單變量時(shí)間序列模型:ARIMA模型多變量時(shí)間序列模型:VAR模型波動(dòng)性模型：ARCH和GARCH模型第一節(jié)時(shí)間序列的平穩(wěn)性及其檢驗(yàn)滯后（LagsTheobservationonthetimeseriesvariableYmadeatdatetisdenotedYt,andthetotalnumberofobservationsisdenotedT.DataFrequency：Theintervalbetweenobservations,thatis,theperiodoftimebetweenobservationtandobservationt+1,issomeunitoftimesuchasweeks,months,quarters(three-monthunits).ThevalueofYinthepreviousperiodiscalleditsfirstlaggedvalueor,moresimply,itsfirstlag,andisdenotedYt-1.Itsjthlaggedvalue(orsimplyitsjthlag)isitsvaluejperiodsago,whichisYt-RateofInflationAnnualRateFirst(Yt-Second(Yt-Third(Yt-......Lag運(yùn)算LLYt=LYt-1=Yt-2LjYt=Yt-一階差分（FirstThechangeinthevalueofYbetweenperiodt-1andperiodtisYt–Yt-1;thischangeiscalledthefirstdifferenceinthevariableYt.Inthetimeseriesdate,“Δ”(delta)isusedtorepresentthefirstdifference,sothat??ΔY???Y???Y??2Y??ΔY??ΔΔY??ΔY??Y????Y??Y??????

??Y??Y?

?ΔY??Y??Y???Y??LY???1?Δ?Y??Y??2Y???Y???Y??2LY??L?Y???1?2L?L??Y??L??Y?Δ?Y???1?自然對(duì)數(shù)lnY??logY??lnTwoManyeconomicseries,suchasgrossdomesticproduct(GDP),exhibitgrowththatisapproximayexponential,thatis,overthelongruntheseriestendstogrowbyacertainpercentageperyearonaverage;ifso,thelogarithmsoftheseriesgrowsapproximaylinearly.1,2,…,Tln?GDP???α? ?1,2,…,Thestandarddeviationofmanyeconomictimeseriesisapproxima proportionaltoitslevel,thatis,thestandarddeviationiswellexpressedasapercentageoftheleveloftheseries;ifso,thestandarddeviationofthelogarithmoftheseriesisapproximayconstant.

??log?Y?log?Y??

?Y?Y?? ???

log ?Y—log 1 1var?log?Y var?Y?? GrowthRate?Y??Y???ΔY?Δln?YY?

Y? Δln?Y???ln?Y???ln?Y?二、平穩(wěn)性（stationary）含義嚴(yán)平穩(wěn)性 stationary）時(shí)間序｛y1，y2,…,yt｝的聯(lián)合概率分布（jointdistribution）｛y1+k，y2+k,…,yt+k｝的聯(lián)合概率分布相同，我們就稱yt是嚴(yán)平穩(wěn)弱平穩(wěn)性 stationary）時(shí)間序yt，如果其均值、方差是不隨時(shí)間而變化，協(xié)方差僅依賴于觀測(cè)E(yt)=var(yt)=E(yt–μ)2=cov(yt,yt+k)=γk=E[(yt-μ)(yt+k-μ)]=E[(yt+m-μ)(yt+m+k-我們通常所說(shuō)時(shí)間序列的平穩(wěn)性是指弱平穩(wěn)性。Onerealizationoftheprocessateachtime三、兩種常見非平穩(wěn)隨機(jī)過(guò)程（stochastic無(wú)漂浮隨機(jī)（RandomWalkwithoutYt=Yt-1+noiseY1=Y0+Y2=Y1+e2=Y0+e1+Y3=Y2+e3=Y0+e1+e2+

Yt=Y0+E(Yt)=E(Y0+Σet)=Y0var(Yt)Yt=Yt-1+012324nois012324--- ---Yt=δ+Yt-1+Yt=1+Yt-1+四、偽回歸 regression）現(xiàn)yt=yt-1+e1te1t~N(0,1)xt=xt-1+e2te2t~N(0,yt與xt之間應(yīng)該沒(méi)有關(guān)系，但是注意STATA回歸結(jié).pwcorre1e2, e1e1||e2-|.regytSource Numberofobs F( 198)=Model| 1 Prob> =Residual| R- = AdjR-squared=Total| 199 Root =yt Std. [95%Conf.+xt _cons .Durbin-Watsond-statistic( 200)=.gendy=yt-(1missingvalue.gendx=xt-(1missingvalue..regdySource Numberofobs F( 197) Model| 1 Prob> =Residual| 197 R- = AdjR-squared=-Total| Root =dy Std. [95%Conf.+dx - _cons - .Durbin-Watsond-statistic( 199)=四、平穩(wěn)性檢驗(yàn)對(duì)時(shí)間序列數(shù)據(jù)的回歸分析中我們一般假設(shè)時(shí)間序列是平穩(wěn)的，但是現(xiàn)實(shí)中的大多數(shù)時(shí)間序列都是非平穩(wěn)的。為了避免偽回歸現(xiàn)象，對(duì)時(shí)間序列首先要 100自協(xié)方差??X,Y?

