时间序列分析及VAR模型Word下载.docx

时间序列分析及VAR模型Word下载.docx

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tneedtopredefinewhetherxoryareendogenous（thedependentvariable）.Infact,wecantestwhetherx（y）isendogenousorexogenoususingGrangercausalitytests•TheideaofGrangercausalityisthatpastobservations（laggeddependentvariables）caninfluencecuiTentobservations一butnotviceversa・Sotheideaisrathersimple:

thepastaffectsthepresent,andthepresentdoesnotaffectthepast.STATAprovidesGrangercausalitytestsafterconductingaVARanalysis,whichisbasedontestingthejointhypothesisthatpastrealisationsdonotGrangercausethepresentrealisationofthedependentvariable.

Inmanyapplications,VARmodelsmakealotofsense,asacleardirectionofcausalitycannotbepredefined・Forinstance,thereisasubstantialliteratureonthebenefitsofinternationalisation（e.g.enteringforeignmarketthroughcross-borderM&

A）.Thereisevidencethatmultinationalsoutperformlocalpeersduetothebenefitsofoperatinginmanycountries・Atthesametime,weknowthathigh-performingcompaniesaremorelikelytoenterforeignmarketsduetotheirownershipspecificadvantages・ThisargumentisbasedontheResource-basedViewandtheOLSframeworkdevelopedbyDunningandRugman（ReadingSchoolofInternationalBusiness）.TheVARmodelallowsyoutoincorporatebotheffects:

infactyoucantestwhetherperformancedrivesinternationalisationorinternationalisationdrivesperformance.

BeforeyoustartusingaVARmodeLyouhavetomakesurethatthetimeseriesarestationary.SothefirststepistocheckwhetherthetimeseriesissStionjryusingDickey-FullertestsandKPSStests.Thesecondstepistospecifytheoptimallaglength（p）ofthemodel.Thisisdonebycomparingdifferentmodelspecificationsusinginformationcriteria.ApartfromusingAkaike（AIC）andBayesianSchwarz（BIC）,theHannan-Quinn（HQIC）iscommonlyused.MostappliedeconometriciansfavourtheHannan-Quinn（HQIC）criterion・STATAwillhelpyoutomakeagoodchoice*Afterspecifyingyourmodel,youneedtocheckstabilityconditions.ThecoefficientmatrixofthereducedformVARhastoensurethattheiterationsequenceconvergestoalong-termvalue・STATAwillhelpyouincheckingstability.

Tobeprecise,youneedtoshowthattheeigenvaluesofthecoefficientmatrixliewithintheunitcircle・Thereasonbehinditcanbeonlyunderstoodwhenyouunderstandthemethodofdiagonalizingamatrix.

VARmodelsofferano什nicefeature:

impulseresponsefunctions・VARmodelscapturethedynamicsoftwo（ormore）stationarytimeseries;

hence,wecanassessthedynamicimpactofamarginalchangeofonevariableonanother・ThestandardOLSregressionprovidescoefficients,andcoefficientsrefertothepartialimpactofanexplanatoryvariableonthedependentvariable・InthecaseofVARmodels,therelationshipbecomesdynamic,asachangeofonevariable

（sayx）intcanaffectxandyint+1•Theimpactonxandyint+linturnaffectsxandyint+2andsoonuntiltheimpactdiesout.Impulseresponsefunctionsareveryusefulinillustratingtheshort-termdynamicsinamodel.

LefslookatanexampletoseehowVARmodellingworks・InLecture5,wetriedveryhardtounderstandgoldprices・Weextendourunivariatemodelbyexploringtherelationshipsbetweengoldandsilverprices・Linkingtwo（similar）assetsorsecuritiesisaveiycommontradingstrategy,whichiscalledpairs-tradin^•

Beforewedoanysophisticatedmodelling,itisalwaysbeneficialtolookatsomelinecharts.Figure1showstheindexedtimeseriesofnominalgoldandsilverpricesfrom1900to2010.

Figure1:

Nominalgoldandsilverprices,indexedA1900-2010

Wecanseethatthereisacertaindegreeofco-movement,whichwemightbeabletoexploitforourtradingstrategy.BeforewecanuseVAR.weneedtoensurethatbothtimeseriesarestationary.ItisobviousfromFigure1thatgoldandsilverpricesarenotstationary.However,aftertakingafirst-differencewecanshowthatpricechangesarestationary・Sobothtimeseriesare1

（1）.

