UNIVERSITEITANTWERPEN

EstimatingtheYieldCurveUsingtheNelsonSiegelModel

ARidge RegressionApproach

JanAnnaert

UniversiteitAntwerpen,Prinsstraat13,2000Antwerp,Belgium

AnoukG.P.Claes

LouvainSchoolofManagement,BoulevardduJardinBotanique43,1000Brussels,Belgium

MarcJ.K.DeCeuster

UniversiteitAntwerpen,Prinsstraat13,2000Antwerp,Belgium

HairuiZhang

UniversiteitAntwerpen,Prinsstraat13,2000Antwerp,Belgium

Abstract

TheNelsonSiegelmodeliswidelyusedinpracticeforfittingthetermstructureofinterestrates.Dueto

theease in linearizingthemodel,agridsearchoranOLSapproachusingafixedshapeparameterare

popularestimationprocedures.Theestimatedparameters,however,havebeenreported(1)tobehave

erraticallyovertime,and(2)tohaverelatively largevariances.WeshowthattheNelsonSiegelmodel

canbecomeheavilycollineardependingontheestimated/fixedshapeparameter.Asimpleprocedure

basedonridgeregressioncanremedythereportedproblemssignificantly.

Keywords:SmoothedBootstrap,RidgeRegression,NelsonSiegel,SpotRates

CorrespondingAuthorEmail:marc.deceuster@ua.ac.be

1

1.Introduction

Goodestimatesofthe termstructureof interestrates (alsoknownasthespotratecurveorthezero

bond yield curve) are of the utmost importance to investors and policy makers. One of the term

structureestimationmethods, initiatedbyBlissandFama(1987), isthesmoothedbootstrap.Blissand

Fama(1987)bootstrapdiscretespotratesfrommarketdataandthenfitasmoothandcontinuouscurve

to thedata.Althoughvariouscurve fittingsplinemethodshavebeen introduced (quadraticandcubic

splines(McCulloch(1971,1975)),exponentialsplines(VasicekandFong(1982)),Bsplines(Shea(1984)

and Steeley (1991)), quartic maximum smoothness splines (Adams and Van Deventer (1994)) and

penalty functionbased splines (Fisher,Nychka and Zervos (1994),Waggoner (1997)), thesemethods

havebeencriticizedontheonehandforhavingundesirableeconomicpropertiesandontheotherhand

forbeing blackboxmodels (Seber andWild (2003)).Nelson and Siegel (1987) and Svensson (1994,

1996) therefore suggested parametric curves that are flexible enough to describe awhole family of

observed term structure shapes. Thesemodels are parsimonious, they are consistentwith a factor

interpretationofthetermstructure(LittermanandScheinkman(1991))andtheyhavebothbeenwidely

used inacademiaand inpractice. Inaddition to the level, slopeandcurvature factorspresent in the

NelsonSiegelmodel, theSvenssonmodel containsa secondhump/trough factorwhichallows foran

evenbroaderandmorecomplicatedrangeoftermstructureshapes.Inthispaper,werestrictourselves

totheNelsonSiegelmodel.TheSvenssonmodelsharesbydefinitionallthereportedproblemsof

theNelsonSiegel approach. Since the source of the problems, i.e. collinearity, is the same for both

models,thereportedestimationproblemsoftheSvenssonmodelmaybereducedanalogously.

TheNelsonSiegelmodel is extensively used by central banks andmonetary policymakers (Bank of

InternationalSettlements(2005),EuropeanCentralBank(2008)).Fixedincomeportfoliomanagersuse

themodel to immunize theirportfolios (Barrett,Gosnell andHeuson (1995) andHodges andParekh

(2006))andrecently,theNelsonSiegelmodelalsoregainedpopularityinacademicresearch.Dullmann

andUhrigHomburg(2000)usetheNelsonSiegelmodeltodescribetheyieldcurvesofDeutscheMark

denominated bonds to calculate the risk structure of interest rates. Fabozzi,Martellini and Priaulet

(2005)andDieboldand Li (2006)benchmarkedNelsonSiegel forecastsagainstothermodels in term

structureforecasts,andtheyfound itperformswell,especiallyfor longerforecasthorizons.Martellini

andMeyfredi (2007) use theNelsonSiegel approach to calibrate the yield curves and estimate the

valueatriskforfixedincomeportfolios.Finally,theNelsonSiegelmodelestimatesarealsousedasan

inputforaffinetermstructuremodels.Coroneo,NyholmandVidavaKoleva(2008)testtowhichdegree

2

theNelsonSiegelmodelapproximatesanarbitragefreemodel.They firstestimate theNelsonSiegel

modelandthenusetheestimatestoconstruct interestratetermstructuresasan inputforarbitrage

freeaffinetermstructuremodels.TheyfindthattheparametersobtainedfromtheNelsonSiegelmodel

are not statistically different from those obtained from the pure noarbitrage affineterm structure

models.

Notwithstanding itseconomicappeal, theNelsonSiegelmodel ishighlynonlinearwhichcausesmany

users to reportestimationproblems.Nelson and Siegel (1987) transformed thenonlinearestimation

problem intoa simple linearproblem,by fixing the shapeparameter that causes thenonlinearity. In

ordertoobtainparameterestimates,theycomputedtheOLSestimatesofaseriesofmodelsconditional

uponagridofthefixedshapeparameter.Theestimatesthat,conditionaluponafixedshapeparameter,

maximizedtheRwerechosen.Werefertotheirprocedureasagridsearch.Othershavesuggestedto

estimate theNelsonSiegelparameterssimultaneouslyusingnonlinearoptimization techniques.Cairns

andPritchard(2001),however,showthattheestimatesoftheNelsonSiegelmodelareverysensitiveto

thestartingvaluesused in theoptimization.Moreover, timeseriesof theestimatedcoefficientshave

been documented to be very unstable (Barrett,Gosnell and Heuson (1995), Fabozzi,Martellini and

Priaulet(2005),DieboldandLi(2006),Gurkaynak,SackandWright(2006),dePooter(2007))andeven

to generate negative long term rates, thereby clearly violating any economic intuition. Finally, the

standarderrorsontheestimatedcoefficients,thoughseldomreported,arelarge.

Althoughtheseestimationproblemshavebeenrecognizedbefore,ithasneverleadtowardssatisfactory

solutions.Instead,itbecamecommonpracticetofixtheshapeparameteroverthewholetimeseriesof

termstructures.1Hurn,LindsayandPavlov (2005),however,pointout that theNelsonSiegelmodel is

verysensitivetothechoiceofthisshapeparameter.dePooter(2007)confirmsthisfindingandshows

thatwithdifferentfixedshapeparameters,theremainingparameterestimatescantakeextremevalues.

Hencefixingtheshapeparameterisanontrivialissue.Inthispaperweuseridgeregressiontoalleviate

theobservedproblemssubstantiallyandtoestimatetheshapeparameterfreely.

Theremainderofthispaperisorganizedasfollows.InSection2,weintroducetheNelsonSiegelmodel.

Section3presentstheestimationproceduresusedintheliterature,illustratesthemulticollinearityissue

which isconditionalontheestimated(orfixed)shapeparameterandproposesanadjustedprocedure

based on the ridge regression. In the subsequent section (Section 4)we present our data and their 1Barrettetal.(1995)andFabozzietal.(2005)fixthisshapeparameterto3forannualizedreturns.DieboldandLi(2006)chooseanannualizedfixedshapeparameterof1.37toensurestabilityofparameterestimation.

3

descriptivestatistics.Sincetheridgeregressionintroducesabiasinordertoavoidmulticollinearity,we

willmainlyevaluatethemeritsofthemodelsbasedontheirabilitytoforecasttheshortandlongendof

thetermstructure.Theestimationresultsandtherobustnessofourridgeregressionarediscussed in

Section5.Finally,weconclude.

2.AfirstlookattheNelsonSiegelmodel

IntheirmodelNelsonandSiegel(1987)specifytheforwardratecurvef()asfollows:

( )( )

0 0 0/

1 1 1/

2 2 2

1,

/

ff e f

e f

= =

(2.1)

whereistimetomaturity,0,1,2andarecoefficients,with>0.

Thismodelconsistsofthreepartsreflectingthreefactors:aconstant(f0),anexponentialdecayfunction

(f1) and a Laguerre function (f2). The constant represents the (longterm) interest rate level. The

exponentialdecayfunctionreflectsthesecondfactor,adownward(1>0)orupward(10)ora trough (2

Figure1d

(respectiv

rate(thes

Fig

PanelA:F

Note:ThisfspotratecuPanelB)rep

Theroleo

time tom

vanishan

0.Based

tobeneg

compone

the slope

slope. Th

compone

depictstheth

velyr0,r1and

spotrate)cur

gure1:Decom

FortheForwa

figureshowstheurve(PanelB)wpresenttheleve

ofthecompo

maturity.Wh

dthelongte

oneconomic

gativeorunre

ntvanishesa

eof the term

e degree of

nt decay to

( )

= r

hreebuilding

r2inPanelB)

rve.

mpositionoft

ardRateCurv

edecomposedcwhentheshapepl,slopeandcurv

onentsbecom

en the time

rmforwarda

cintuition,w

ealisticallyhi

andtheforwa

m structure,w

the curvatu

o zero. Final

((

0

1

2

1

1 e

blocksofthe

)representth

theNelsonS

ve

componentsoftparameterisfixevaturecompone

mesclearwhe

tomaturity

andspotrate

weassume0

gh.Whenth

ardandthesp

whereby ane

re is control

lly, the loca

4

))

/

/

1

1 /

/

e

e e

eNelsonSieg

helevel,slope

iegelModelw

Pane

theNelsonSiegeedat3.Thecurentsoftheforwa

enwelookat

grows to inf

willconverge

tobecloset

etimetoma

potrateconv

egative (posit

led by 2, th

ation of the

=

0

1

/ 2

gelmodel.Th

eandcurvatu

withtheSha

elB:fortheS

elmodelforthervesf0,f1andf2ardrate(thespo

theirlimiting

finity, the slo

etothesame

otheempiric

aturityappro

vergeto(0+

tive) 1 repre

he rate at w

e maximum/

0

1

2

.rrr

hecurvesf0,f

urecompone

peParamete

SpotRateCur

eforwardratecinPanelA(respotrate)curve.

gbehaviourw

ope and curv

econstantlev

callongterm

oacheszero,o

1).Thespre

esents anup

which the slo

minimum va

f1andf2inPa

entsofthefor

erFixedat3

rve

urve(PanelA)apectivelyr0,r1an

withrespectt

vature compo

velofinterest

mspotratean

only thecurv

ead,1,mea

pward (downw

ope and curv

alue of curv

(2.3)

anelA

rward

andthendr2in

tothe

onent

trate,

ndnot

vature

asures

ward)

vature

vature

compone

the hump

functionw

structure.

