to interest rate data
DietmarBauer 1
and Martin Wagner 2
1
Department of ElectricalEngineering
Linkoping University, SE-58183 Linkoping,Sweden
WWW: http://www.control.isy.l iu.s e
Email: Dietmar.Bauer@tuwien.ac.at
2
Department of Economics
University of Berne, CH-3012Berne
Email: mwagner@vwi.unibe.ch
June 6,2000
REG
LERTEKNIK
AUTO
MATIC CONTR
OL
LINKÖPING
Report no.: LiTH-ISY-R-2264
Submitted to the 15 th
International Workshop onStatisticalModelling,Bilbao,
2000
TechnicalreportsfromtheAutomaticControlgroupinLinkopingareavailablebyanonymousftp
to interest rate data
DietmarBauer
Inst. forEconometrics,
OperationsResearchandSystemTheory
TUWien
Argentinierstr. 8,A-1040Wien
Austria e-mail: Dietmar.Bauer@tuwien.ac.at Martin Wagner DepartmentofEconomics UniversityofBerne Gesellschaftsstrasse49, CH-3012Berne Switzerland e-mail: mwagner@vwi.unibe.ch June 6,2000 Abstract
In this paper the application of so called subspace methods for the speci cation and
estimation of cointegrated systemsis examined. This method,which isbased onthe state
space representation,issuited fortheanalysis ofgeneralcointegratedsystemsoforderone,
i.e. is not limited to autoregressive models, as is e.g. Johansen's method. To assess the
empiricalusefulnessofthemethodweapplyittoperformacointegration analysisoftheUS
termstructureofinterestrates.
Keywords: Cointegration analysis;Subspacealgorithms;Termstructure.
1 Introduction
Overthepast15yearscointegrationanalysishasbecomeoneofthemostpopular eldsinmodern
econometrics. By now avariety ofmethods isavailable,but themajorityof analyses iscarried
outusingthemethodsdevelopedbyJohansenand hisco-authors. Thismethod howeverhasone
limitation: ItisrestrictedtotheanalysisofVARmodels. Althoughthisassumptionmaybeagood
approximationin manycases, thepossibility ofamoregeneraldata generatingprocessdeserves
someattention. Thiscane.g. bedoneusingsubspacemethods,inparticularthemethodpresented
inLarimore(1983),which isdealtwithhere,combinedwithacointegrationanalysisashasbeen
examined in Bauer and Wagner (1999a). This method is suited for estimation of cointegrated
ARMAmodels. Therearealreadyacoupleofrelatedresultsavailableintheliterature. E.g. Yap
andReinsel(1995)derivetheMLestimateforcointegratedGaussianARMAsystemsintegratedof
orderoneandgivethedistributionoftheestimates,whichallowstotesthypothesis. Themethod
presentedinthispaperisbasedonthestatespacerepresentationandderivesconsistentestimates
ofthe cointegratingspace aswellasthetransferfunction. From thelatteronecaneasilyderive
e.g. anARMArepresentationifthisisthepreferredsystemrepresentation. Alsotestprocedures
for the number of common trends are derived. We apply the method to test the expectations
hypothesis ofthetermstructureonUS data,whichstatesthattheyieldto maturityattimetof
akperiodpurediscountbondr
t;t+k
isrelatedtotheyieldonabondwithoneperiodtomaturity
r t;t+1 viaequation(1): r t;t+k = 1 k k 1 X j=0 E t (r t+j;t+j+1 )+L(t;k) (1)
withL(t;k)beingtheriskpremiumandE
t
denotingtheconditionalexpectationgiventhe
t;t+k t;t+1
stationary. As equation(1) holdsforall k, inasystemwith ninterestrates n 1cointegrating
relationsoccur,thusonlyonecommonfactorr
t;t+1
isdrivingthesystem. 1
Inapplications often
lessthann 1cointegratingvectorsarefound. Thepossiblereasonsforthisincludesizedistortions
duetomultipletests,problemswithhighdimensional systemswithmanycointegratingrelations,
orsimplyanundermodelingduetotheuseofatoosimplemodellikee.g. aloworderVARmodel.
Themethodpresentedheredoesnotsu erfromthetwoabovementionedpossibleproblems,thus
itmayconstituteavaluableadditionalorcomplementarytool.
Thestructureof thepaperisas follows: Section 2startswith averybriefdescriptionof the
method, Section3thenappliesthemethodtotheUS interestratedataandSection4concludes.
