chapter 6时间序列.docx
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chapter 6时间序列.docx
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chapter6时间序列
CHAPTER6
COINTEGRATIONANDERROR-CORRECTIONMODELS
AnswerstoQuestions
1.Letequations(6.14)and(6.15)containintercepttermssuchthat:
yt=a10+a11yt-1+a12zt-1+εytandzt=a20+a21yt-1+a22zt-1+εzt
A.Showthatthesolutionforytcanbewrittenas:
yt=[(1-a22L)εyt+(1-a22)a10+a12Lεzt+a12a20]/[(1-a11L)(1-a22L)-a12a21L2]
Answer:
Uselagoperatorstorewritethesystemas:
(1-a11L)yt-a12Lzt=a10+εyt
-a21Lyt+(1-a22L)zt=a20+εzt
Nowwritethesysteminmatrixformas:
UsingCramer'sRuleormatrixinversion,wecanobtainthesolutionsforytas:
Since(1-a22L)a10=(1-a22)a10,anda12La20=a12a20,thesolutionforytisverified.
B.Findthesolutionforzt.
Answer:
UseyouranswertoPartAandapplyCramer'sRuleormatrixinversiontoobtain:
C.SupposethatytandztareCI(1,1).Usetheconditionsin(6.19),(6.20),and(6.21)towritetheerror-correctingmodel.Compareyouranswerto(6.22)and(6.23).Showthattheerror-correctionmodelcontainsaninterceptterm.
Answer:
ImposingtherestrictionsnecessarytoensurethatytandztareCI(1,1),theequationsforytandztcanbewrittenas:
∆yt=-[a12a21/(1-a22)]yt-1+a21zt-1+εyt+a10
∆zt=a21yt-1-(1-a22)zt-1+εzt+a20
Normalizingthecointegratingvectorwithrespecttoyt-1:
∆yt=αy(yt-1-βzt-1)+εyt+a10
∆zt=αz(yt-1-βzt-1)+εzt+a20
where:
αy=-a12a21/(1-a22);αz=a21;andβ=(1-a22)/a21.
Thus,theerror-correctingequationsfor∆ytand∆zteachcontainadriftterm.AnotherwaytoanswerthequestionistonotethatthesolutionsforytandztobtainedinPartsAandBcontainthedeterministicexpressions[(1-a22)a10+a12a20]and[a21a10+(1-a11)a20],respectively.Sincethedenominatorcontainsacharacteristicrootequaltounity,thesolutionforeachcontainsadeterministictrend.
D.Showthat{yt}and{zt}havethesamedeterministictimetrend(i.e.,showthattheslopecoefficientofthetimetrendsareidentical).
Answer:
Theconstantinthenumeratorofthesolutionforytis:
[(1-a22)a10+a12a20].Since1-a22=a12a21/(1-a11),thisconstantcanberewrittenas:
[a12/(1-a11)][a21a10+(1-a11)a20].Uptotheexpression[a12/(1-a11)],thisdeterministicnumeratorexpressionisthesameasthatinthesolutionforzt.Giventhatthedenominatorsareidentical,ytandztcanbesaidtohavethesamedeterministictimetrend.
E.Whatistheconditionsuchthattheslopeofthetrendiszero?
Showthatthisconditionissuchthattheconstantcanbeincludedinthecointegratingvector.
Answer:
The{yt}sequencedoesnothaveaslopeif(1-a22)a10+a12a20=0.Solvingfora10yieldsa10=-a12a20/(1-a22).Usingthisrelationship,theerror-correctionequationfor∆ytis:
∆yt=αy(yt-1-βzt-1)+εyt-a12a20/(1-a22)
=αy(yt-1-βzt-1+a20/a21)+εyt
Sinceαz=a21,theerror-correctionmodelfor∆ztcanbewrittenas:
∆zt=αz(yt-1-βzt-1+a20/a21)+εzt.