In Figure

paramete

maximum

determine

PanelA:i

Note:This fvarious shawhereasthaEquation2.3

Alternativ

tomaturi

facilitates

themaxim

maximizin

hump/tro

ntisdetermi

p/trough of

with respect

.

2 the curvat

ers ranging fr

mwhen=,

edbysimply

intheForwar

figureshows thepeparameters.atinthespotra3withfixed.

vely,wecanf

ty,byfixingt

sestimation.

mumofthec

ng the curva

ough become

nedby.Not

the term str

to ,we ar

ture compon

rom 1 to 10

,whereas th

maximizingr2

Figure

rdRateCurve

eshapesof theThe curvature

atecurve(Panel

forcetheloca

theshapepa

Severalautho

curvaturecom

ature compo

es approxima

tethatdete

ructure. By m

e able to de

nent in the f

0. The curvat

at in thespo

2inEquation

e2:Shapesof

e

ecurvaturecomcomponent in tB)reachesitsm

ationoftheh

rametertoa

orsusethisid

mponentinth

onent in th

tely 1.37wit

5

erminesboth

maximizing th

etermine the

forward rate

ture compon

otratecurve

2.3withfix

ftheCurvatu

Pane

ponent in the fthe forward ratmaximumwhen

hump/trough

specificvalu

dea.Intheire

hespotratef

e spot rate

th annualized

theshapeof

he curvature

location of

and the spo

ent in the f

reaches itsm

xed.

ureCompone

elB:intheSp

forward rateante curve (Panel>,whichisd

oftheterm

ue.Thisalsol

empiricalwor

functionata

e function, t

d data. In Fa

fthecurvatu

e component

the hump/tr

ot rate are d

orward rate

maximumwh

nt

potRateCurv

d thespotrateA) reaches itsmdeterminedbys

structureto

inearizesthe

rk,Dieboldan

amaturityof

the shape p

abozzi et al. (

recomponen

t in the spot

rough of the

depicted for s

curve reach

hen >,wh

ve

curvewithrespmaximumwhensimplymaximizin

beatagiven

emodelandh

ndLi(2006)e

2.5years.Th

parameter o

(2005),wher

ntand

t rate

term

shape

es its

hich is

pect ton = ,ngr2in

ntime

hence

e.g.fix

husby

f the

re the

6

shapeparameterwassettobe3,thehumpislocatedapproximatelyatamaturityof5.38onthespot

ratecurve.

3.Estimationprocedures

In order to obtain all the parameter estimates simultaneously, we could use nonlinear regression

techniques. Ferguson and Raymar (1998) and Cairns and Pritchard (2001), however, show that the

nonlinear estimators are extremely sensitive to the starting values used and that the probability of

gettinglocaloptimaishigh.Takingthesedrawbacksintoaccount,mostresearchershavefixedtheshape

parameterandhaveestimateda linearizedversionoftheNelsonSiegelmodel.Theparametersofthe

NelsonSiegelmodelhavetypicallybeenestimatedbyminimizingthesumofsquarederrors(SSE)using

(1)OLSoveragridofprespecifieds(NelsonandSiegel(1987)),and(2)alinearregression,conditional

onachosenfixedshapeparameter(DieboldandLi(2006),dePooter(2007),andFabozzietal.(2005)).

Werefertothesemethodsasthetraditionalmeasures.Theestimatedparametersusingthetraditional

methods, however, are reported (1) to behave erratically in time, and (2) to have relatively large

variances.We will first show that these problems result frommulticollinearity problems. Next, we

introducearidgeregressionapproachtoremedythereportedproblemsandhowtojudiciouslyfixthe

shapeparameterinordertoestimatethelinearizedNelsonSiegelmodel.

3.1.Thenatureofthemulticollinearityproblem

Researchershavebeen awareofpotentialmulticollinearity issueswhile estimating theNelsonSiegel

model.DieboldandLi(2006)e.g.indicatethatthehighcorrelationbetweenthefactorsoftheNelson

Siegel(1987)modelmakes itdifficulttoestimatetheparameterscorrectly.Whatseemstohavegone

unnoticed,however,isthefactthatthecorrelationbetweenthetworegressorsofthemodeldepends

on(thetimestomaturityof)thefinancialinstrumentschoseninthebootstrap.Inordertoillustratethis

point,weconsiderfourdifferentsetsoftimestomaturity:

1. 3and6months,1,2,3,4,5,7,10,15,20and30years;

2. 3,6,9,12,15,18,21,24,30months,310years;

3. 1week,112month,and210years;

4. 1week,6months,and110years.

7

ThefirstsetofmaturitieswasusedbyFabozzietal.(2005).DieboldandLi(2006)optedforthesecond

maturity vector. Since researchersare inclined touseall thedata they can find,we study twoextra

vectorsthatalsoincludeadditionalshortertimetomaturities.

Table1summarizesthecorrelationsbetweentheregressorsforthevalueschosenbyDieboldandLi

(2006)andFabozzietal.(2005),usingthefourvectorsoftimetomaturitythatweconsider.Thetable

showsthatthecorrelationbetweentheslopeandthecurvaturecomponentoftheNelsonSiegelmodel

heavily depends on the choice of the shape parameter. The second maturity vector e.g. implies

correlationsvaryingbetween5%and87%dependingontheshapeparameterchosen.Thecorrelation

alsodependsseverelyonthechoiceofthetimetomaturityvector.Using=1.37,thecorrelationvaries

from0.549to0.256,forthematurityvectorschosen.Setting=3producescorrelationsfrom0.324to

0.931.Thevectorcontainingtheseriesofshortmaturities(thethirdvector)turnsouttobethemost

sensitivetothecollinearityissue.

Table1:CorrelationbetweenRegressorsUsingAlternativeTimetoMaturityVectors

MaturityVector DieboldandLi(=1.37) Fabozzietal.(=3)1 0.256 0.3242 0.051 0.8713 0.549 0.9314 0.352 0.872

Note:ThecorrelationsbetweentheregressorsintheNelsonSiegelspotratefunctionconditionedontheshapeparameterandthevectoroftimestomaturityarepresented.Vector1usesthefollowingtimestomaturity:3and6months,1,2,4,5,7,10,15,20and30years(asinFabozzietal.(2005));vector2uses3,6,9,12,15,18,21,24,30monthsand310years(asinDieboldandLi(2006));vector3isbasedon1week,112monthsand210years;andvector4on1week,6monthsand110years.

Figure3givesamorecompletepicturebyplottingthecorrelationbetweenthetworegressorsovera

rangeof valuesusing the four time tomaturity vectors studied.Wenotice that the choiceof the

maturityvector influencesthesteepnessofthecorrelationcurve. ItappearsthatFabozzietal.(2005)

andDieboldandLi(2006)chosetheirvaluesjudiciouslyconditionalontheirmaturityvector,although

they both motivate their choice differently. It is clear that for empirical work it is of the utmost

importanceforalloftheestimationmethodstotakethispotentialmulticollinearityissueintoaccount.

Note:Thisfratecurveo6months,1and310ye6monthsan

3.2Tradit

3.2.1Grid

Toavoidn

estimate

values ra

paramete

the time

(1995),Ca

anddePo

ofthisins

3.2.2OLS

Someres

curvature

Figur

figureplotstheovertheshapep1,2,4,5,7,10,ears(asinDiebond110years.

tionalestima

dsearchbase

nonlinearest

Equation2.3

nging from0

erset.Inprac

seriesofest

airnsetal.(2

ooter(2007).

stability.More

withfixedsh

earchersfixt

eofthespotr

re3:Correlat

correlationsbeparameterusin15,20and30yldandLi(2006)

tionmethod

edOLS

imationproc

3withordina

0.027 to1. T

ctice,itiswe

timates. This

001),Fabozzi

Farlessobse

eover,highm

hapeparame

theshapepa

rates.

tionsbetwee

tweentheslopengfourtimetomyears(asinFabo);vector3isbas

s

edures,Nelso

ry leastsqua

heestimates

llknownthat

hasbeenpo

ietal.(2005)

ervedisthat

multicollineari

eters

rameterwhic

8

ntheSlopea

eandcurvaturematurityvectorsozzietal.(2005)sedon1week,

onandSiegel

ares.Thispro

swith thehig

tgridsearch

ointedoutby

),Dieboldeta

multicollinea

itycanalsoin

chtheytypica

andCurvature

ecomponentofs.Vector1uses));vector2uses112monthsan

l(1987)linea

ocedurewas

ghestRwer

basedOLSle

ymany resea

al.(2006and

rityamongth

nflatethevari

allymotivate

eComponen

theNelsonSiegthefollowingtims3,6,9,12,15,nd210years;an

arizetheirmo

repeated for

re then chos

eadstoparam

archers inclu

d2008),Gurka

hetworegre

ianceofthee

ebypriorkno

t

gelmodelforthmestomaturity18,21,24,30mndvector4on1

odelbyfixing

rawholegrid

sen as theop

meterinstabi

dingBarrett

aynaketal.(

ssorsisthes

estimators.

owledgeabou

hespoty:3andmonths1week,

and

dof

ptimal

lityin

et al.

