The Affine Arbitrage-Free Class of Nelson-Siegel Term Structure Models

39 Pages·2009·0.31 MB·English

JournalofEconometrics164(2011)4–20
ContentslistsavailableatScienceDirect
Journal of Econometrics
journalhomepage:www.elsevier.com/locate/jeconom
The affine arbitrage-free class of Nelson–Siegel term structure models
JensH.E.Christensena,FrancisX.Dieboldb,∗,GlennD.Rudebuscha
aFederalReserveBankofSanFrancisco,UnitedStates
bUniversityofPennsylvaniaandNBER,UnitedStates
a r t i c l e i n f o a b s t r a c t
Articlehistory: Wederivetheclassofaffinearbitrage-freedynamictermstructuremodelsthatapproximatethewidely
Availableonline26February2011 used Nelson–Siegel yield curve specification. These arbitrage-free Nelson–Siegel (AFNS) models can
be expressed as slightly restricted versions of the canonical representation of the three-factor affine
JELclassification:
arbitrage-freemodel.ImposingtheNelson–Siegelstructureonthecanonicalmodelgreatlyfacilitates
E43
estimationandcanimprovepredictiveperformance.Inthefuture,AFNSmodelsappearlikelytobea
G12
usefulworkhorserepresentationfortermstructureresearch.
G17
©2011ElsevierB.V.Allrightsreserved.
Keywords:
Yieldcurve
Interestrate
Bondmarket
Factormodel
Forecasting
1. Introduction Unfortunately, the canonical affine AF models often exhibit
poorempiricaltimeseriesperformance,especiallywhenforecast-
Understandingthedynamicevolutionoftheyieldcurveisim- ingfutureyields(Duffee,2002).Inaddition,andcrucially,thees-
portantformanytasks,includingpricingfinancialassetsandtheir timationofthosemodelsisknowntobeproblematic,inlargepart
derivatives, managing financial risk, allocating portfolios, struc- becauseoftheexistenceofnumerouslikelihoodmaximathathave
turing fiscal debt, conducting monetary policy, and valuing cap- essentiallyidenticalfittothedatabutverydifferentimplications
italgoods.Toinvestigateyieldcurvedynamics,researchershave foreconomicbehavior.Theempiricalproblemsappeartoreflectan
produced a vast literature with a wide variety of models. How-
underlyingmodelover-parameterization,andasasolution,many
ever,thosemodelstendtobeeithertheoreticallyrigorousbutem-
researchers(e.g.Duffee,2002;DaiandSingleton,2002)simplyre-
piricallydisappointing,orempiricallysuccessfulbuttheoretically
strict to zero those parameters with small t-statistics in a first
lacking.Inthispaper,weintroduceatheoreticallyrigorousyield
round of estimation. The resulting more parsimonious structure
curve model that simultaneously displays empirical tractability,
is typically somewhat easier to estimate and has fewer trouble-
goodfit,andgoodforecastingperformance.
somelikelihoodmaxima.However,theadditionalrestrictionson
Becausebondstradeindeepandwell-organizedmarkets,the
model structure are not well motivated theoretically or statisti-
theoretical restrictions that eliminate opportunities for riskless
cally,andtheirarbitraryapplicationandthecomputationalbur-
arbitrage across maturities and over time hold powerful appeal,
denofestimationeffectivelyprecluderobustmodelvalidationand
and they provide the foundation for a large finance literature
thoroughsimulationstudiesofthefinite-samplepropertiesofthe
on arbitrage-free (AF) models that started with Vasiček (1977)
estimators.
and Cox et al. (1985). Those models specify the risk-neutral
evolution of the underlying yield curve factors as well as the Inparttoovercometheproblemswithempiricalimplementa-
dynamics of risk premia. Following Duffie and Kan (1996), the tionofthecanonicalaffineAFmodel,wedevelopinthispapera
affineversionsofthosemodelsareparticularlypopular,because newclassofaffineAFmodelsbasedontheworkhorseyieldcurve
yieldsareconvenientlinearfunctionsofunderlyinglatentfactors representation introduced by Nelson and Siegel (1987) and ex-
(statevariablesthatareunobservedbytheeconometrician)with tendedtodynamicenvironmentsbyDieboldandLi(2006).(Were-
parameters, or ‘‘factor loadings’’, that can be calculated from a fertotheDiebold–LiextensionasdynamicNelson–Siegel,orDNS.)
simplesystemofdifferentialequations. Thus,fromoneperspective,wetakethetheoreticallyrigorousbut
empiricallyproblematicaffineAFmodelandmakeitempirically
tractablebyincorporatingDNSelements.