Thus,thenormalizedlong-runequilibriumrelationshipisyt-1-βzt-1+a20/a21.Thecointegratingvectorhasaninterceptalthoughthe{∆yt}and{∆zt}sequencesdonotcontaindeterministictrends.
2.ThedatafileCOINT6.PRNcontainsthethreesimulatedseriesusedinsections5and9.Thefollowingprogramswillreproducetheresults.
SampleProgramforRATSUsers
all100
opendataa:
coint6.prn;*Thedatadiskisindrivea:
\
data(format=prn,org=obs)/yzw
table;*Producesummarystatisticsfory,zandw
setdy=yy(t1);*Takefirst-differences
setdz=zz(t1)
setdw=ww(t1)
linregdy;*PerformDickey-Fullertest
#constanty{1}
linregdy;*PerformAugmentedDickey-Fullertest
#constanty{1}dy{1to4}
*Repeatthefourlinesaboveforzandw.Alternatively,youcanusetheprocedureentitled
*DFUNIT.SRC.TouseDFUNIT.SRC,typethestatements
sourcec:
\rats\dfunit.src;*Theprocedureisassumedthedfunit.srcprocedureisinthe:
*RATSdirectoryondrivec:
dfunit(lags=4)/y
linregy/residy;*Estimatethelong-runequilibriumrelationshipusingyas
#constantzw;*theleft-hand-sidevariable.Savetheresidualsas"residy"
setdresidy=residyresidy{1};*Obtainfirst-differenceoftheresiduals
linregdresidy;*PerformtheDickey-Fullertestoftheresiduals
#residy{1}
linregdresidy;*PerformtheAugmentedDickey-Fullertest
#residy{1}dresidy{1to4}
*Repeatthe7linesaboveforzandw
system1to3;*Setupthesystemfortheerror-correctionmodel
variablesdydzdw
lags1to2;*Use2lagsofdy,dz,anddw
detconstantresidy{1};*Includeaconstantandtheerror-correctionterm.Youcan
end(system);*usetheresidualsfromtheothertwoequilibriumrelations
estimate(outsigma=vsigma)*Estimatethemodel.Vsigmaisthevariance/covariancematrix
errors(impulses)324vsigma;*Performinnovationaccountingusingtheerror-correction
#1;#2;#3;*model
*ToreproducetheresultsinSection9,usetheCATSprocedureorthedownloadablefile
*entitledjohansen.src.NotethattheJohansenprocedureinRATSdoesnotallowyoutouse
*thevariablenamew.Redefinewusingthefollowingstatement
setx=w
sourcec:
\rats\johansen.src;*Itisassumedthejohansen.srcprocedureisintheRATSdirectoryondrivec:
\
@johansen.src(lags=2)/
#yzx
Notethatjohansen.srcmayinappropriatelyaddseasonaldummyvariablestoyourmodel.Moreover,thereisnosimplewaytochoosetheformoftheinterceptterm.IfyouuseRATS,youranswerswillbeslightlydifferentfromthosereportedinthetext.Forexample,theλmaxandλtracestatisticswillbereportedas:
lambda,lambdamaxandtracetest
0.324960.134010.02536
38.5127214.100612.51767
2.5176716.6182955.13101
3.ThefileCOINT_PPP.XLScontainsquarterlyvaluesofGerman,Japanese,andCanadian
wholesalepricesandbilateralexchangerateswiththeUnitedStates.ThefilealsocontainstheU.S.wholesalepricelevel.Thenamesontheindividualseriesshouldbeself-evident.Forexample,p_usistheU.S.pricelevelandex_gistheGermanexchangeratewiththeUnitedStates.Allvariablesexceptthemark/dollarexchangeratesrunfrom1973:
Q4to2001:
Q4andallhavebeennormalizedtoequal100in1973:
Q4.