2006)

ource

utthe

9

DieboldandLi (2006)set to16.4withmonthlycompounded returns,orapproximately1.37with annualized data. Their choice implies that the curvature component in the spot rate

functionwillhave itsmaximalvalueatthetimetomaturityof2.5years.Theymotivatedtheir

choicebystatingthatmostofthehumps/troughsarebetweenthesecondandthethirdyear.As

we have shown, this choice, also turned out to avoid multicollinearity problems for their

maturityvector.

Fabozzietal.(2005)fixedto3withannualizeddata,asarguedbyBarrettetal.(1995).Barrettetal. (1995)performedagrid searchby fixing for thewholedatasetandobtainedaglobal

optimalshapeparameterinonego.Inafootnote,Fabozzietal.(2005)mentionthatwhenthe

shapeparameterisfixedat3,thecorrelationbetweenthetworegressorswillnotcausesevere

problems for their data. They, however, do not offer a universal procedure to tackle the

multicollinearityissue.

Fromtheprevioussection,weknowthatthedegreeofmulticollinearitydependsonthechoiceofthe

fixed shapeparameterand the choiceof the vectorof times tomaturity.The shapeparameter that

minimizesthesquarederrorsmayalsovaryovertime.Figure4plotsthespotratecurveontwodays(i.e.

April18,2001andJanuary4,1999)wherefixingtheoptimalshapeparametertoeither1.37or3visually

underperformscomparedtothegridsearch.Thevaluesestimatedusingthegridsearcharereported

inthefigure.Onthelefthandside,theshapeparameterisestimatedtobe1.35.Theuseofafixedof

1.37willthereforeresultinabetterfitthanthemodelwithashapeparameterfixedat3.Ontheright

handside,however,thegridsearchestimateofis2.9.Themodelusingafixedof3willthusfitthe

databetterthanthatusingashapeparameterof1.37.

PanelA(

Note:Thisf(2.9)whichparametero

3.2.3Grid

Whereas

optimal e

Nelsonan

paramete

ridge reg

therefore

discussed

implemen

3.2.3.1M

In order t

collinearit

number.

TheVIFis

Fig

Date:April18

figuregivesaneminimizesthe fof1.37(3)result

dsearchwith

linear regres

estimators, th

ndSiegelsap

er.Conditiona

gression whe

need to tes

below. Su

ntationofthe

Measuringthe

to address th

tymeasures

definedas

ure4:AnExa

8,2001)

exampleofthelfittingerrors intsinabetterfit

conditional

ssionsdonot

heydo suffer

pproachbyco

alon the th

never the d

st the degre

bsequently,

eridgeregres

degreeofm

hemulticollin

include thev

ampleofthe

imitationoffixePanelA (Panelthanthemodel

ridgeregress

t require star

r from instab

ombiningthe

hat results in

egree of mu

ee of multico

we discuss

sionfortheN

ulticollineari

nearity issue,

variation infla

10

Limitationof

Pane

edshapeparameB).Asaconseqwithafixedsha

sion

rtingvalues f

bility inparam

egridsearch

n thehighest

ulticollinearit

ollinearity of

the nature

NelsonSiegel

ty

,we need to

ation factor (

fFixedShape

elB(Date:Ja

eter.Thegridsequence,theNelapedparameter

for theestim

meter estima

withtheOLS

R, thepara

ty among th

the two fac

e of ridge

termstructu

o verify the d

(VIF), the tole

eParameter

nuary4,199

earchgivesashsonSiegelmodof3(1.37).

matorsandal

ation. In this

Sregressiont

ametersare r

e regressors

ctors. The m

regression

ureestimation

degree of co

erance level

9)

apeparameterelwitha fixeds

waysgiveglo

paper,we f

tofreethes

reestimated

s is too high

measure we u

and present

n.

ollinearity. Po

and thecon

of1.35shaped

obally

follow

shape

using

h. We

use is

t the

opular

dition

11

21 ,

1 iR

whereRi isthecoefficientofmultipledeterminationofthe independentvariableXionallotherXs in

themodel.TheintuitionofVIFforasimpleregressionisstraightforward:ifthecorrelationbetweenthe

regressorsishigh,thentheVIFwillbelarge.ThetolerancelevelisthereciprocalofVIF.

Belsley(1991)pointsoutthemaindrawbacksofthesetwomeasures:(1)highVIFsaresufficientbutnot

necessary to collinearity problem, and (2) it is impossible to determinewhich regressors are nearly

dependent on each other by usingVIFs. The thirdmeasure, the condition number, is based on the

eigenvalues of the regressors. Since the condition number avoids the shortcomings of the

aforementionedmethodologies(seeBelsley(1991)andDeMaris(2004)),weusetheconditionnumber

asthecollinearitymeasure.Assumethereisastandardizedlinearsystemy=X+.Denote(kappa)as

theconditionnumberandtheeigenvaluesofXX.TheconditionnumberofXisdefinedas

( ) = maxmin

1.vv

X (3.1)

IfXiswellconditioned(i.e.theregressorcolumnsareuncorrelated),thentheconditionnumberisone,

which impliesthatthevariance isexplainedequallybyalltheregressors.Ifcorrelationexists,thenthe

eigenvaluesarenolongerequalto1.Thedifferencebetweenthemaximumandminimumeigenvalues

willgrowasthecollinearityeffectincreases.AssuggestedbyBelsley(1991),weuseaconditionnumber

of10asameasureofthedegreeofmulticollinearity.2

3.2.3.2Remedyofhighcollinearity

Oncecollinearity isdetected,weneedtoremedytheproblem.ToovercomeOLSparameter instability

duetomulticollinearity,we implementanalternativetothe linearregression, i.e.theridgeregression

technique.Thisestimationprocedurecansubstantiallyreducethesamplingvarianceoftheestimator,

by adding a small bias to the estimator. Kutner,Nachtsheim,Neter and Li (2004) show that biased

estimatorswithasmallvariancearepreferabletotheunbiasedestimatorswithlargevariance,because

2As thereareonly two regressors in theNelsonSiegelmodel,wecanplotaonetoone relationshipbetween theconditionnumber and the correlation between the two regressors. In our dataset, a condition number above 10 is equivalent to acorrelationwithanabsolutevalueabove0.8.

12

the small variance estimators are less sensitive tomeasurement errors.We therefore use the ridge

regressionandcomputeourestimatesasfollows:

[ ] = + 1* ,k X X I X y (3.2)wherekiscalledtheridgeconstant,whichisasmallpositiveconstant.Astheridgeconstantincreases,

thebiasgrowsand theestimatorvariancedecreases,alongwiththeconditionnumber.Clearly,when

k=0theridgeregressionisasimpleOLSregression.

3.2.3.3Implementation

AspointedoutbyKutneretal.(2004),collinearity increasesthevarianceoftheestimatorsandmakes

the estimated parameters unstable. However, even under high collinearity, the OLS regression still

generatestheunbiasedestimates.Asaresult,weimplementacombinationofthegridsearchandthe

ridgeregressionusingthefollowingsteps:

1. PerformagridsearchbasedontheOLSregressiontoobtaintheestimateofwhichgenerates

thelowestmeansquarederror.

2. Calculatetheconditionnumberfortheoptimal.

3. Reestimatethecoefficientsbyusingridgeregressiononlywhentheconditionnumberisabove

aspecificthreshold(e.g.10).Thesizeoftheridgeconstantischosenusinganiterativesearching

procedure3that finds the lowestpositivenumber, k,whichmakes the recomputed condition

numberfallbelowthethreshold.Byaddingasmallbias,thecorrelationbetweentheregressors

willdecreaseandsowilltheconditionnumber.

4.Dataandmethodology

To illustrateourNelsonSiegel termstructure fittingprocedure,weuseEuriborratesmaturing from1

weekup to12monthsandEuroswap rateswithmaturitiesbetween2yearsand10years.TheEuro

OvernightIndexAverage(EONIA),andthe20,25and30yearEuroswaprateswerealsocollectedto

assess the outofsample prediction quality of the selected estimation procedures. The Euribor and

3Westartwithk=0andweiterativelyrecomputetheconditionnumberafterincreasingtheridgeconstantwith0.001.Westopiteratingwhentherecomputedconditionnumberislowerthantheprespecifiedthreshold(i.e.10).

13

EONIA rateswereobtained from theirofficialwebsite,and theEuro swap ratesweregathered from

Thompson DataStream. Our dataset spans the period from January 4, 1999 toMay 12, 2009 and

includes2644days.

Weusethesmoothedbootstraptoconstructthespotratecurves:

1. AstheEuriborrates,R(),withmaturitieslessthanoneyearusesimpleinterestrateswiththe

actual/360daycountconvention,weconvertthemto

( ) ( ) = + 365/1 / 360 ,ActualDaysactualR R Days (4.1)whereR()istheannuallycompoundedEuriborrateusingactual/365dateconvention.

2. Swapratesareparyields,sowebootstrapthezerosrates.DenoteS()astheswaprate,timeto

maturitybeing.Thefollowingequationhelpsustoextractthespotratesfromtheswaprates:

( ) ( ) ( )

=

= +

1/

1

11 1.1

jj

R SR j

(4.2)

Here=2,...,10andtheEuroswapratesusetheactual/365daycountconvention.