∗ Correspondingauthor.Tel.:[phone];fax:[phone]. Fromanalternativeperspective,wetaketheDNSmodel,which
E-mailaddress:[email protected](F.X.Diebold). is empirically successful but theoretically lacking, and make it
0304-4076/$–seefrontmatter©2011ElsevierB.V.Allrightsreserved.
doi:10.1016/j.jeconom.2011.02.011
J.H.E.Christensenetal./JournalofEconometrics164(2011)4–20 5
(τ) τ
rigorousbyimposingabsenceofarbitrage.Thisrigorisimportant wherey isthezero-couponyieldwith monthstomaturity,
β ,β ,β λ
becausetheNelson–Siegelmodelisextremelypopularinpractice, and 0 1 2,and areparameters.
among both financial market practitioners and central banks As noted earlier, this representation is commonly used by
(e.g., Svensson, 1995; Bank for International Settlements, 2005; central banks and financial market practitioners to fit the cross-
Gürkaynak et al., 2007). DNS’s popularity stems from several section of yields. Although such a static representation is useful
sources, both empirical and theoretical, as discussed in Diebold for some purposes, a dynamic version is required to understand
the evolution of the bond market over time. Hence Diebold and
andLi(2006).Empirically,theDNSmodelissimpleandstableto
β
Li (2006) suggest allowing the coefficients to vary over time,
estimate, and it is quite flexible and fits both the cross-section
inwhichcase,giventheirNelson–Siegelloadings,thecoefficients
andtimeseriesofyieldsremarkablywell,inmanycountriesand
may be interpreted as time-varying level, slope and curvature
periods,andformanygradesofbonds.Theoretically,DNSimposes
factors.Toemphasizethis,were-writethemodelas
certain economically desirable properties, such as requiring the
discountfunctiontoapproachzerowithmaturity,andDieboldand y (τ)=L +S 1−e−λτ+C 1−e−λτ −e−λτ. (2)
Li(2006)showthatitcorrespondstoamodernthree-factormodel t t t λτ t λτ
of time-varying level, slope and curvature. However, despite its
DieboldandLiassumeanautoregressivestructureforthefactors,
goodempiricalperformanceandacertainamountoftheoretical
which produces the DNS model, a fully dynamic Nelson–Siegel
appeal,DNSfailsonanimportanttheoreticaldimension:itdoes
specification. Indeed, it is a state-space model, with the yield
not impose the restrictions necessary to eliminate opportunities
factorsasstatevariables,asemphasizedinDieboldetal.(2006).
for riskless arbitrage (e.g. Filipović, 1999; Diebold et al., 2005).
Empirically,theDNSmodelishighlytractableandtypicallyfits
This motivates us in this paper to introduce the class of AF well.Theoretically,however,itdoesnotrequirethatthedynamic
Nelson–Siegel(AFNS)models,whichareaffineAFtermstructure evolution of yields cohere such that arbitrage opportunities are
modelsthatmaintaintheDNSfactorloadingstructure. precluded. Indeed, the results of Filipović (1999) imply that
In short, the AFNS models proposed here combine the best whatever stochastic dynamics are chosen for the DNS factors, it
of the AF and DNS traditions. Approached from the AF side, isimpossibletoprecludearbitrageatthebondpricesimplicitin
they maintain the AF theoretical restrictions of the canonical theresultingNelson–Siegelyieldcurve.Inthenextsubsection,we
affine models but can be easily estimated, because the dynamic showhowtoremedythistheoreticalweakness.
