chapter 6时间序列Word格式.docx

chapter 6时间序列Word格式.docx

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C.SupposethatytandztareCI（1,1）.Usetheconditionsin（6.19）,（6.20）,and（6.21）towritetheerror-correctingmodel.Compareyouranswerto（6.22）and（6.23）.Showthattheerror-correctionmodelcontainsaninterceptterm.

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）.

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.

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}

*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}

*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

\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

#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

\int_rates.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;

linregdresids;

#resids1{1}dresids1{1to9}

*Repeatusingthe10-yearrateasthe‘dependent’variable

linregr10/resids10

#constantr3tbill

difresids10/dresids10

lindresids10;

#resids10{1}dresids10{1to4};

*Notethelaglengthof4

*Notethatsomewoulduse12lags

#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