3. AstherelationshipbetweenEquation2.1and2.3onlyholdsforcontinuouslycompoundedrates,

weneedtoconverttheannualizedspotratestocontinuouslycompoundedrates:

( ) ( ) = + log 1 .r R (4.3)Table2summarizesthedescriptivestatisticsforthetimeseriesofcontinuouslycompoundedspotrates

weusetofitthespotratecurve.Thetableshowsthatthevolatilityofthetimeseries increasesfrom

0.98%forweeklyratesto1.03%for3monthrates,andthengoesdownto0.69%for10yearspotrate.

Theaveragespotrateincreasesastimetomaturitygrows,from3.16%fortheoneweekrate,to4.53%

fora10yearmaturity.Autocorrelationishighforratesofallmaturities,fromabove0.998witha5day

lagtoabove0.915witha255daylag.

14

Table2:DescriptiveStatisticsofSpotRates(RatesAreinPercentage)

Maturity Mean Std.Dev. Min. Max. ( ) 5 ( ) 25 ( ) 255 1Week 3.186 0.982 0.710 5.240 0.998 0.972 0.9261Month 3.238 0.995 0.866 5.257 0.999 0.975 0.9262Months 3.292 1.013 1.111 5.299 0.999 0.980 0.9353Months 3.334 1.030 1.307 5.431 0.999 0.983 0.9414Months 3.351 1.027 1.379 5.441 0.999 0.984 0.9445Months 3.364 1.026 1.435 5.442 0.999 0.984 0.9446Months 3.377 1.025 1.501 5.449 0.999 0.984 0.9457Months 3.388 1.022 1.535 5.448 0.999 0.984 0.9458Months 3.400 1.020 1.566 5.445 0.999 0.984 0.9459Months 3.413 1.019 1.590 5.448 0.999 0.984 0.94410Months 3.426 1.017 1.613 5.444 0.999 0.984 0.94311Months 3.439 1.015 1.636 5.440 0.999 0.983 0.94312Months 3.453 1.014 1.656 5.451 0.999 0.983 0.9422Years 3.568 0.911 1.724 5.435 0.998 0.974 0.9233Years 3.742 0.845 2.160 5.513 0.998 0.972 0.9184Years 3.895 0.796 2.383 5.563 0.998 0.970 0.9155Years 4.029 0.762 2.599 5.614 0.998 0.971 0.9166Years 4.153 0.740 2.729 5.692 0.998 0.971 0.9207Years 4.267 0.726 2.845 5.755 0.998 0.973 0.9258Years 4.369 0.716 2.951 5.820 0.998 0.974 0.9289Years 4.456 0.705 3.047 5.891 0.998 0.975 0.93210Years 4.530 0.696 3.134 5.957 0.998 0.975 0.934Note:Spotratesareexpressedinpercentagewithcontinuouscompounding.ThesampleperiodrunsfromJanuary4,1999toMay12,2009, totaling to2644days.Thespotrateswithmaturities less thanoneyearareretrieved from theEuriborrates,whereasthosewithamaturityofmorethanoneyeararebootstrappedfromEuroswaprates.Bothareconvertedtoobtainrateswithaccordingtotheactual/365(ISDA)dateconvention. ( ) n isthendaylagautocorrelation.

Note:This f2009, totalimaturities lbootstrappeconvention.

Figure5p

datasetc

shortterm

0.7% in2

3.134%an

theyieldc

compared

Fig

figureplots theng to2644dayessthanoneyeed from Euro s.

plots the tim

containsa lot

mspotrates

2009,due to

ndalmost6%

curves,while

dtotheSsha

gure5:TimeS

timeseriesof tyswith the folloearareretrievedswap rates. Bo

eseriesofm

tofvariation

(1week)var

the financial

%.Theyieldc

atothertime

apedcurvesin

Seriesofthe

theEurospotraowing times tomdfromtheEuribth are converte

monthlyspot

inthe level,

yfromappro

crisis.The lo

curveismost

estherearet

notherperiod

15

EuroSpotRa

atecurve.Thesmaturity:1weeborrates,whereed to obtain ra

rates in3di

theslopean

oximately3.7

ongtermspo

oftenupwar

troughs.Betw

ds.

ateCurves(1

sampleperiodrek,112monthseasthosewithaates with acco

mensions.W

ndthecurvat

5%to5%in

ot rate is rela

rdsloping.Ar

ween2003an

9992009)

runs from Januasand210yearamaturityofmrding to the a

Wecanclearly

tureofthete

2008,andde

ativelystable

round2006t

nd2005they

ary4,1999 toMrs.Thespot ratemorethanoneyectual/365 (ISDA

yobserve tha

ermstructure

ecreasestoa

e,varyingbet

herearehum

ieldcurveisf

May12,eswithearareA) date

atour

e.The

lmost

tween

mpsin

flatter

16

5.EmpiricalComparisonoftheEstimationMethods

Inordertocomparethedifferentestimationmethods,weevaluatetheestimationproceduresbasedon

themean absoluteerrorof their forecastingperformance (theMeanAbsolutePredictionErroror in

shorttheMAPE).Theabsoluteerrorismeasuredinbasispointswhichgivesusagoodindicationofthe

economic importanceof the results. For everyday inour time serieswe estimate theNelsonSiegel

modelbasedontheproposedestimationmethods,i.e.forthegridsearch,theOLSwiththefixedshape

parameterandthegridsearchusingtheconditionalridgeregression.Wethenusetheestimatedterm

structures to forecast the spot rates used in the estimation (insample forecasting) and the

contemporaneous EONIA, 20, 25 and 30year Euro swap rates (outofsample forecasting). The

estimationprocedurewhichproduces(overtheavailabletimeseries)the lowestMAPEswinstherat

race.

5.1TheTimeSeriesoftheEstimatedParameters

First,we discuss the results for the grid search, and subsequentlywe comment on the parameter

estimatesfortheOLSapproachwheretheshapeparameterisfixed.Finally,wepresenttheparameters

forthegridsearchusingtheconditionalridgeregression.

5.1.1Thegridsearch

Figure6graphicallyrepresentsthetimeseriesof0,1,2and(0+1)estimates,basedonagridsearch

usingOLS.Atsomepointsintime,thethreecoefficientsareclearlyquiteerratic.Moreover,forthelong

term interestrate level0,somenegativevaluesareobtained, therebyclearlyviolatinganyeconomic

intuition.Theshortendofthetermstructure,denotedby(0+1),isalwayspositive.Thisisexplained

byhighlynegativecorrelationbetweenthetimeseriesof0and1estimates.ForAugust21,2007e.g.

the high 0coefficient (9.737) is accompaniedwith a low 0 coefficient (5.096)which leads to an

estimateoftheshortrateof5.477%.

17

Figure6:TimeSeriesofEstimatedParameterswiththeGridSearchBasedonOLS

Note:Thisfigureplotsthetimeseriesof0,1,2and(0+1)estimatesoftheNelsonSiegelmodelonEurospotratesbasedonagridsearchusingOLSovertheperiodfromJanuary4,1999toMay12,2009.

Inanattempttounderstandthesourceoftheerratictimeseriesbehaviour,wereportthehistogramof

theestimatedshapeparameters (Figure7).Thevariationof the shapeparameterestimates indicates

that thisparametercannotbeassumedconstantover time.Thisconfirms thevisual inspectionof the

timeseriesplotofourdataset(Figure5)whichrevealedthattheshapeandthepositionofthehumps

changedovertime.Morethan55%oftheestimatedshapeparametersarelocatedwithintherangeof0

to2,approximately20%withintherangeof2to4,andalittlemorethan19%withintherangeof8to

01/15/00 05/29/01 10/11/02 02/23/04 07/07/05 11/19/06 04/02/08-5

0

5

10

15

20

25

21-Aug-20079.737

Date

0 (Grid Search Based on OLS)

01/15/00 05/29/01 10/11/02 02/23/04 07/07/05 11/19/06 04/02/08-20

-15

-10

-5

0

5

21-Aug-2007-5.095

Date

1 (Grid Search Based on OLS)

01/15/00 05/29/01 10/11/02 02/23/04 07/07/05 11/19/06 04/02/08-30

-25

-20

-15

-10

-5

0

5

10

15

21-Aug-2007-7.339

Date

2 (Grid Search Based on OLS)

01/15/00 05/29/01 10/11/02 02/23/04 07/07/05 11/19/06 04/02/081

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Date

0 + 1 (Grid Search Based on OLS)

10.Forth

boundof

Note:Thisfgridsearchperformedw

Thereare

bound of

hump/tro

4WeperforThe shapesearchinter

heshapepar

thesearchin

Figure7:His

figureshowsthebasedonOLSowiththeshapep

etwoexplana

f the search

ough.Bothsit

rmedgridsearchparameters thatrvalwhenthisup

ameterswith

ntervali.e.at

stogramofO

ehistogramtheoverthesampleparameterrangi

ationsforthe

interval: the

uationsoccu

hwiththeshaptareestimatedpperboundis20

hintherange

10.4

OptimalShape

estimatedshapperiodfromJangfrom0to10

erelatively la

e absence o

rinourdatas

eparameterran tobeat theu0or30.