Nelson–Siegel structure helps to identify the latent yield curve
factors and delivers analytical solutions (which we provide) for 2.2. Thearbitrage-freeNelson–Siegelmodel
zero-coupon bond prices. Approached from the DNS side, they
maintain the simplicity and empirical tractability of the popular Our derivation of the AFNS model starts from the standard
DNS models, while simultaneously enforcing the theoretically continuous-timeaffineAFstructureofDuffieandKan(1996).1To
desirablepropertyofabsenceofrisklessarbitrage. representanaffinediffusionprocess,defineafilteredprobability
space(Ω,F,(F ),Q),wherethefiltration(F ) = {F : t ≥ 0}
AfterderivingthenewclassofAFNSmodels,weexaminetheir t t t
satisfiestheusualconditions(Williams,1997).Thestatevariable
in-sample fit and out-of-sample forecast performance relative
X isassumedtobeaMarkovprocessdefinedonasetM ⊂Rnthat
to standard DNS models. For both the DNS and the AFNS t
solvesthestochasticdifferentialequation(SDE),2
models,weestimateparsimoniousandflexibleversions(withboth
independent factors and more richly parameterized correlated dX =KQ(t)[θQ(t)−X ]dt+Σ(t)D(X ,t)dWQ, (3)
t t t t
factors). We find that the flexible versions of both models are
whereWQ isastandardBrownianmotioninRn,theinformation
preferred for in-sample fit, but that the parsimonious versions ofwhichiscontainedinthefiltration(F ).Thedriftsanddynamics
t
exhibit significantly better out-of-sample forecast performance. θQ :[0,T]→RnandKQ :[0,T]→Rn×narebounded,continuous
As a final comparison, we also show that an AFNS model can functions.3 Similarly, the volatility matrix Σ : [0,T] → Rn×n
outperformthecanonicalaffineAFmodelinforecasting. is a bounded, continuous function, while D : M × [0,T] →
We proceed as follows. First we present the main theoretical Rn×n has a diagonal structure with ith diagonal entry given by
results of the paper; in Section 2 we derive the AFNS class of 
γi(t)+δi(t)x1+···+δi(t)xn.
models,andinSection3wecharacterizethepreciserelationship 1 t n t
γ( ) δ( )
betweentheAFNSclassandthecanonicalrepresentationofaffine Tosimplifythenotation, t and t aredefinedas
AFmodels.Wenextprovideanempiricalanalysisoffourleading γ1(t) δ1(t) ··· δ1(t)
DfleNxSibalendverAsFioNnSs;minodSeelcst,ioinnc4orwpoeraetxinagmibnoethin-psaarmsipmleonfiito,uasndanind γ(t)= ...  and δ(t)= 1... ... n... ,
Section 5 we examine out-of-sample forecasting performance. γn(t) δn(t) ··· δn(t)
1 n
WeconcludeinSection6,andweprovideproofsandadditional where γ : [0,T] → Rn and δ : [0,T] → Rn×n are bounded,
technicaldetailsinseveralappendices. continuous functions. Given this notation, the SDE of the state
variablescanbewrittenas
2. Nelson–Siegeltermstructuremodels dX =KQ(t)[θQ(t)−X ]dt+Σ(t)
t t
γ1(t)+δ1(t)X ··· 0 
AFHafefirneewteerrmevisetwructhtuerDeNmSomdeoldsetlhaantdminatirnotdaiuncethteheNAeFlsNoSnc–lSaisesgoefl ×  ... t ... ... dWtQ,
factorloadingstructure. 0 ··· γn(t)+δn(t)X
t
2.1. ThedynamicNelson–Siegelmodel
1 Krippner(2006)derivesaspecialcaseoftheAFNSmodelwithconstantrisk
TheoriginalNelson–Siegelmodelfitstheyieldcurvewiththe
premiums.
simplefunctionalform 2 Notethat(3)referstotherisk-neutral(‘‘Q’’)dynamics.
y(τ)=β +β 1−e−λτ+β 1−e−λτ −e−λτ, (1) ei3geSntvaatliuoensaroitfyKoQf(tt)hearestpatoesivtiavrei;absleeesAishnenestuarle.d(2i0f0t2h)e.Hreoawlecvoemr,psotanteionntsaroitfyailsl
0 1 λτ 2 λτ notanecessaryrequirementfortheprocesstobewelldefined.
6 J.H.E.Christensenetal./JournalofEconometrics164(2011)4–20
where δi(t) denotes the ith row of the δ(t) matrix. Finally, the where the state variables X = (X1,X2,X3) are described by the
t t t t
instantaneousrisk-freerateisassumedtobeanaffinefunctionof followingsystemofSDEsundertherisk-neutralQ-measure
thestatevariables dXt1 0 0 0 θ1Q Xt1
rt =ρ0(t)+ρ1(t)′Xt, dX2 = 0 λ −λ θQ−X2dt
 t λ  2  t
where ρ : [0,T] → R and ρ : [0,T] → Rn are bounded, dX3 0 0 θQ X3
0 1 t 3 t
continuousfunctions. dW1,Q
Duffie and Kan (1996) prove that zero-coupon bond prices t
in this framework are exponential affine functions of the state +ΣdWt2,Q, λ>0.