A.Formthelogofeachvariable.Estimatethelong-runrelationshipbetweenCanada
andtheUnitedStatesas
log(ex_ca)=4.12+0.937log(p_ca)–0.830log(p_us)
Dothepointestimatesoftheslopecoefficientsseemtobeconsistentwithlong-runPPP?
Answer:
Althoughthepointestimatesseemtobeconsistentwithlong-runPPP,youneedtobeabitcareful.Thereisanaturaltendencytothinkthat0.937isapproximatelyequaltounityand0.830isapproximatelyequaltominusone.However,inferenceonthecointegratingisunwarrantedsincetheresidualsfromtheregressionareseriallycorrelatedandpricesarenotnecessarilyweaklyexogeneous.
B.Sincetheresidualsfromtheequilibriumregressioncontainaunitroot,shockstotherealexchangerateneverdecay.Hence,long-runPPPfails.
C.ARATSprogramthatcanperformtheindicatedtestsis
cal197344;*Thedatasetbeginsin1973Q4andendsin2004Q4
all2001:
4
opendataa:
\coint_ppp.xls
data(org=obs,format=xls)
*Next,takethelogofeachvariable
logex_g/lex_g;logex_ca/lex_ca;logex_j/lex_j
logp_g/lp_g;logp_j/lp_j;logp_ca/lp_ca;logp_us/lp_us
*Youshouldnowtesteachforaunitroot
*Thelong-runrelationshipfortheCanadian-U.S.ratecanbeobtainedusing
linlex_ca/resids;#constantlp_calp_us
*Now,testtheresidualsforaunitroot
difresids/dr
lindr;#resids{1}dr{1to3}
*Similarly,PPPfortheGerman-U.S.ratecanbetestedusing
linlex_g/resids;#constantlp_glp_us
difresids/dr
lindr;#resids{1}dr{1to4}
4.Thesecond,fourth,andfifthcolumnsofthefilelabeledINT_RATES.XLScontaintheinterestratespaidonU.S.3-month,3-year,and10-yearU.S.governmentsecurities.Thedatarunfrom1954:
7to2002:
12.ThesecolumnsarelabeledTBILL,r3,andr10,respectively.
RATSPROGRAM
cal1954712;*ThedatasetrunsfromJuly1954toDecember2002
all2002:
12
opendataa:
\int_rates.xls
data(org=obs,format=xls)
*Totesteachseriesforaunitrootusingdfunit.src
source(noecho)c:
\winrats\dfunit.src
@dfunit(ttest,lags=12)tbill
@dfunit(ttest,lags=12)r3
@dfunit(ttest,lags=12)r10
*Thelong-runrelationshipcanbeestimatedusingtheT-billrateasthe‘dependent’variable.
linregtbill/resids;*Savetheresidualsasresids
#constantr3r10
*PerformtheEngle-Grangertestonresids
diffresids/dresids;*Obtainfirst-differenceoftheresiduals
linregdresids;*PerformtheDickey-Fullertestoftheresiduals
#resids1{1}dresids1{1to9}
*Repeatusingthe10-yearrateasthe‘dependent’variable
linregr10/resids10
#constantr3tbill
difresids10/dresids10
lindresids10;#resids10{1}dresids10{1to4};*Notethelaglengthof4
*Notethatsomewoulduse12lags
lindresids10;#resids10{1}dresids10{1to12}
*Toestimatetheerror-correctionmodelyouneedtodifferencethevariables
difftbill/dtbill;diffr3/dr3;diffr10/dr10
*BeginningwithRATS5.0,youcanestimatetheerror-correctionmodelasasystemofequations.NotethattheresisualsfrompartB(i.e.,resids)areusedastheerror-correctionterms
system1to3
variablesdtbilldr3dr10
lags1to12
detresids{1}
end(system)
estimate(noftests,outsigma=v)/1
*ThemultivariateAICandSBCarecalculatedusing
computeaic=%nobs*%logdet+2*(38*3)
computesbc=%nobs*%logdet+38*3*log(%nobs)
display'aic='aic'sbc='sbc
*No
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