18

eof8to10,

eParameters

peparametersnuary4,1999to.

rgeamount

f a hump/tr

setasinshow

ngingfrom0topperbound (e.g

472outof5

sUsingGridS

oftheNelsonSoMay12,2009

ofshapepar

rough or the

wninFigure8

10,0to20(nog.10)arealsoe

506areestim

SearchBased

Siegelmodelon9,totalingto264

rameterestim

e presence o

8.

otreported)or0estimatedat th

matedattheu

donOLS

nEurospotrate44days.Gridse

matesattheu

of more than

0to30(notrepheupperbound

upper

susingearchis

upper

n one

ported).of the

Figure8

Pa

Note:Thisfparametersclearevidenflexibleeno

Figure8

estimate

ispresent

tobeat t

humpatt

2007.Her

inspection

optimals

boundsha

from the

potential

moreapp

8:TheTermS

anelA(Date

figuredepictsthbasedongridsnceofhump/troughtocapturet

plots the ter

attheupper

ted.Sincethe

theupperbo

theveryend

rethetermst

nshowsthat

hapeparame

apeparamete

NelsonSieg

multicollinea

propriatecand

tructureofS

:May20,200

hetermstructursearch inbothPough.InPanelBtheshapeofthis

m structure

boundofthe

etermstructu

oundof the s

oftheterm

tructureisto

onehumpis

eter iscompu

erestimates.

el specificati

arity. In the l

didatetodesc

potRateson

04)

eofspotratesoPanelAandPanthetermstructustermstructure

onMay20,

esearchinter

uredoesnot

search interv

structure.Fig

oocomplicate

notsufficien

uted tobe10

Intheforme

on to fit the

attercase,a

cribetheterm

19

May20,200

Pane

onMay20,2004nelBarebindingurehasonehume.

2004andon

rval.Inpanel

showahump

al.Theoptim

gure8,Panel

edtobedescr

nttocloselyf

0.Bothexam

ercase,thecu

e curve, wit

more flexibl

mstructure.

04(PanelA)a

elB(Date:Au

4(PanelA)andgat10.InPanempandonetrou

nAugust21,

AofFigure8

p/trough,the

mizationproc

lBdepictsth

ribedbythe

fitthetermst

mples illustrat

urvaturecom

h the additio

lemodelsuc

andAugust2

ugust21,200

August21,200elAthetermstrgh,buttheNels

2007witha

8,thesituatio

eshapeparam

cedure theref

hetermstruc

NelsonSiege

tructureont

teproblems

mponentcans

onal benefit

hasSvensso

1,2007(Pane

07)

07(PanelB).TheructuredoesnosonSiegelmode

a shapeparam

onofMay20,

meterisestim

foreestimate

ctureofAugu

elmodel.Gra

hisday.Agai

resulting inu

simplybedro

of eliminati

on (1994)wil

elB)

eshapeotshowelisnot

meter

,2004

mated

es the

st21,

phical

n,the

upper

opped

ng of

lbea

20

Figure9:LongTermSpotRateEstimatesandtheirStandardErrorsGridSearch

Note:Thisfigureplotsthelongtermspotrateestimates(solidline)andtheirstandarderrors(dashedline)basedongridsearchoverthesampleperiodfromJanuary4,1999toMay12,2009,totalingto2644days.TheerraticbehaviouroftheNelsonSiegelmodelbasedongridsearchisillustrated.

Gridsearchnotonlyresults inerratictimeseriesof factorestimates,theprecisionoftheestimates is

alsoverytimevarying.Figure9redraws(asanexample)theestimatesofthelongtermspotrate(solid

line)anditsstandarderrors(dashedline).Whereasthestandarderrorsaresmallattimes,manyperiods

ofturbulenceareshowninwhichthestandarderrorsbecome1%andmore!

5.1.2TheOLSwithfixedshapeparameter

Figure10plotsthetimeseriesoftheestimatedparametersconditionalonafixedshapeparameter.In

the leftcolumnwepresent theestimatesusinga fixedshapeparameterof1.37 (as inDieboldandLi

(2006)),whereasontherightsidethosewithafixedshapeparameterof3(as inFabozzietal.(2005))

aredepicted.Generally,thetimeseriesoftheestimatesarenotasvolatileasthosebasedonthegrid

search and the time series look smoothed. The higher shape parameter, however, results in a less

smoothed timeseriesof theestimated0,1and2.The longterm interest rate level impliedby the

NelsonSiegel model is always positive, and the short end does not show negative interest rate

01/01/00 01/01/02 01/01/04 01/01/06 01/01/080

5

10

15

20

25

0 (

%)

01/01/00 01/01/02 01/01/04 01/01/06 01/01/08

0

1

2

3

4

5

6

7

Sta

ndar

d Er

ror(

%)

0Standard Error

21

estimates either. However, the economic interpretation of the coefficients as factors, remains

problematic.Atthestartofourseriese.g.,thelongtermrateisestimatedasbeing4.53%(=1.37)and

5.74%(=3)whereasthe30yearswapratewas4.99%.So,whichshapeparametershouldweconsider

tobethemostappropriate?

Theprecisionoftheestimatesimprovesdramaticallycomparedtothegridsearch.Figure11showsthe

standarderrorsforthelongtermrateusing=1.37.Duringthefinancialcrisis,thestandarderrorwent

uptomorethan35basispoints.Thisisaseriousimprovement,comparedtothestandarderrorsofthe

grid search,where the standard errorswent up to 500 basis points andmore.Whether the shape

parametercanbestbefixedto1.37orto3,however,remainsanopenquestion.Especiallytheresults

for2arequitedifferentforbothchoicesinshapeparameter,ascanbeseeninFigure10.Ourestimates

alsosuggest that theeconomiccharacteristicsofthetimeseriesoftheestimatedcoefficientsmaybe

quitedifferent.RecallthattheNelsonSiegelmodelcanbeinterpretedintermsofathreefactormodel,

theestimated coefficientsbeingweights.Taking thevariabilityof the shapeparameterestimateswe

obtained inourgridsearch intoaccount, itcanbequestionedwhether the timevariation incanbe

ignoredatall!

22

Figure10:TimeSeriesofEstimatedParameterswithFixedShapeParameters

Note:Thisfigureplotsthetimeseriesof0,1,2and(0+1)estimatesoftheNelsonSiegelmodelonEurospotratesbasedonfixedshapeparametersof1.37(PanelA)and3(PanelB).ThesampleperiodrunsfromJanuary4,1999toMay12,2009.

01/15/00 05/29/01 10/11/02 02/23/04 07/07/05 11/19/06 04/02/080

1

2

3

4

5

6

7

8

9

10

Date

0 (Diebold and Li)

01/15/00 05/29/01 10/11/02 02/23/04 07/07/05 11/19/06 04/02/080

1

2

3

4

5

6

7

8

9

10

Date

0 (Fabozzi et al)

01/15/00 05/29/01 10/11/02 02/23/04 07/07/05 11/19/06 04/02/08-5

-4

-3

-2

-1

0

1

2

3

4

5

Date

1 (Diebold and Li)

01/15/00 05/29/01 10/11/02 02/23/04 07/07/05 11/19/06 04/02/08-5

-4

-3

-2

-1

0

1

2

3

4

5

Date

1 (Fabozzi et al)

01/15/00 05/29/01 10/11/02 02/23/04 07/07/05 11/19/06 04/02/08-10

-8

-6

-4

-2

0

2

4

6

8

10

Date

2 (Diebold and Li)

01/15/00 05/29/01 10/11/02 02/23/04 07/07/05 11/19/06 04/02/08-10

-8

-6

-4

-2

0

2

4

6

8

10

Date

2 (Fabozzi et al)

01/15/00 05/29/01 10/11/02 02/23/04 07/07/05 11/19/06 04/02/081

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Date

0 + 1 (Diebold and Li)

01/15/00 05/29/01 10/11/02 02/23/04 07/07/05 11/19/06 04/02/081

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Date

0+1 (Fabozzi et al)

23

Figure11:EstimatesoftheLongTermRateandtheirStandardErrors(=1.37)

Note:Thisfiguredepictstheestimatesofthe longtermrate(0)andtheirstandarderrors(StandardError)basedonafixedshapeparameterof1.37overthesampleperiodfromJanuary4,1999toMay12,2009.

5.1.3Gridsearchwithconditionalridgeregression

Adrawbackoftheridgeregressiontechniqueisthelackofstandarderrors,whichprohibitsanykindof

significance testson theestimatedcoefficients (DeMaris (2004)).However,wecan (visually)examine

thestabilityofthetimeseriesestimates.Figure12againconfirmsthatthetimeseriesof0,conditional

onafixedshapeparameterof3,almostperfectlycoincideswiththegridsearch0,timeseries.A low

shapeparametersmoothestheextremejumpsinthecoefficientsseriesalmostcompletely,whereasthe

ridgeregressiontakesamiddleposition.Whenevercanbefreelyestimated(i.e.whenissufficiently

low), no multicollinearity problems occur and the ridge regression is redundant. Whenever the

correlation between the regressors exceeds our threshold, the ridge regression has a moderate

smoothingeffect.However,thevariation inthecoefficients,e.g.0 inFigure12retainsthestrainsput

onthetermstructure.

01/01/00 01/01/02 01/01/04 01/01/06 01/01/080

1

2

3

4

5

6

7

0 (

%)

01/01/00 01/01/02 01/01/04 01/01/06 01/01/08

0

0.1

0.2

0.3

0.4

Sta

ndar

d Er

ror(

%)

0Standard Error

Note:This fregression,periodfrom

Figure13

therights

Theestim

Thereare

leveljump

reached i

Ridgereg

longterm

shortend

Figure12:Th

figureplots thefixed shapepar

mJanuary4,1999

showstheti

sidetheresul

matesfromth

enonegative

pedupfrom

tspeakof8

ression impro

minterestrate

oftheterms

heEstimated

estimated longrameterof1.379toMay12,200

meseriesof

ltsfromboth

heridgeregre

evalues inth

4.80%inJune

.99% inOcto

ovesthestab

elevelcompl

structureaga

LongTermR

g term ratesbas7 (DieboldandL09.

alltheestim

thegridsear

essionaremo

e longterm

e2007to5.8

ober2008.Af

bilityofthee

lieswiththe

ainisalwaysp

24

RateBasedon

sedon the folloLi),and fixed sh

atedcoefficie

rchbasedon

orestablecom

interestrate

85%inAugust

fterwards the

estimatescalc

economicint

positive,cons

nAlternative

owingestimationhapeparameter

entsusingth

OLSandther

mparedtoth

levelanymo

t2007,then8

e longterm i

culatedbyth

tuitionbehind

sistentwithre

eEstimationM

nmethods: theof3 (Fabozzie

eridgeregre

ridgeregress

heresultsfro

re.The long

8.14%inMar

interest rate

hegridsearch

dtheNelson

eality.