variables, dW3,Q
t
[  ∫ T ] Thenzero-couponbondpricesare
P(t,T)=EQ exp − r du =exp(B(t,T)′X +A(t,T)),
t t u t [  ∫ T ]
P(t,T) = EQ exp − r du
where B(t,T) and A(t,T) are the solutions to the system of t u
t
ordinarydifferentialequations(ODEs) = exp(B1(t,T)X1+B2(t,T)X2+B3(t,T)X3+A(t,T)),
t t t
( , )
dB t T =ρ +(KQ)′B(t,T) whereB1(t,T),B2(t,T),B3(t,T),andA(t,T)arethesolutionstothe
1
dt systemofODEs:
−1−n (Σ′B(t,T)B(t,T)′Σ)j,j(δj)′, B(T,T)=0, (4) dB1(t,T)
2
dA(dtt,T)j==1ρ0−B(t,T)′KQθQ dB2dd(ttt,T)=110+000 −λ0λ λ00BBB123(((ttt,,,TTT))) (6)
dB3(t,T)
−1−n (Σ′B(t,T)B(t,T)′Σ)j,jγj, A(T,T)=0, (5) dt
2
j=1 and
ainndththeenpootsastiibolne.tiTmheedperpiceinndgenfucnecotfiothnespimarpamlyettherastiszesruop-pcroeuspsoend dA(t,T) =−B(t,T)′KQθQ − 1−3 (Σ′B(t,T)B(t,T)′Σ)j,j, (7)
yieldsare dt 2 j=1
y(t,T)=−T −1 t logP(t,T)=−BT(t−,Tt)′Xt − AT(t−,Tt). wA(iTth,Tb)ou=nd0a.rTyhecosnolduittiioonnstoBt1h(iTs,sTys)tem=ofBO2(DTE,sTis): = B3(T,T) =
Given the pricing functions, for a three-factor affine model with B1(t,T)=−(T −t),
Xfutnc=tio(nXit1s,aXyt2i,eXldt3)f,utnhcetiocnloosefstthemfaotrcmh4to the Nelson–Siegel yield B2(t,T)=−1−eλ−λ(T−t),
y(t,T) = Xt1+ 1λ−(Te−−λ(Tt−)t)Xt2 B3(t,T)=(T −t)e−λ(T−t)− 1−eλ−λ(T−t),
+ [1−e−λ(T−t) −e−λ(T−t)]X3− A(t,T), A(t,T)=(KQθQ)2∫ TB2(s,T)ds+(KQθQ)3∫ TB3(s,T)ds
λ(T −t) t T −t t t
withODEsfortheB(t,T)functionsthathavethesolutions +1−3 ∫ T(Σ′B(s,T)B(s,T)′Σ)j,jds.
2
B1(t,T)=−(T −t), j=1 t
1−e−λ(T−t) Finally,zero-couponbondyieldsare
B2(t,T)=− λ , 1−e−λ(T−t)
B3(t,T)=(T −t)e−λ(T−t)− 1−e−λ(T−t). y(t,T) = Xt1+ λ(T −t) Xt2
λ + [1−e−λ(T−t) −e−λ(T−t)]X3− A(t,T).
InthiscasethefactorloadingsexactlymatchNelson–Siegel,but λ(T −t) t T −t
there is an unavoidable ‘‘yield-adjustment term’’, −A(t,T), which
T−t
dependsonlyonthematurityofthebond,notontime.Asdescribed Proof. SeeAppendixA. (cid:3)
inthefollowingproposition,thereexistsaclassofaffineAFmodels
TheexistenceofanAFNSmodel,asdefinedinProposition1,is
thatsatisfiestheaboveODEs.
relatedtotheworkofTrolleandSchwartz(2009),whoshowthat
thedynamicsoftheforwardratecurveinageneralm-dimensional
Proposition1. Supposethattheinstantaneousrisk-freerateis
Heath–Jarrow–Morton (HJM) model can always be represented
r =X1+X2, byafinite-dimensionalMarkovprocesswithtime-homogeneous
t t t volatilitystructureifeachvolatilityfunctionisgivenby
σi(t,T)=pn,i(T −t)e−γi(T−t), i=1,...,m,
(τ) τ
4 Onecouldofcoursedefine‘‘closest’’inotherways.Ourstrategyistofindthe wherepn,i isannth-orderpolynomialin .Becausetheforward
affineAFmodelwithfactorloadingsthatmatchNelson–Siegelexactly. ratesintheDNSmodelsatisfythisrequirement,thereexistssuch
J.H.E.Christensenetal./JournalofEconometrics164(2011)4–20 7
an AF three-dimensional HJM model. However, the simplicity of theyield-adjustmenttermcanbederivedinanalyticalform(see
thesolutioninthecaseoftheNelson–Siegelmodelpresentedin AppendixB)as
Proposition1isstriking.