???Y??N? ?????vY, ?

?Y????Y?? ?

T1

?

?

?tYt-Yt-1..2.345678自相關(guān)系數(shù)（autocorrelationorserialcorrelationρ?cov?X,? ??

????∑?

?????rr ? ?

?j??autocorrelation?ρY???var?Y??var?Y?STATA:.corrytL.yt +yt--. L1. .corrytL2.yt| +yt--.

L2. .corrytL20.yt| +yt--.

L20. .corrgramyt,- 1- Prob>Q[Autocorrelation][Partial1|------|------2-|-------3|------4-|------|5|------|6-|------|7-|------|8-|------|9-|-----||-----|-----||-----||-----|-|-----|-|-----||-----|-|----|-|----|-|----|-|----|---Autocorrelationsofyt=yt-1+Autocorrelationsofwhitenoise白噪聲（WhiteAtimeseriesetiscalledwhitenoiseifetisasequenceofindependentandidenticallydistributed(i.i.d)randomvariableswithfinitemeanandvariance.Inparticular,ifetisnormallydistributedwithmeanzeroandvarianceσ2,theseriesiscalledGaussianwhitenoise.Forawhitenoiseseries,alltheACFsarezero.PortmanteauTestforWhiteBoxandPierce(1970)proposethePortmanteau????asateststatisticforthenullhypothesisHo:ρ??ρ???ρ??0againsttealternativehypothesisHo:ρ?0forsomei??1,2,…,m?.Undertheassumptionthat{et}isaniidsequencewithcertainmomentconditions,Q*(m)isasymptoticallyachi-squaredrandomvariablewithmdegreesoffreedom.LjungandBox(1978)modifytheQ*(m)statisticasbelowtoincreasethepowerofthetestinfinitesamples: ?????T?T?2????ThedecisionruleistorejectHoifQ?m??χ?,whereχ?denotesthe100(1- percentileofachi-squareddistributionwithmdegreesoffreedom.Mostsoftwarepacketswillprovidethep-valueofQ(m).ThedecisionruleisthentorejectHoifthep-valueislessthanorequaltoα,thesignificancelevel..wntestqe1,lags(50)PortmanteautestforwhitenoisePortmanteau(Q)=Prob>=.wntestqe2,lags(50)PortmanteautestforwhitenoisePortmanteau(Q)=Prob>=.wntestqlny,lags(50)PortmanteautestforwhitenoisePortmanteau(Q)statistic=3865.0100Prob> .wntestqdlny,lags(50)PortmanteautestforwhitenoisePortmanteau(Q)statistic= Prob> AutocorrelationsofAutocorrelationsof-0.200.000.20ACFoflog(exchange ACFofAutocorrelationsofAutocorrelationsof-0.200.000.20平穩(wěn)時(shí)間序列ACF特征:當(dāng)k增大時(shí)，衰減非?？?。單位根（unitroot）及其檢驗(yàn)單位yt=ρyt-1+et，-1≤ρ≤?ρ

??ρ?ρy???e???ρ?

?ρe?

??ρ?ρy?

??ρ?y???ρ?e???ρe?

??e??ρe?

?ρ?e?

?ρ?e???E(yt)=?var?y???cov?y

?? ? （2）-（Dickey-Fuller）單位根檢驗(yàn)思yt=ρyt-1+方程兩邊同時(shí)減去Yt-1，得ytyt1(1)yt1或

ytyt1注意：不是用t檢驗(yàn)，因?yàn)樵谠僭O(shè)為真的情況下，δ從t分布，而服注意：原假設(shè)是具有單位根STATA結(jié).dfullerDickey-Fullertestforunit Numberof 1%5%10%1%5%10% - - - -*MacKinnonapproximatep-valueforZ(t)=.dfullerDickey-Fullertestforunit Numberof 1%5%10%1%5%10% - - - -*MacKinnonapproximatep-valueforZ(t)=(3)AugmentedDickey-Fuller(ADF)通過(guò)以下三個(gè)模型完成：??δ?