2 Description of the method
Inthissectionwebrie ydescribethesubspacealgorithmcointegrationanalysispresentedinBauer
andWagner(1999a). Thestartingpointofourmethodisthestatespacerepresentationof
nite-dimensional,timeinvariant,discretetimesystems
x t+1 = Ax t +K" t ; y t = Cx t +E" t (2) wherey t
;t=0;1;:::;T denotes thes-dimensional observedseries. "
t
denotesanergodic,strictly
stationarywhitenoisesequencewithzeromean,nonsingularinnovationvarianceand nitefourth
moments. For detailedassumptions in amartingaledi erence framework seeBauerand Wagner
(1999a). Furthermore we restrictourselvesto systemsthatare strictly minimum-phase,i.e. the
eigenvalues of (A KE 1
C) have an absolute value smaller than one. The system poles, i.e.
theeigenvaluesof A, are restrictedto beinside the openunit discorat z =1. 2
Thegeometric
multiplicities of theeigenvaluesat z =1are restrictedto beequalto one, this assumption
cor-respondsto anorderofintegration ofone. Theresultsof BauerandWagner(1999b)implythat
y t =C 1 K 1 P t 1 j=1 " t +k st (L)" t ,where k st (L)=E+LC st (I LA st ) 1 K st
is astableandstrictly
minimum-phase transferfunction, L denoting the backward shift operator. In order to achieve
identi ability(fordetailsseeBauerandWagner,1999b)C
1
ischosentobepartofanorthonormal
matrix, i.e. C 1 2R sr ;C 0 1 C 1 =I r
. Therefore there exists a matrixC
2 with C 0 2 C 2 =I s r and C 0 2 C 1 = 0, i.e. C 2
is in the orthogonal complement of C
1
. This representation coincides with
Granger's. The rstcomponentcorrespondsto thecommontrendsand thecolumns of C
2 span
the cointegrating space. Therefore the cointegrating rank is equal to s r and the numberof
commontrendsequalsthenumberofeigenvaluesofA atone. 3
Thebasis of thealgorithm is found in the interpretation of the statevector: For given
pos-itiveintegersf and pde ne Y + t;f =[y 0 t ;y 0 t+1 ;:::;y 0 t+f 1 ] 0 and Y t;p = [y 0 t 1 ;y 0 t 2 ;:::;y 0 t p ] 0 . F ur-ther let E + t;f = [" 0 t ;" 0 t+1 ;:::;" 0 t+f 1 ] 0 . Let O f = [C 0 ;A 0 C 0 ;:::;(A f 1 ) 0 C 0 ] 0 and K p = [K ;(A KE 1 C)K ;:::;(A KE 1 C) p 1 K]. Finally de ne E f
as the matrix, whose i-th block rowis
equalto thematrix[CA i 1
K ;;CK ;E;0]. Thenitfollowsfromthesystemequations(2), that
Y + t;f =O f K p Y t;p +O f (A KE 1 C) p x t p +E f E + t;f
Here for notational simplicity y
t
= 0;t < 0;x
t
= 0;t 0. Nowthe subspace algorithm canbe
describedasfollows: 1) Ina rststepregressY + t;f onY t;p toobtainanestimate ^ f;p ofO f K p . 2) Typically ^ f;p
hasfullrank,whereasO
f K
p
isofranknforf;pn. Thusapproximate ^
f;p
byaranknmatrixwithdecomposition ^ O f ^ K p . 1
Itcanfurthermorebeshownthatanysetoflinearlyindependentspreadsisformingabasisforthecointegrating
space.
2
Thisassumptionexcludesunitrootsatotherpointsthanz=1.
3
Also incase of higherorders of integration, the structure of the eigenvaluesat 1 (i.e. their algebraic and
0 . 0 0
0 . 0 4
0 . 0 8
0 . 1 2
0 . 1 6
5 5
6 0
6 5
7 0
7 5
8 0
8 5
9 0
9 5
Y R 1
0 . 0 0
0 . 0 4
0 . 0 8
0 . 1 2
0 . 1 6
5 5
6 0
6 5
7 0
7 5
8 0
8 5
9 0
9 5
Y R 2
0 . 0 0
0 . 0 4
0 . 0 8
0 . 1 2
0 . 1 6
5 5
6 0
6 5
7 0
7 5
8 0
8 5
9 0
9 5
Y R 3
0 . 0 0
0 . 0 4
0 . 0 8
0 . 1 2
0 . 1 6
5 5
6 0
6 5
7 0
7 5
8 0
8 5
9 0
9 5
Y R 4
0 . 0 0
0 . 0 2
0 . 0 4
0 . 0 6
0 . 0 8
0 . 1 0
0 . 1 2
0 . 1 4
0 . 1 6
5 5
6 0
6 5
7 0
7 5
8 0
8 5
9 0
9 5
Y R 5
1952
1957
1962
1967
1972
1977
1982
1987
1992
-0.027
-0.018
-0.009
0.000
0.009
0.018
0.027
Figure1: Leftpartofthis gure: interestratesforthedi erentmaturities,fromoneto veyears.