threPerosptoatseitivoanri1aballesosharaessGeavuesrsailainntOerrenssttienign–imUhplleicnabteiocnksp.Friorcset,stshees AT(t−,Tt) = 21T −1 t ∫tT−j=31(Σ′B(s,T)B(s,T)′Σ)j,jds
wrthaitetehciausrcvtohanetustrsaeunmtfavcootfloarlte’islviet(ylX3am)nadstroislxleoΣpreo.5lefaTchitseoriasnss(taXatn1sttaaonncedhoauXsstt2i)ci,ntwteimrheielse-t =A(T −6t)2 +B[21λ2 − λ131−Te−−λ(tT−t) + 41λ31−Te−−2λt(T−t)]
t
varyingmeanfortheslopefactorundertheQ-measure.Second, 
Proposition1onlyimposesstructureonthedynamicsoftheAFNS +C 1 + 1 e−λ(T−t)− 1 (T −t)e−2λ(T−t)− 3 e−2λ(T−t)
2λ2 λ2 4λ 4λ2
model under the Q-measure and is silent about the dynamics
under the P-measure. Still, the very indirect role of curvature 2 1−e−λ(T−t) 5 1−e−2λ(T−t)
generallyaccordswiththeempiricalliteraturewhereithasbeen − +
difficult to find sensible interpretations of curvature under the λ3 T −t 8λ3 T −t
Pun-mit-eraosoutrepr(oDcieesbsouldndeetratlh.,e2Q00-6m).eaSsimuriela,rwlyh,itchhealcecvoerldsfawctiotrhitshea +D[ 1 (T −t)+ 1 e−λ(T−t)− 1 1−e−λ(T−t)]
2λ λ2 λ3 T −t
usualfindingthatoneormoreoftheinterestratefactorsareclose
tPλorobpeoisnigtionnon1stpartoiovnidaersyipnrsoigchetssienstounthdeernathtuereP-omfetahseurpea.r6amTheitredr, +Eλ32e−λ(T−t)+ 21λ(T −t)
. Although a few authors (e.g. Koopman et al., 2010) have
caonndsihdaesretdhetiminet-evraprryeitnagtioλn, iatsisthaecmonesatna-nrtevinersthioenArFaNteS omfotdheel +1(T −t)e−λ(T−t)− 3 1−e−λ(T−t)
λ λ3 T −t
curvatureandslopefactorsaswellasthescalebywhichadeviation
of the curvature factor from its mean affects the mean of the 
slope factor. Fourth, and crucially, AFNS contains an additional +F 1 + 1 e−λ(T−t)− 1 e−2λ(T−t)
maturity-dependent term −A(t,T) relative to DNS. This ‘‘yield- λ2 λ2 2λ2
T−t
adjustment’’termisakeydifferencebetweenDNSandAFNS,and 3 1−e−λ(T−t) 3 1−e−2λ(T−t)
wenowexamineitindetail. − + ,
λ3 T −t 4λ3 T −t
2.3. Theyield-adjustmentterm
whereA=σ2 +σ2 +σ2,B=σ2 +σ2 +σ2,C =σ2 +σ2 +
11 12 13 21 22 23 31 32
TheonlyparametersinthesystemofODEsfortheAFNSB(t,T) σ323,D=σ11σ21+σ12σ22+σ13σ23,E =σ11σ31+σ12σ32+σ13σ33,
functionsareρ1andKQ,i.e.,thefactorloadingsofrtandthemean- andF =σ21σ31+σ22σ32+σ23σ33.
reversionstructureforthestatevariablesundertheQ-measure. Thisresulthastwoimplications.First,thefactthatAFNSzero-
ThedrifttermθQ andthevolatilitymatrixΣdonotappearinthe coupon bond yields are given by an analytical formula greatly
ODEs,butratherintheyield-adjustmentterm−A(t,T).Henceinthe facilitatesempiricalimplementationofAFNSmodels.Second,the
AFNSmodelthechoiceofthevolatilitymatrixΣT−atffectsboththe nine underlying volatility parameters are not identified. Indeed,
, , , ,
P-dynamicsandtheyieldfunctionthroughtheyield-adjustment onlythesixtermsA B C D E,andF canbeidentified;hencethe
term. In contrast, the DNS model is silent about the real-world maximally flexible AFNS specification that can be identified has
dynamics of the state variables, so the choice of P-dynamics is triangularvolatilitymatrix7
irrelevantfortheyieldfunction. σ 0 0 
11
Asdiscussedinthenextsection,weidentifytheAFNSmodelsby Σ = σ σ 0 .