Methods

gridsearch, theetal.)over the s

ession,where

ionareplotte

mthegridse

term interes

rch2008,and

levelwentd

h.Andthepo

Siegelmode

e ridgesample

eason

ed.

earch.

strate

dthen

down.

ositive

el.The

25

Figure13:EstimatedParameterswiththeRidgeRegressionandGridSearch

Note: This figure depicts the estimated parameters of theNelsonSiegelmodel based on both grid search (line) and ridgeregression(dot).0representsthelongterminterestratelevelimpliedbytheNelsonSiegelmodel.0+1representstheshortterminterestratewhentimetomaturityiszero.

Inordertomeasurethestabilityofthetimeseriesofestimatedcoefficientsmoreformally,wecompute

thestandarddeviationoftheirfirstdifferences(Table3).Wenoticethatridgeregressionshavealower

volatility forall threeparameters.Moreover,anFtestat the95%confidence intervalshows that the

standard deviations of the ridge regression coefficients changes are significantly lower than those

obtainedthroughthegridsearch.Theridgeregression,hence,cansubstantiallyreducetheinstabilityof

the estimates in the grid search. Compared with the fixed shape parameter procedures, the ridge

regressionallows theshapeparameter tovaryover time.Whetherornot this isadesirableproperty

fromaneconomicpointofview,stillhastobeseen.Therefore,weexaminetheinandoutofsample

predictionperformanceofthevariousmethods.

01/15/00 05/29/01 10/11/02 02/23/04 07/07/05 11/19/06 04/02/08-5

0

5

10

15

20

25

Date

0 (Ridge vs Grid Search)

Grid Search Based on OLS

Ridge Regression

01/15/00 05/29/01 10/11/02 02/23/04 07/07/05 11/19/06 04/02/08-20

-15

-10

-5

0

5

Date

1 (Ridge vs Grid Search)

Grid Search Based on OLSRidge Regression

01/15/00 05/29/01 10/11/02 02/23/04 07/07/05 11/19/06 04/02/08-30

-25

-20

-15

-10

-5

0

5

10

15

Date

2 (Ridge vs Grid Search)

Grid Search Based on OLSRidge Regression

01/15/00 05/29/01 10/11/02 02/23/04 07/07/05 11/19/06 04/02/081

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Date

0 + 1 (Ridge vs Grid Search)

Grid Search Based on OLSRidge Regression

26

Table3:TheStandardDeviationsofFirstDifferencesintheEstimates(inPercentage)

GridSearch RidgeRegression =1.37 =30 0.602 0.141 0.058 0.0791 0.584 0.138 0.068 0.0802 1.594 1.106 0.126 0.225

Note:Thistableshowsthestandarddeviationsofthefirstorderchangesinthetimeseriesoftheestimatedparameters.Thedatasetusedtoestimatetheparametersiscomposedof1week,1to12month,and1to10yearspotrates.AnFtestata95%confidenceintervalshowsthatallthestandarddeviationsaresignificantlydifferentfromeachother.

5.2Insampleperformance

Inorder toexamine the insampleperformance,wecompute themeanabsoluteerrorsbetween the

predictedand thebootstrappedspot rate (Table4).To testwhetherMAPEsarestatisticallydifferent

fromeachother,wecompute:

1 2 ,Method Method = 1 r r r r i (5.1)

wherer isthevectorofempiricalratesforacertaintimetomaturity,1 isavectorofones, 1Methodr and

2Methodr arethevectorsofestimatedspotratesusingdifferentmethods(e.g.grid,ridge,DieboldandLi(DL)andFabozzietal. (FMP)). Iftheestimatedcoefficient issignificantlynegative,thenweconsiderMethod1tobebetterthanMethod2.TheNeweyWestcorrectionisusedtoremoveserialcorrelation

fromtheresiduals.

AlthoughtheoverallMAPEisminimizedbyconstructionfortheGridSearch,theinsampleMAPEsper

maturity show how the fixed shape parametermethods perform compared to the ridge regression

technique.TheMAPEofFMP(DL)isfor13(9)outofthe22maturitiesthehighest.For8outofthe22

maturities,theridgeregressiongeneratesthelowestMAPE,butitneverproducesthehighestMAPEs.

Thegrid search is formore thanhalfof thematurities theMAPEminimizingmethod.The insample

MAPEs reveal the limitationsof reducing themodel flexibilityby fixing the shapeparameters.Ridge

regressionoutperformsthesetwomethodsconvincingly.

27

Table4:InSampleMeanAbsolutePredictionErrors

Maturity Grid Ridge DL(=1.37) FMP(=3)1Week 0.113 0.103* 0.130 0.170+1Month 0.067 0.059* 0.076 0.114+2Months 0.035 0.031* 0.033 0.062+3Months 0.031* 0.035 0.034 0.042+4Months 0.030* 0.034 0.036+ 0.0345Months 0.031* 0.035 0.039+ 0.0366Months 0.033* 0.035 0.043 0.047+7Months 0.033* 0.034 0.043 0.055+8Months 0.034* 0.034 0.044 0.063+9Months 0.035 0.035* 0.045 0.070+10Months 0.036 0.035* 0.045 0.075+11Months 0.038 0.037* 0.045 0.079+12Months 0.041 0.039* 0.046 0.083+2Years 0.055* 0.068 0.073+ 0.0573Years 0.048* 0.065 0.077+ 0.0554Years 0.033* 0.052 0.072+ 0.0655Years 0.028* 0.044 0.065+ 0.0616Years 0.024* 0.030 0.046 0.047+7Years 0.015 0.013* 0.017 0.026+8Years 0.013 0.018 0.022 0.008*9Years 0.020* 0.037 0.054+ 0.03410Years 0.033* 0.055 0.083+ 0.066Note: InsamplemeanabsoluteerrorsbetweenproduceddatafromtheNelsonSiegelandempiricaldataarepresented.Thedatasetusedtoestimatetheparametersiscomposedby1week,1to12month,and1to10yearspotrates.*ThisapproachyieldsthelowestMAPEforthistimetomaturity.+ThisapproachyieldsthehighestMAPEforthistimetomaturity.Exceptthefollowingpairs,alltheMAPEsaresignificantlydifferentfromeachother(at95%confidenceinterval)basedonthettestwithNeweyWestcorrectedstandarderrors:2month:gridandDL;3month:ridgeandDL;4month:ridgeandFMP;5month:ridgeandFMP;2year:gridandFMP;5year:DLandFMP;6year:DLandFMP;9year:ridgeandFMP.ThemeanabsoluteerrorsaretestedusingEquation5.1.

5.3Outofsampleperformance

SincetheridgeregressionaddsasmallbiastotheOLS,weconsideranoutofsampleperformancetest

superior in order to judge the appropriateness of the various estimation methods. If we denote

lr r = as the residualbetween thespot rate r observed from themarketand theestimatedspotratebytheNelsonSiegelmodel lr foracertaintimetomaturity,thebiasisestimatedastheaverageofthes.Theoutofsamplebiasoneachday ismeasuredbyaveragingtheresidualsafterfittingthe

28

EONIA, 20, 25 and 30year swap rates using four methods. Table 5 clearly shows that the bias

introducedintheridgeregressiondoesnotaffectthebiasintheforecastedspotratesinamaterialway

comparedtotheotherestimationmethods.

Table5:OutofSampleBias

Grid Ridge DL(=1.37) FMP(=3)

Mean 0.070 4.00E4 0.099 0.089

Std.Dev. 0.384 0.152 0.102 0.214

Minimum 2.216 0.925 0.356 0.820

Maximum 0.703 0.369 0.393 0.558

Note:Thebiasisexpressedinpercentage.ThesampleperiodrunsfromJanuary4,1999toMay12,2009,totalingto2644daysforwhich theoutofsampleestimationbias ismeasured.Theoutofsamplebiasoneachday ismeasuredbyaveraging theresidualsafterfittingtheEONIA,20,25and30yearswapratesusingfourmethods.AllthebiasesaresignificantlydifferentfromeachotherbasedonattestwithNeweyWestcorrectedstandarderrors.

To investigate the ability of these four approaches to forecast the long and short end of the term

structure,wewillcomparetheestimatedratestotheEONIA,the20,25and30yearEuroswaprates.

TheMAPEwillagainbeourcriterion.

Theoreticallyspeaking,theestimated0+1representstheshortendofthetermstructure.However,

theEONIAratesaretheshortesttermspotratesthatcanbeobservedinthemarket.Consequently,ifa

modelcanpredicttheEONIArateswiththehighestaccuracy,itwillbesuperiorinpredictingtheshort

endoftheyieldcurve.

TheestimatedEONIAratesarecalculatedbypluggingtimetomaturity=1/365intoEquation2.3along

with the estimated parameters for all fourmethodologies.Afterwards, the differences between the

empirical and the estimated rates are examined following the same procedure as for the insample

MAPEstests.