L?

??y??α?δy?

?

?

??第二節(jié)單變量時(shí)間序列模ARIMA模型經(jīng)濟(jì)知識(shí)，只有具備感變量的歷史數(shù)據(jù)就夠了，因而模型的制定和數(shù)據(jù)的一、ARIMA模型的特點(diǎn)ARIMA（AutoregressiveIntegratedMovingAverage）Box-Jenkins方法，是上世紀(jì)七十年代建立起來(lái)的。ARIMA模型兩個(gè)特點(diǎn)：

?其中et是白噪

?2.AR模型的識(shí)別ACF（AutocorrelationFunction）PACF（PartialAutocorrelationρ?γ??cov?y?,y? PACFytyt-kyt-1,….,yt-k+1所帶來(lái)的間接相關(guān)性之后，yt與yt－k之間的直接相關(guān)性。yt,yt-1,yt-2,….,yt-k+1,yt-PACF可以通過(guò)以下Yule-Walker方程求y???????y???y???????y?????y???y???????y?????y?????y???…Autocorrelationsof-0.200.000.200.40Partialautocorrelationsof-0.200.00Autocorrelationsof-0.200.000.200.40Partialautocorrelationsof-0.200.000.200.40

ARmodel：ACF：衰減 AutocorrelationsofAutocorrelationsofPartialautocorrelationsof 利用AIC、SIC等信息準(zhǔn) AIC?T

?T??#ofSIC??2ln?likelihood??#ofparametersln AR模型

?AR模型1-stepahead???1??????y????y?????y??? ???y??????1?var????1???2-stepahead???2??????y?????y????y??? ???y??????????y???y????????e??var????2????σ????σ? yprediction, yprediction,462三、MA（MovingAverage）模型模

?????

?θ?e?

??θ?e?MA模型的識(shí)別AutocorrelationsofAutocorrelationsof-0.200.000.200.40Partialautocorrelationsof-0.40-0.200.000.200.40AutocorrelationsofPartialautocorrelationsofAutocorrelationsofPartialautocorrelationsof-0.40-0.200.00 MA模型的估計(jì)e0=MA模型的預(yù)測(cè) yprediction, yprediction,-05--05 yprediction,one--Autocorrelationsof-0.40-0.200.000.200.40PartialAutocorrelationsof-0.40-0.200.000.200.40Partialautocorrelationsof-0.200.000.200.40 .arima.arimaa,ARIMASample:1-Numberof Wald Loglikelihood=-Prob> |a Std.z [95%Conf.

| | |||||| -.-.-+/sigma Note:Thetestofthevarianceagainstzeroisonesided,andthetwo-sidedconfidenceintervalistruncatedatzero.五、ARIMA(p,d,q)模型建模一般對(duì)估計(jì)的ARMAARMAAICSIC等第三節(jié)多變量時(shí)間序列VAR模型TheMotivationbehindVARsinThemotivationbehindVARsinmacroeconomicsrunsdeeperthanthestatisticalissues.Thelargestructuralequationsmodelsofthe1950sand1960swerebuiltonatheoreticalfoundationthathasnotprovedsatisfactory.ThattheforecastingperformanceofVARssurpassedthatoflargestructuralmodels—someofthelatercounterpartstoKlein’sModelIrantohundredsofequations—signaledtoresearchersamorefundamentalproblemwiththeunderlyingmethodology.TheKeynesianstylesystemsofequationsdescribeastructuralmodelofdecisions(consumption,investment)thatseemlooselytomimicindividualbehavior.Intheend,however,thesedecisionrulesarefundamentallyadhoc,andthereislittlebasisonwhichtoassumethattheywouldaggregatetothemacroeconomiclevelanyway.Onamorepracticallevel,thehighinflationandhighunemploymentexperiencedinthe1970swereverybadlypredictedbytheKeynesianparadigm.Fromthepointofviewoftheunderlyingparadigm,themosttroublingcriticismofthestructuralmodelingapproachcomesintheformof“theLucascritique”(1976)inwhichtheauthorarguedthattheparametersofthe“decisionrules”embodiedinthesystemsofstructuralequationswouldnotremainstablewheneconomicpolicieschanged,eveniftherulesthemselveswereappropriate.Thus,theparadigmunderlyingthesystemsofequationsapproachtomacroeconomicmodelingisarguablyfundamentallyflawed.Morerecentresearchhasreformulatedthebasicequationsofmacroeconomicmodelsintermsofamicroeconomicoptimizationfoundationandhas,atthesametime,beenmuchlessambitiousinspecifyingtheinterrelationshipsamongeconomicvariables.(FromGreene2003,p587)VectorAutoregressionVAR???