21 22
fixingthemeanlevelsofthestatevariablesundertheQ-measure σ σ σ
at0,i.e.θQ = 0.Thisimpliesthattheyield-adjustmenttermisof 31 32 33
Laterwewillquantifytheyield-adjustmenttermandexamine
theform:
howitaffectsempiricalperformanceinleadingspecifications,to
−AT(t−,Tt) =−12T −1 t −3 ∫ T(Σ′B(s,T)B(s,T)′Σ)j,jds. whichwenowturn.
j=1 t 2.4. FourspecificNelson–Siegelmodels
Givenageneralvolatilitymatrix
In general, the DNS and AFNS models are silent about the P-
σ σ σ 
11 12 13 dynamics,sothereareaninfinitenumberofpossiblespecifications
Σ = σ σ σ ,
21 22 23 thatcouldbeusedtomatchthedata.However,forcontinuitywith
σ σ σ
31 32 33 theexistingliterature,wefocusontwoversionsoftheDNSmodel
thathavefeaturedprominentlyinrecentstudies,examiningthe
effectsofimposingabsenceofarbitrage.
Intheindependent-factorDNSmodel,thethreestatevariables
5 Proposition1canbeextendedtoincludejumpsinthestatevariables.Aslong areindependentfirst-orderautoregressions,asinDieboldandLi
asthejumparrivalintensityisstateindependent,theNelson–Siegelfactorloading
structureintheyieldfunctionismaintainedbecauseonlyA(t,T)isaffectedbythe (2006).Thestatetransitionequationis
iOinm6DcplEulWifseoiisorthtnAha(oatfut,wsnTuii)tctrhhinoajoutnthmiuinsnptcbsha.oesSueelne.edvDeelduffhafcioetroiezrt,o−anl.TA(T(2tt−,h0Tte)00m→)ofdo−erl∞tihsenansoetmeaadrbteuidtrrimatygoeidnficrfericeeaa.tTsiehosne,rwoeffhotirhcehe, CSLttt−−−µµµSCL=a0011 a0022 a0033CSLttt−−−111−−−µµµSCL+ηηηttt(((CSL))),
as is often done in theoretical discussions, we impose an arbitrary maximum
horizon.Alternatively,wecouldmodifythemean-reversionmatrixKQtoincludea
sufficientlysmallε>0intheupperleft-handpositiontoobtainanAFmodelthat
isindistinguishablefromtheAFNSmodelinProposition1. 7 Thechoiceofupperorlowertriangularisirrelevant.
8 J.H.E.Christensenetal./JournalofEconometrics164(2011)4–20
η ( ),η ( ) η ( )
where the stochastic shocks L S , and C have covari- Inthecorrelated-factorAFNSmodel,thethreestatevariablesmay
t t t
ancematrix interactdynamicallyand/ortheirshocksmaybecorrelated,
 
Q =q0211 q02 00 . dXt1 κ1P1 κ1P2 κ1P3θ1P Xt1
0 022 q2 dXt2 = κ2P1 κ2P2 κ2P3θ2P−Xt2dt
33 dX3 κP κP κP θP X3
In the correlated-factor DNS model, the state variables follow a t 31 32 33 3 t
first-ordervectorautoregression,asinDieboldetal.(2006).The σ 0 0 dW1,P
11 t
traLnts−itiµonLequatiao1n1is a12 a13Lt−1−µL ηt(L) + σσ2311 σσ2322 σ033 ddWWtt23,,PP.
St −µS = a21 a22 a23 St−1−µS + ηt(S) ,
Ct −µC a31 a32 a33 Ct−1−µC ηt(C) ThisisthemostflexibleAFNSmodelwithallparametersidentified.
η ( ),η ( ) η ( ) Inboththeindependent-factorandcorrelated-factorAFNSmodels,
where the stochastic shocks L S , and C have covari-
ancematrixQ =qq′,where t t t themeasurementequationis
q=qq11 q0 00 . 1 1−λeτ−λτ1 1−λeτ−λτ1 −e−λτ1
Itnhebomtheqa23ts11huereiqnm23d22eenpteqen3qd3ueantti-ofnacitsorandcorrelated-factorDNSmodels, yytt((...ττ12)) = 1... 1−λeτ...−12λτ2 1−λeτ−12λτ2... −e−λτ2XXXtt123
yyytt(((...τττ12))) = 11... 11−−λλeeττ...−−12λλττ12 11−−λλeeττ−−12λλττ12... −−ee−−λλττ12 yt(τN) 1AA1((τττ−121λ))eτ−NλτNεε1t((−ττ1λ))eτ−NλτN −e−λτN t
t N 1 1−λeτ−NλτN 1−λeτ−NλτN −e−λτN −  τ...2 +εt(...τ2), (10)
ε (τ ) A(τ ) t N
× CSLttt+εεtt((...ττ12)), (8) wherethemeasuremτeNnNterrorsεt(τi)arei.i.d.noise.