Rightpart: thefourspreadsr
t;t+i r
t;t+1 .
3) Usethe estimate ^
K
p
to estimatethestateasx^
t = ^ K p Y t;p
. Giventheestimateof thestate,
thesystemmatrices(A;K ;C ;E)canbeestimatedbyOLSusingthesystemequations. 4
InBauerandWagner(1999a)consistencyfortheestimatesofthecointegratingspace,thesystem
order and the transfer function estimates is derived. Also a method for the estimation of the
dimension of the cointegrating space is developed, which is basedon estimated singular values:
Theapproximationinstep2oftheprocedureoutlinedaboveisperformedusingthesingularvalue
decompositionof( ^ + f ) 1=2 ^ f;p ( ^ p ) 1=2 = ^ U n ^ n ^ V 0 n + ^ R ,where ^ + f
denotesthesamplecovarianceof
Y + t;f and ^ p
thesamplecovarianceofY
t;p . Here
^
U
n
isthematrix,whichcontainsthe rstnright
singularvectorsascolumnsand ^
n
isadiagonalmatrix,whosediagonalentriesaretheestimated
singular values in decreasingorder. ^
R accounts for the neglected singular values. In the case,
where thereare rcommon trends,the rst rsingularvaluesof thelimitof( ^ + f ) 1=2 ^ f;p ( ^ p ) 1=2
areequaltoone. InBauerandWagner(1999a)theasymptoticdistribution ofthesingularvalues
isderivedandatestprocedurebasedontheasymptoticdistribution issuggested.
3 An application to interest rate data
The interest rate data we use are the yields on 1 to 5 year US government bonds, they are
displayedintheleftpartofFigure1. Thereturnsarecomputedfrom thebondpricesunderlying
theanalysis ofFamaand Bliss(1987). The datarange from June 1952to December1994 with
monthlyfrequency.
Unitroottestsperformedforalltheseriesleadtotheconclusionthatallofthemareintegrated
of order one. Furthermore univariate unit root testing also leadsto the conclusion that all the
spreadsr
t;t+i r
t;t+1
fori=2;:::;5arestationary. Thesespreadsaredisplayedintherightpicture
inFigure1andareseento quiteresemblestationary timeseries. Hence univariateinvestigations
lendstrongsupportto theexpectationshypothesisofthetermstructure. Clearresultslikethose
just mentionedare only obtained for theUS market, e.g. for German data the evidence is not
that supportivefor theexpectationshypothesis. Theresultsobtainedby applying thesubspace
procedurescon rmthetheoryandpreliminaryinvestigations. TheorderestimateaccordingtoAIC
isequalto5,thereforef =p=10arechosen. InTable1wepresentthe rst3estimatedsingular
4 Firstregressy t onx^ t toobtainan estimate ^ C T andresiduals"~ t . Then ^ = 1 T P T t=1 ~ " t ~ " 0 t isanestimatefor
theinnovationvariance. Thus ^
E
T
canbecalculatedasthelowertriangularCholeskyfactorof ^ and"t^ = ^ E 1 T ~ "t.
Finallyregressxt+1^ onxt^ and"t^ toobtainestimates ^ A T and ^ K T respectively.
i ^ i T(1 r j=1 ^ 2 j =r) c.v.(true) c.v.(N) c.v. (asymp.) 1 0.999 0.741 10.713 11.326 59.509 2 0.954 23.385 20.263 20.189 36.727 3 0.766 85.999 { { {
Table1: The rst3 estimated singularvalues and thestatistic described in Bauer and Wagner
(1999a). Third column: valueofthestatistic. Fourthcolumn: bootstrappedcriticalvalue,using
thedistributionoftheresiduals. Fifthcolumn: usesnormalinnovationshavingthesamecovariance
matrix. Lastcolumn: asymptoticvalues. Theunderlyingtest isone-sided.