????

?????

?????whereutisavectorofnonautocorrelateddisturbances(innovations)withzeroandcontemporaneouscovariancematrixE(utut’)=∑.Thisequationsystemisavectorautoregression,orVAR.?

?

??

? ? ???? ?? ??..

??

???????

?? ?

..?..

??

? ? ???? ..? .. ..

?

.....?

??

???

???RolesofForecasting:researchershavefoundthatsimple,small-scaleVARswithoutapossiblyflawedtheoreticalfoundationhaveprovedasgoodasorbetterthanlarge-scalestructuralequationsystems.StudyingtheeffectsofpolicythroughimpulseresponseGrangerDefinitionsofGranger(1969)hasdefinedaconceptofcausalitywhich,undersuitableconditions,isfairlyeasytodealwithinthecontextofVARmodels.Thereforeithas equitepopularinrecentyears.Theideaisthatacausecannotcomeaftertheeffect.Thus,ifavariablexaffectsavariablez,theformershouldhelpimprovingtheofthelatterAnecessaryandsufficientconditionforxtbeingnotGranger-causalforzt,thatis,ztisnotGranger-causedbyxtifandonlyifA12,i=0fori=1,...,For???????doesnotGranger-causez=ybecauseA=0

Ontheotherhand,ztGranger-causesInappliedwork,itisoftenofinteresttoknowtheresponseofonevariabletoanimpulseinanothervariableinasystemthatinvolvesanumberoffurthervariablesaswell.Thus,onewouldliketoinvestigatetheimpulseresponserelationshipbetweentwovariablesinahigherdimensionalsystem.Tracingaunitshockinthefirstvariableinperiodt=0inthissystemwet= ?????,??????,??? t=

??????,???????? 0.3??0?? t= ??????,???????????????????? 0.12??0?? t ??????,?????????0.0370.0310.057??0??

y3‐>y3‐>y2y3‐>10 VAR→LetusconsidertheVAR(1)Ifthisgenerationmechanismstartsatsometimet=1,say,weThisformoftheprocessiscalledthemovingaverage(MA)Anotherwayto