t N
ε (τ)
wherethemeasurementerrors arei.i.d.whitenoise.
t i 3. TheAFNSsubclassofcanonicalaffineAFmodels
ThecorrespondingAFNSmodelsareformulatedincontinuous
time, and the relationship between the real-world dynamics
undertheP-measureandtherisk-neutraldynamicsundertheQ- BeforeproceedingtoanempiricalanalysisofthevariousDNS
measureisgivenbythemeasurechange andAFNSmodels,wefirstanswerakeytheoreticalquestion:What,
dWQ =dWP +Γ dt, precisely,aretherestrictionsthattheAFNSmodelimposesonthe
t t t canonicalrepresentationofthree-factoraffineAFmodels?8
whereΓtrepresentstheriskpremium.Topreserveaffinedynamics DenotingthestatevariablesbyYt,thecanonicalA0(3)modelis
undertheP-measure,welimitourfocustoessentiallyaffinerisk
premiumspecifications(seeDuffee,2002),inwhichcaseΓt takes rt =δ0n+(δ1n)′Yt
theformγ0 γ1 γ1 γ1X1 dYt =KnP[θnP −Yt]dt+ΣndWtP
Γt =γ210+γ21111 γ21122 γ21133Xtt2. dYt =KnQ[θnQ −Yt]dt+ΣndWtQ,
γ30 γ311 γ312 γ313 Xt3 with δn ∈ R,δn,θP,θQ ∈ R3, and KP,KQ,Σ ∈ R3×3.9 If the
0 1 n n n n n
WiththisspecificationtheSDEforthestatevariablesundertheP- essentially affine risk premium specification Γ = γ0 + γ1Y
measure, t n n t
is imposed on the model, the drift terms under the P-measure
dreXmt a=inKsPa[fθfiPne−.DXut]edtto+thΣefdleWxitPb,lespecificationofΓt,wearefree(9to) (QK-nPm,eθanPs)ucraen(KbenQc,hθonQs)e.nindependentlyofthedrifttermsunderthe
chooseanymeanvectorθP andmean-reversionmatrixKP under Becausethelatentstatevariablesmayrotatewithoutchanging
theP-measureandstillpreservetherequiredQ-dynamicstructure the probability distribution of bond yields, not all parameters
describedinProposition1.HencewefocusonthetwoAFNSmodels in the above model can be identified. Singleton (2006) imposes
thatcorrespondtothetwoDNSmodelsabove. identifyingrestrictionsundertheQ-measure.Specifically,hesets
Intheindependent-factorAFNSmodel,thethreestatevariables themeanθnQ = 0,thevolatilitymatrixΣn equaltotheidentity
areindependentundertheP-measure, matrix, and he sets the mean-reversion matrix KQ equal to a
n
dX1 κP 0 0 θP X1
t 11 1 t
dX2 =  0 κP 0 θP−X2dt
t 22 2 t
dXt3 0 0 κ3P3 θ3P Xt3 8 BythiswemeantheA0(3)representationwiththreestatevariablesandzero
+ σ001 σ002 σ003dddWWWtt123,,,PPP. sasqur9eupaedNrrieosffc-terreroieptohnttsatptofrrYfoot‘c‘mnde’es’tsndheoeesnt,XeoasttsestdhtcaeeotteesatfivalfeiatcderiieavinnbatlrSseiiasinnbiglntleehsttehooecnfaAt(nh2FoeN0n0cSia6cmn,aoClonrhdeiaecpplars.le,1rsa2eenp)nd.rteathstieaontnts.autbiosncr,iwpthsicohr
t
J.H.E.Christensenetal./JournalofEconometrics164(2011)4–20 9
Table1
AFNSparameterrestrictionsontheA0(3)canonicalrepresentation.