Componentnumber 2 3 4 5 Hausdor
lowerbound(true) 0.908 0.731 0.624 0.450 0
upperbound(true) 1.170 1.692 1.786 2.274 0.147
lowerbound (normal) 0.971 0.953 0.940 0.929 0
upperbound(normal) 1.130 1.319 1.331 1.370 0.082
estimatedvalues 1.0375 1.0648 1.0857 1.0974 0.0332
Table 2: Bootstrapped con dence regions for the entries of the column of C corresponding to
thecommontrend. Heretruestandsfortheprocedure,whichre-samplestheestimatedresiduals,
whereasnormalusesnormalinnovationswiththesamecovariancematrix.
values^
i
,theteststatisticT(1 P r j=1 2 j
=r),criticalvaluesobtainedby2di erentbootstrapping
proceduresandtheasymptoticcriticalvalues,whicharegeneratedusingtheestimatedparameters.
Thecritical valuesforthebootstrapshavebeengeneratedby 1000replications oftheestimated
modelunderthenullhypothesis,wherethe rstprocedurere-samplestheestimatedresidualsand
thesecondusesGaussianresidualswith thesamecovariancematrix.
Theresultsareasexpected: Onecommontrendisfound. Thegap betweenthesecondandthird
estimatedsingularvalueleadsto theorderestimaten=2. Notethatthecriticalvaluesobtained
fromtheasymptotictheory,aredi eringsubstantiallyfromthebootstrappedcriticalvalues. This
isevidenceinfavorofusingthelatter. Alsonotethatthetestsforthenumberofcommontrends
areonlycomputeduptoorder 2,sincethestatedimensionisanupperbound forthenumberof
common trends. Also the estimated eigenvalues,which arez =0:9121 and z =0:9992, con rm
thehypothesisofonlyonecommontrend.
Finally also the hypothesis that the spreads are forming abasis for the cointegrating space
canbe tested. This hypothesis is equivalent to the hypothesis that the common trend is C
1 =
[1;1;1;1;1] 0
. Again bootstrapping methods can be used to generate con dence intervals, see
Table2. WetestthishypothesisusingtheHausdor distance. Thetestisthereforeone-sidedand
the hypothesis is rejected ifthe Hausdor distance between theestimated and thehypothetical
space is larger than the critical value. Also for the individual components of the vector, after
normalizingthe rst component to 1, con dence intervalsaround 1 canbe generated. Noneof
the nullhypothesescan berejected, and again the di erent distributions for the bootstrapping
procedureleadtoverysimilarresults.
ForcomparisonwehavealsoappliedtheJohansen procedureto thisdataset. Forall
speci -cationstheconclusionisalwaysa4-dimensionalcointegratingspace. Howeverthehypothesisthat
thespreadsspanthatspaceisrejectedthroughout,althoughtestsfortheindividualspreadstobe
containedinthecointegratingspaceleadtoanacceptanceofthesehypotheses.
Finallynote that the state space systemis preferred to theautoregressivesystemusing the
AICcriterion. Thebestautoregressivemodelleadstoavalueof 64:3731,whereasthestatespace
model withtwostatesresultsin 64:5548,whichis anupperbound ofthe AICvalue, sincethe
In this paperwedealt with the application of socalled subspacemethods to the estimation of
cointegrated systems. The methods havebeen applied to the Fama-Bliss data set. The results
oftheanalysis areacon rmationof theexpectationshypothesisof thetermstructure. Alsothe
structureofthecointegratingspacehasbeeninvestigated,showingthatthespreadsbetweenthe
interestrates seemto be stationary, in accordancewiththe univariatestatistics. Note, however
that the assumptions in the multivariate setting of course are di erent, as we also model the
interdependenciesbetweenthevariousinterestrates. Theapplicationdemonstrates,thatthestate
spacemodelleadstogoodmodelsasmeasuredbytheAIC.Furthermorethetestingofhypotheses
onthecommontrendsisalsoeasilyincorporatedin thisframework. It hasto benotedhowever,
thatsimilarresultshavenotbeenachievedforotherdatasetsandthattheproceduresseemupto
nowlackaprofoundtheoreticaljusti cationwhenexogenousinputsandnonzeromeansandtime
trendsarepresent. Alsotheaccuracy forsmalldata setsseemsto bepoorin somecases,ashas
beennotedwhenanalyzingGermaninterestratedata.
Acknowledgments
The rstauthorwouldliketoacknowledgethe nancialaidof theEUTMR project'SI'in form
ofapost-docpositionattheUniversityofLinkoping,Sweden.
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