Thenextstepistodeterminetheoptimallaglengthusinginformationcriteria・Table1showsdifferentspecificationsusingthevarsoccommand・

Table1:

Determiningtheoptimallaglengthusinginformationcriteria

lag

LL

LR

df

P

FPE

AIC

HQIC

SBIC

104・118

・00049

■1.94511

•丄•92463

■JL-89456

1

113.551

JIR-866

4

0・00l

・000442

■2-0486

-1.98714

■JI・89694*

2

120.015

12.927

0.01?

•00042?

ifK

-2.09552*

■1.9931*

-1.R4276

3

120-ft19

1-6086

0-807

・00044R

■2.03465

■:

I・89126

-1.6R079

124-R26

8.0136

0・09l

・00044ft

-2.03478

-1.85042

-1.5798?

5

130.126

10.599*

0.031

・000438

-2.05954

-1.83421

-1.50347

Numberofobs

Endogenous:

return.greturn^sExogenous:

_cons

Selectioreordercriteria

Sample:

1906-2010

105

BasedontheAICandHQIC,twolagsareoptimal;

however,the（S）BICprefersonlyonelag.IwouldpreferHQICandtrytwolagsfirst.Ifthesecondlagdoesnotexhibitsignificantcoefficient,wecouldtrytoreducethelagleng什】inlinewith（S）BIC-

WerunaVARwithtwolagstoexplaincurrentpricechangesingoldandsilver-Table2providestheOLSestimates.

Table2:

VARmodelwithtwolags

1903-2010

IISI#I:

1^X%^1

dnoAdOO

・0004

Det（Sigma_ml）

二・OOOM323

NO-ofobsAIC

=108=・2・148455二・2・04776=•丄•90Oil

Parms

Fcmmlien

return.g5

RMSE

・1269270.2425

chi2P>

chi2

34.57860.0000

Vectorautoregression

return^s

5・1965690.130616.227630-0027

coef.

Std・Err.

P>

|Z|

[95%Conf.

Interval]

return^g

return.a

11-

・4ftR4in7

•l29QREA

2・QA

nnon

・7274SR1

L2-

-0139809

・122817

-0.11

0-909

-.2546979

・2267361

return.s

I:

I

-006R126

•6R05903

-0.0R

0.933

-164766R

・1511415

L2・

■・207786

・0807151

-2.57

0.010

-.3659847

■・0495874

_cons

・0277213

・0124857

2.22

O・026

・0032497

・0521929

return^sreturnAg

・如427弘

・1Q04R4ft

1AS

nOQQ

-0SRQ9S7

・AR7AR9A

・1085011

・1902038

0.57

0.568

-2642915

・4812937

・1094293

・124B0阳

0・RR

0.3R1

-.1351905

・354049

-3201805

・1250015

-2.56

-.5651789

■・0751821

・024511

.0193363

1.27

0-205

-.0133875

・0624095

Weseethatsilverprices（lag2）affectcurrentgoldprices,andwecanestablishautocorrelationinbothtimeseries*TotestwhethergoldGrangercausessilverorviceversa,werunGrangercausalitytestsreportedinTable3.

Table3:

Grangercausalitytests

GrangercausalityWaldtests

Equation

Excluded

chi2

dfProb>

chi2

return_g

returns

6.765

0.034

ALL

3-9615

0.138

Hence,weconfirmthatpastchangesinsilverpricescanpredictfuturegoldpricechanges.Thisisveryinteresting,asitcanbeusedtodevelopatradingstrategy.Finally,weneedtoshowthattheVARisstable（seeTable4）.

Table4:

StabilityconditionoftheVAR

Eigenvaluestabilitycondition

Eigenvalue

Modulus

・2367286+

・362415?

.43288

•2367286-

・3624157

・43288

・06119136+

・3747777?

•37974

・06119136-

•37477771

・37974

Alltheeigenvalueslieinsidetheunitcircle・VARsatisfiesstabilitycondition・

Finally,wecanillustratetheimpactofsilverpricechangesonfuturegoldpricechangesusinganimpulseresponsefunction*Figure2showstheimpulseresponsefunctionandconfidenceintervalsderivedfrombootstrapping.Ifsilverpricesincreasetodayby1%,weshouldexpectasignificantdeclineingoldpricesintwoyearsby0.2%•

Figure2:

Impulseresponsefunction

r,return_s,return_g2•

024

step

95%Climpulseresponsef