Inaddition,wealsocheckwhichmodelcanbestfitthelongtermendofthetermstructure.However,

longtermspotratesarenotdirectlyobservableinthemarketeither.Moreover,duetothelackofswap

ratesforcertaintimestomaturity(e.g.11or29yearswaprate)tobootstrapthe longtermspotrate

(e.g.30yearspotrate),wewillusethefollowingproceduretochecktheforecastingability:

1. CombiningEquation2.3and3.4,wecalculatetheestimatedswapratesusing

29

( ) ( )( )1

1 1,

1

1j

j

RS

R j

=

+ = +

(5.2)

where=2,...,30arethetimestomaturity,

( ) ( ) 1rR e =

is the estimated year annually compounded spot rate, ( )r is the estimated continuouslycompoundedspotrate,and ( )S istheestimatedswaprate.

2. Wecalculateand test theMAPEsbetween theempiricaland theestimated swap ratesusing

thesameprocedureaspreviously.

TheresultsaresummarizedinTable6.TheridgeregressionalwaysyieldsthestatisticallylowestMAPEs.

TherestrictionDLandFMPputontheshapeparametermakesthemunderperformcomparedtoboth

the ridge and theOLS regression.Moreover, the ridge regression has lower prediction errorswhen

predictingthelongend,comparedtotheshortendofthetermstructure.

Table6:OutofSampleMeanAbsolutePredictionErrors

Maturity Grid Ridge DL(=1.37) FMP(=3)OvernightEONIA 0.254 0.246* 0.287+ 0.27820yearswaprate 0.201 0.142* 0.234+ 0.22125yearswaprate 0.262 0.150* 0.236 0.268+30yearswaprate 0.304 0.152* 0.227 0.307+Note:OutofsampleMAPEsbetweenproduceddatafromNelsonSiegelandempiricaldataarepresented.Thedatasetusedtoestimate theparameters is composedby1week,1 to12month,and1 to10year spot rates.*Thisapproach yields thelowestMAPEforthistimetomaturity.+ThisapproachyieldsthehighestMAPEforthistimetomaturity.Exceptthefollowingpairs,alltheMAPEsaresignificantlydifferentfromeachother(at95%confidence interval)basedonthettestwithNeweyWestcorrectiononstandarderrors:20year:gridandFMP,DLandFMP;25year:DLandFMP,gridandFMP;30year:gridandFMP.ThemeanabsoluteerrorsaretestedusingEquation5.1.

The superiority of the ridge regression procedure is not only statistically significant. Looking at the

MAPEsforthe30yearswaprate,theMAPEsareloweredwith7to15basispoints.Ontheshortend,

thegainismoremoderate,upto4basispoints.

30

5.4Robustnesscheckontheforecastingability

Weconsidertworobustnesscheckstoconfirmtheoutperformanceoftheconditionalridgeregression

procedurewepropose. First,wewant to verifywhetherornotour results aremainlydrivenby the

20082009financialcrisiswhichispartofourdataset.Second,wehaveshownthatthemulticollinearity

problemisseverelyaffectedbythechoiceofthematurityvector.Wewillexaminewhetherourresults

arerobusttothechoiceofadifferentmaturityvector.

5.4.1Theimpactofthefinancialcrisis

AsseeninFigure13,thefinancialcrisishashadasubstantialimpactonparameterestimation.Wethus

divideourdataset into two subsets, theprecrisisperiod from January4,1999 to July2,2007 (2169

days),andthecrisisperiodfromJuly3,2007toMay12,2009(475days).Theoutofsamplepredicting

powerofallfourapproaches(Grid,Ridge,DLandFMP)ispresentedinTable7andTable8.

AttestbetweentheprecrisisandcrisisperiodMAPEsat95%confidence levelshowsthatduringthe

financialcrisisthepredictingabilityofallfourmethodsissignificantlylowered.Thepredictingabilityof

thegridsearchdropsdramaticallyforboththeshortandthe longendofthetermstructure,whilefor

theothermethodologies,thefinancialcrisishasmoreimpactontheshortendthanonthelongend.

Nevertheless, theridgeregressionperformsconsistently inbothperiods,while thegridsearchclearly

doesnot.Duringthefinancialcrisisthegridsearchpredictionsareworstforthe longendoftheyield

curvewhereasintheprecrisisperiod,theFMPmodelisworstforthesematurities.

Theresultsshowninthesetwotablesagainconfirmourpreviousfindingsthattheridgeregressionhas

superiorpredictabilityinforecastingbothendsoftheyieldcurve.

31

Table7:OutofSampleMeanAbsolutePredictionErrors(PrecrisisPeriod)

Maturity Grid Ridge DL(=1.37) FMP(=3)OvernightEONIA 0.161 0.161* 0.195+ 0.16920yearswaprate 0.142 0.137* 0.206 0.217+25yearswaprate 0.161 0.140* 0.218 0.260+30yearswaprate 0.169 0.131* 0.211 0.296+Note:OutofsampleMAPEsbetweenproduceddata fromNelsonSiegelandempiricaldataarepresented for theprecrisisperiodfromJanuary4,1999toJuly2,2007(2169days).Thedatasetusedtoestimatetheparametersiscomposedby1week,112month,and1to10yearspotrates.*Thisapproachyieldsthe lowestMAPEforthistimetomaturity.+ThisapproachyieldsthehighestMAPEforthistimetomaturity.Exceptthefollowingpairs,alltheMAPEsaresignificantlydifferentfromeachotherbasedonthettestwithNeweyWestcorrectiononstandarderrors:EONIA:gridandridge,gridandFMP,ridgeandFMP;20year:gridandridge,DLandFMP.

Table8:OutofSampleMeanAbsolutePredictionErrors(CrisisPeriod)

Maturity Grid Ridge DL(=1.37) FMP(=3)OvernightEONIA 0.676 0.635* 0.704 0.775+20yearswaprate 0.470+ 0.165* 0.363 0.24025yearswaprate 0.721+ 0.198* 0.317 0.30630yearswaprate 0.919+ 0.251* 0.296 0.361Note:OutofsampleMAPEsbetweenproduceddatafromNelsonSiegelandempiricaldataarepresentedforthecrisisperiodfromJuly3,2007toAugust12,2009(475days).Thedatasetusedtoestimatetheparameters iscomposedby1week,112month,and1to10yearspotrates.*ThisapproachyieldsthelowestMAPEforthistimetomaturity.+ThisapproachyieldsthehighestMAPEforthistimetomaturity.Exceptthefollowingpairs,alltheMAPEsaresignificantlydifferentfromeachotherbasedonthettestwithNeweyWestcorrectiononstandarderrors:EONIA:gridandDL;20year:gridandDL;25year:DLandFMP;30year:ridgeandDL,DLandFMP.

5.4.2ADifferentMaturityVector

Inordertotesttherobustnessofourresults,weestimatetheyieldcurvesagainusingonly1week,6

month,1to10yearspotrates.Theresults,showninTable9,alsoconfirmourpreviousfindingsthatthe

ridgeregressionhassuperiorpredictabilityinforecastingbothendsfortheyieldcurve.However,unlike

fortheotherdataset,nowthepredictabilityofFMPon25and30yearswapratesisnottheworst,but

the grid search is.Here the correlationbetween the slope andhump factorusing theDLestimation

recipe is0.35 insteadof0.55.ThehighestMAPEsforthe25and30yearswapratesobtainedusing

thegridsearchbasedontheOLSregression,canbeblamedontheinstableestimates.

32

Table9:OutofSampleMeanAbsolutePredictionErrors

Maturity Grid Ridge DL(=1.37) FMP(=3)OvernightEONIA 0.293 0.217* 0.255+ 0.24320yearswaprate 0.171 0.128* 0.205+ 0.17925yearswaprate 0.226+ 0.140* 0.207 0.22230yearswaprate 0.266+ 0.147* 0.200 0.259Note:OutofsampleMAPEsbetweenproduceddatafromNelsonSiegelandempiricaldataarepresented.Thedatasetusedtoestimate theparameters is composedby1week,6month,and1 to10year spot rates.*Thisapproachyields the lowestMAPEforthistimetomaturity.+ThisapproachyieldsthehighestMAPEforthistimetomaturity.Exceptthefollowingpairs,alltheMAPEsare significantlydifferent fromeachotherbasedon the ttestwithNeweyWest correctionon standarderrors:EONIA:gridandDL;20year:gridandFMP;25year:gridandDL,gridandFMP,DLandFMP;30year:gridandDL.

Again,we test the influenceof the financialcrisison theperformanceof theproposedmethods, this

timeonourlimitedsample.Theresultsfromthisanalysiscomparingestimatesoftheprecrisisandcrisis

periodaresummarizedinTable10andTable11.

At the95% confidence level,a ttestbetween theprecrisisand crisisperiodMAPEs shows that the

performanceofthefourmethodsbehavessimilartothatoftheotherdataset.Thehighvolatilityofthe

estimatesinthegridsearchmakesitsperformancedropsubstantiallyduringthefinancialcrisis.

Theridgeregressionalwayshas thehighestpredictingabilityexcept for the30yearswaprateduring

thecrisisperiod,wheretheDLhasthehighestpredictingability.However,thedifferencebetweenthese

twomethodsforthe30yearswaprateduringthefinancialcrisisisnotstatisticallysignificant.

Table10:OutofSampleMeanAbsolutePredictionErrors(PrecrisisPeriod)

Maturity Grid Ridge DL(=1.37) FMP(=3)OvernightEONIA 0.216+ 0.156* 0.186 0.16020yearswaprate 0.128 0.120* 0.191+ 0.17325yearswaprate 0.150 0.124* 0.201 0.209+30yearswaprate 0.161 0.121* 0.192 0.242+Note:OutofsampleMAPEsbetweenproduceddata fromNelsonSiegelandempiricaldataarepresented for theprecrisisperiodfromJanuary4,1999toJuly2,2007(2169days).Thedatasetusedtoestimatetheparametersiscomposedby1week,6month,and1to10yearspotrates.*ThisapproachyieldsthelowestMAPEforthistimetomaturity.+ThisapproachyieldsthehighestMAPEforthistimetomaturity.Exceptthefollowingpairs,alltheMAPEsaresignificantlydifferentfromeachotherbasedonthettestwithNeweyWestcorrectiononstandarderrors:25year:DLandFMP.