????????????????????????????????????????????????????? ???????????????????????????? Notethat??????????????????????????? ResponsestoOrthogonalAproblematicassumptioninthistypeofimpulseresponseysisisthatashockoccursonlyinonevariableatatime.Suchanassumptionmaybereasonableiftheshocksindifferentvariablesareindependent.Iftheyarenotindependentonemayarguethattheerrortermsconsistofalltheinfluencesandvariablesthatarenotdirectlyincludedinthesetofyvariables.Thus,inadditiontoforcesthataffectallthevariables,theremaybeforcesthataffectvariable1,say,only.IfashockinthefirstvariableisduetosuchforcesitmayagainbereasonabletointerprettheΦicoefficientsasdynamicOntheotherhand,correlationoftheerrortermsmayindicatethatashockinvariableislikelytobe paniedbyashockinanothervariable.Inthatcase,settingallotherresidualstozeromayprovideamisleadingpictureoftheactualdynamicrelationshipsbetweenthevariables. If∑isapositivedefinite(k×k)matrix,thenthereexistsalower(upper)triangularmatrixPwithpositivemaindiagonalsuchthat?Σ??????orΣ?For ? 0?? 0?? positionA=PP’wherePislowertriangularwithpositivemaindiagonal,issometimescalledCholesky ConsideraVARwithoutexogenousTheVARrepresentsthevariablesinytasfunctionsofitsownlagsandseriallyuncorrelatedinnovationsut.AlltheinformationaboutcontemporaneouscorrelationsamongtheKvariablesinytiscontainedin∑.Toseehowtheinnovationsaffectthevariablesinytafter,say,iperiods,rewritethemodelinitsmoving-averageformwhereμistheK*1time-invariantmeanofWecanthususeP-1toorthogonalizetheutandrewritetheaboveequationChoosingaPissimilartoplacingidentificationrestrictionsonasystemofdynamicsimultaneousequations.ThesimpleIRFsdonotidentifythecausalrelationshipsthatwewishto yze.ThusweseekatleastasmanyidentificationrestrictionsasnecessarytoidentifythecausalIRFs.So,wheredowegetsuchaP?Sims(1980)popularizedthemethodofchoosingPtobetheCholesky theorthogonalizedIRFs.ChoosingPtobetheCholesky positionof?isequivalenttoimposingarecursivestructureforthecorrespondingdynamicstructuralequationmodel.TheorderingoftherecursivestructureisthesameastheorderingimposedintheCholesky position.Becausethischoiceisarbitrary,someresearcherswilllookattheOIRFswithdifferentorderingsassumedintheCholeskyeeGrangerCausality.GrangercausalityWaldProb>e224e2e2e42e24 enotGrangercausingdlinverstment Ho:dlconsumptionnotGrangercausing FailtoHo:dlinvestmentnotGrangereHo:dlconsumptionnotGrangereHo:dlinvestmentnotGrangercausingdlconsumption enotGrangercausingdlconsumption RejectInvestment Consumption ImpulseResponse95% orthogonalizedGraphsbyirfname,impulsevariable,andresponse005005results1, .varirftableoirf, e)Resultsfrom |step | | - | |- - | - | - | - | - | - | - 95%lowerandupperbounds(1)irfname=results1,impulse e,andresponse=Granger,C.W.J.,1969.Investigatingcausalrelationsbyeconometricmodelsandcross-spectralmethods,Econometrica37:424–438.Lütkepohl,H.2005.NewIntroductiontoMultipleTimeSeriesysis.NewYork:Springer.Juselius,K.2006.TheCointegratedVARModel:MethodologyandApplications.OxfordUniversityPress.Stock,J.H.,andM.W.Watson.2001.Vectorautoregressions.JournalofEconomics15:101–115.Sims,C.A.1980.Macroeconomicsandreality,Econometrica48:[6].Watson,M.W.1994.Vectorautoregressionsandcointegration.InVol.IVofHandbookofEconometrics,ed.R.F.EngleandD.L.McFadden.Amsterdam:第四節(jié)波動(dòng)性模型：ARCHGARCH模一、金融時(shí)間序列數(shù)據(jù)特征原數(shù)據(jù)是隨機(jī)（randomwalk）過(guò)程（非平穩(wěn)的；波動(dòng)性現(xiàn)象（volatilityFirstdifferenceofUS/UKexchange（January1973–October二、自回歸條件異方差(AutoregressiveConditionalHeteroscedasticity，ARCH(1)Yt2var(| ) 1STATA結(jié)ARCHfamilySample:2to Numberof Wald Loglikelihood= Prob> | +

Std. [95%Conf. |- - - +

|L1 EngleARCHp

)0.0006Yt2var(| )2 ... 1 2 p三、廣義自回歸條件異方差模型(GeneralizedAutoregressiveConditionalHeteroscedasticity,GARCH)GARCH（1，1）模Yt2var(| )2 1 1GARCH（1，1）模ARCH()Lag運(yùn)算：

Lyt=yt-L2yt=yt-1L2L23L3.... GARCH11)

122 1 12L2 1(1L)212

(1

(11L)2 (1L2L23L3 (1

2L22L22 3L32 (11L) 1t1 11 11 11 22 22 32 (11L) 1t1 11t2 11t3 11t4ARCHfamilyregressionSample:2to286Numberof=Loglikelihood=Prob>chi2==..|||Std.z[95%Conf. |- - - + L1 L1 - ln(exchange_rate)20.00010.22202 第七講板數(shù)(PanelData第一節(jié)面板數(shù)據(jù)及面板數(shù)據(jù)模型一、面板數(shù)據(jù)（PanelDataorLongitudinalPanelData含PanelData包括兩個(gè)維度，對(duì)每一橫截面觀測(cè)多橫截面：i=1,2,…,時(shí)間序列：t=1,2,itKL………兩個(gè)重要的面板數(shù)據(jù)集（PanelData（1）U.S.NationalLongitudinalSurveysofLaborMarketExperienceTheNationalLongitudinalSurveys(NLS)areasetofsurveysdesignedtogatherinformationatmultiplepointsintimeonthelabormarketactivitiesandothersignificantlifeeventsofseveralgroupsofmenandwomen.Formorethan4decades,NLSdatahaveservedasanimportanttoolforeconomists,sociologists,andotherresearchers.NLSGeneralNationalLongitudinalSurveyofYouth1997(NLSY97)--Surveyofyoungmenandwomenbornintheyears1980-84;respondentswereages12-17whenfirstinterviewedin1997.NationalLongitudinalSurveyofYouth1979(NLSY79)--Surveyofmenandwomenbornintheyears1957-64;respondentswereages14-22whenfirstinterviewedin1979.NLSY79ChildrenandYoungAdults--Surveyofthebiologicalchildr