33

Table11:OutofSampleMeanAbsolutePredictionErrors(CrisisPeriod)

Maturity Grid Ridge DL(=1.37) FMP(=3)OvernightEONIA 0.642+ 0.499* 0.571 0.62420yearswaprate 0.365+ 0.164* 0.268 0.20525yearswaprate 0.578+ 0.213* 0.236 0.28030yearswaprate 0.749+ 0.264 0.236* 0.337Note:OutofsampleMAPEsbetweenproduceddatafromNelsonSiegelandempiricaldataarepresentedforthecrisisperiodfromJuly3,2007toAugust12,2009(475days).Thedatasetusedtoestimatetheparametersiscomposedby1week,6month,and1 to10year spot rates.*Thisapproachyields the lowestMAPE for this time tomaturity.+Thisapproach yields thehighestMAPE for this time tomaturity.Except the followingpairs,all theMAPEsaresignificantlydifferent fromeachotherbasedonthettestwithNeweyWestcorrectiononstandarderrors:EONIA:gridandDL,gridandFMP;25year:ridgeandDL,DLandFMP;30year:ridgeandFMP.

One thingworthmentioning is that for the alternative dataset, not only the long end of the term

structurebasedonthegridsearchisextremelyvolatileandsometimesnegative,soistheshortendof

thetermstructure,asshowninFigure14.Thisagainshowsthatthegridsearchisverysensitivetothe

underlyingdataset.

Figure14:TheShortEndoftheTermStructurebytheGridSearchBasedonOLS

Note:ThisfigureplotstheshortendofthetermstructureimpliedbythegridsearchedbasedonOLSoverthesampleperiodfromJanuary4,1999toMay12,2009,totalingto2644days.

01/15/00 05/29/01 10/11/02 02/23/04 07/07/05 11/19/06 04/02/08-20

-15

-10

-5

0

5

10

15

20

25

30

Date

0 + 1 (Grid Search Based on OLS)

34

6.Conclusion

Many researchers have reported problems in estimating the NelsonSiegel (1987)model.We have

shown thatmulticollinearity between the slope and hump factor are causing both instability of the

regressioncoefficientsovertimeandlargestandarderrorsonthecoefficients.Totempertheestimation

problems,weapplyridgeregressiontechnique,wheneverthegridsearchbasedestimateoftheshape

parameter results in highly correlated slope and hump factors. For euro spot rate curves, over the

period1992009,wecomparethegridsearchestimatesoriginallyproposedbyNelsonandSiegelto

ourridgeregressionestimates.TheDieboldandLi(2006)andtheFabozzietal.(2005)estimateswere

alsocalculatedasabenchmark.

The insample comparison shows that the grid search produces erratic time series estimateswhich

sometimesviolatetheeconomicintuitionbehindtheNelsonSiegelmodel.Thedistributionofthefreely

estimatedshapeparametersandthe insamplefittingerrorsrevealthe limitationoftheuseofafixed

shapeparameter.Theoutofsamplepredictabilityatthetwoendsofthetermstructureshowsthatthe

ridgeregressionalwaysproducesthe lowestmeanabsolutepredictionerrors.Forthe longendofthe

termstructure,theeconomicgain intheMAPEmountedupto15basispoints.Fortheshortend,the

differencesinMAPEswerestatisticallysignificantbuteconomicallysmaller.

Robustnesschecksshowthattheridgeregressionperformsrobustlyandconsistentlybetterindifferent

economicenvironments (precrisisandcrisisperiod)andwithdifferentchoicesof thematurityvector

(i.e.thesetoffinancialinstrumentsusedtobootstraptheyieldcurve).

Basedonourfindings,fixingtheshapeparameterinordertoavoidmulticollinearity,isastatisticaltrick

thatdoesreducethecorrelationbetweentheregressorswhenthefixingisjudiciallychosen.Ithowever

ignoresthebarefactthat inpractice,termstructuresdotakeallkindof(humpy)shapesandthatthe

humpsimply isnot fixedover the time tomaturityspectrum.The loss in flexibilitycomesatasevere

priceespeciallyforthepredictionsof longtermspotrates.Thispaperadvancesaprocedurebasedon

ridgeregressiontoreducethepredictionerrorssignificantly.

35

7.References

Adams,K.SmoothInterpolationofZeroRates,AlgoResearchQuarterly,Vol.4,12,(2001)1122.

Adams, K. J. and VanDeventerD. R. Fitting Yield Curves and Forward Rate CurveswithMaximum

Smoothness,JournalofFixedIncome,Vol.4,No.1(June1994),5262.

Bankof InternationalSettlements.ZeroCouponYieldCurvesTechnicalDocumentation,BISPaper

No.25,2005.

Barrett,W.R.,Gosnell,T.F.Jr.andHeuson,A.J.,YieldCurveShiftsandtheSelectionofImmunization

Strategies,JournalofFixedIncome,Vol.5,No.2(September1995),5364.

Belsley,D.A. ConditioningDiagnostics Collinearity andWeak Data in Regression,Wiley Series in

ProbabilityandmathematicalStatistics,1991.

Bliss,R.,andFama,E.,TheInformationinLongMaturityForwardRates,AmericanEconomicReview,

Vol.77(1987),680692.

Cairns,A.J.G.,andPritchard,StabilityofDescriptiveModelsfortheTermStructureof InterestRates

withApplicationstoGermanMarketData,BritishActuarialJournalVol.7(2001),467507.

Causey,B.D.,AFrequentistanalysisofaClassofRidgeRegressionEstimators, JournalofAmerican

StatisticalAssociation,Vol.75,No.371(September1980),736738.

Clikff,A.D.andOrd,J.K.,SpatialProcessesModels&Applications,PionLimited,1981.

Cohen,J.,Cohen,P.,West,S.G.,andAiken,L.S.,AppliedMultipleRegression/CorrelationAnalysisfor

theBehavioralSciences(3rdEdition),LawrenceErlbaumAssociates,Publishers,2003.

Coroneo, L., Nyholm, K., and VivadaKoleva, R., How ArbitrageFree Is the NelsonSiegelModel?,

EuropeanCentralBankWorkingPaperSeries,No.874(February2008).

de Pooter, M., Examining the NelsonSiegel Class of Term Structure Models, Tinbergen Institute

DiscussionPaper,IT2007043/4(2007).

DeMaris,A.,RegressionwithSocialData:ModelingContinuousandLimitedResponseVariables,Wiley

SeriesinProbabilityandStatistics2004.

36

Diebold, F. X. and Li, C., Forecasting the Term Structure of Government Bond Yields, Journal of

Econometrics,Vol.130(2006),337364.

Diebold,F.X.,Li,C.,andYue,V.Z.,GlobalYieldCurveDynamicsandInteractions:AGeneralizedNelson

SiegelApproach,JournalofEconometrics,Vol.146(2008),351363.

Dullmann,K.andUhrigHomburg,M.ProfitingRiskStructureofInterestRates:AnEmpiricalAnalysisfor

DeutschemarkDenominatedBonds,EuropeanFinancialManagement,Vol.6,No.3(2000),367388.

EuropeanCentralBank,TheNewEuroAreaYieldCurves,MonthlyBulletin,(February2008),95103.

Fabozzi,F.J.,Martellini,L.AndPriaulet,P.,PredictabilityintheShapeoftheTermStructureofInterest

Rates,JournalofFixedIncome,Vol.15,No.1(June2005)4053.

Fisher,M.,Nychka,D., and Zervos,D. Fitting the Term Structure of Interest Rateswith Smoothing

Splines,FinanceandEconomicsDiscussionSeries,FederalReserveBoard,September,1994.

Gurkaynak, R. S., Sack, B. andWright, J. H., The U.S. Treasury Yield Curve: 1961 to the Present,

Working paper in the Finance and Economics Discussion Series (FEDS), Federal Reserve Board,

WashingtonD.C.200628,2006.

Hurn, A.S., K.A. Lindsay, and V. Pavlov, Smooth Estimation of Yield Curves by Laguerre Function,

QueenslandUniversityofTechnologyWorkingPaper(2005),10421048.

Kutner,M.H.,Nachtsheim,C. J.,Neter, J.,andLi,W.,AppliedLinearStatisticalModels (5thEdition),

McGrawHill,2004.

Litterman,R.and J.Scheinkman,CommonFactorsAffectingBondReturns, JournalofFixed Income,

Vol.1,No.1(June1991),5461.

Martellini, L, andMeyfredi, J. C., A Copula Approach to ValueatRisk Estimation for FixedIncome

Portfolios,JournalofFixedIncome,Vol.17,No.1(June2007),515.

Nelson, C., and Siegel, A. F., ParsimoniousModeling of Yield Curves, Journal of Business, Vol. 60

(October1987),473489.

Seber,G.A.F.,andWild,C.J.,NonlinearRegression,WileySeriesinProbabilityandStatistics,2003.

37

Steeley, J. M., Estimating the GiltEdged Term Structure: Basis Splines and Confidence Intervals,

JournalofBusinessFinance&Accounting,Vol.18,No.4(June1991),513529.

Svensson, L. E. O., Estimating and Interpreting Forward Interest Rates: Sweden 19921994, IMF

WorkingPaper,WP/94/114(September1994),149.

Svensson, L. E. O., Estimating the Term Structure of Interest Rates forMonetary Policy Analysis,

ScandinavianJournalofEconomics,Vol.98(1996),163183

Vasicek,O.A.,andFong,H.G.,TermStructureModelingUsingExponentialSplines,JournalofFinance,

Vol.73(May1982),339348.