AFNSmodel δn,δn KQ KP θP Restrictions
0 1 n n n
Independentfactor δδ01nn,3==00 κκ12nn,,,,12QQ ==κκ13nn,,,,23QQ =κ1n,,3Q =0 KnPisdiagonal Norestriction 12
Correlatedfactor δ0n=0 κκ1nn,,,1QQ ==κ0n,Q Norestriction Norestriction 3
2,2 3,3
triangular matrix.10 The canonical representation then has Q- form solution and, as described in the next section, eliminate in
dynamics anappealingwaythesurfeitoftroublesomelikelihoodmaximain
dY1 κn,Q κn,Q κn,QY1 estimation.
dYt2 = − 101 κ1n2,Q κ1n3,QYt2dt
dYt3 0 202 κ2n3,Q Yt3 4. Estimationandin-samplefitofDNSandAFNSmodels
t 33 t
1 0 0dW1,Q ThusfarwehavederivedtheaffineAFclassofNelson–Siegel
t
+ 00 10 01 ddWWt23,,QQ, treersmtricsttirouncstutrheamt oitdepllsa,caensdownethheavceanexopnliiccailtlyA0c(h3a)ramctoedreizle.dHtehree
t we undertake estimation of the AFNS model and illustrate its
andP-dynamics relativesimplicity.Weproceedinseveralsteps.Firstweintroduce
dY1 κn,P κn,P κn,Pθn,P Y1 the state-space/Kalman-filter maximum-likelihood estimation
dYt2 = κ1n1,P κ1n2,P κ1n3,Pθ1n,P−Yt2dt frameworkthatweemploythroughout.Second,weestimateand
dYt3 κ2n1,P κ2n2,P κ2n3,P θ2n,P Yt3 compare independent- and correlated-factor DNS models. Third,
t 31 32 33 3 t we estimate independent- and correlated-factor AFNS models,
1 0 0dW1,P whichwecomparetoeachotherandtotheirDNScounterparts,
t
+ 0 1 0 dW2,P. devoting special attention to the estimated yield-adjustment
t
0 0 1 dW3,P terms. Throughout, our estimates are based on monthly US
t TreasurybillandbondyieldsfromJanuary1987toDecember2002.
Theinstantaneousrisk-freerateis The data are end-of-month, unsmoothed (Fama and Bliss, 1987)
rt =δ0n+δ1n,1Yt1+δ1n,2Yt2+δ1n,3Yt3. zero-couponyieldsatsixteenmaturities:3,6,9,12,18,24,36,48,
60,84,96,108,120,180,240,and360months.
Hencethereare22freeparametersinthecanonicalrepresentation
oftheA (3)modelclass.11
0
In the AFNS class, the mean-reversion matrix under the Q- 4.1. Estimationframework
measure is triangular, so it is straightforward to derive the
restrictions that must be imposed on the canonical affine Wefirstdisplaythestate-spacerepresentationsoftheDNSand
representationtoobtaintheclassofAFNSmodels.Theprocedure AFNSmodels.FortheDNSmodels,thestatetransitionequationis
throughwhichtherestrictionsareidentifiedisbasedontheso- Xt =(I−A)µ+AXt−1+ηt,
calledaffineinvarianttransformations.AppendixCdescribessuch
transformations and derives the restrictions associated with the whereXt =(Lt,St,Ct),andthemeasurementequationis
AFNSmodelsconsideredinthispaper.Theresultsaresummarized y =BX +ε . (11)
t t t
inTable1,whichshowsthatforthecorrelated-factorAFNSmodel
therearethreekeyparameterrestrictionsonthecanonicalaffine For the continuous-time AFNS models, the conditional mean
model.First,δn = 0,sothereisnoconstantintheequationfor vectorandtheconditionalcovariancematrixare
0
tbheecainusstea,nwtaitnhetohuessreicsko-nfdrereesrtartiec.tTiohnerκe1n,,i1Qsn=o0n,etehdeffoirrstthfiascctoornmstaunstt EP[XT|Ft]=(∫I−(cid:49)texp(−KP(cid:49)t))θP +exp(−KP(cid:49)t)Xt,
beaunit-rootprocessundertheQ-measure,whichalsoimplies VP[X |F ]= e−KPsΣΣ′e−(KP)′sds,
thatthisfactorcanbeidentifiedasthelevelfactor.Finally,κ2n,,2Q = T t 0
κ3n,,3Q, so the own mean-reversion rates of the second and third where(cid:49)t =T −t.Wecomputeconditionalmomentsofdiscrete
factorsundertheQ-measuremustbeidentical.Theindependent- observationsandobtaintheAFNSstatetransitionequation
faadcdtosrnAinFeNoStmheordseulnmdearinbtoaitnhstthheePt-harenedpQa-rmameaesteurrerse.s1t2rictionsand Xt =(I−exp(−KP(cid:49)t))θP +exp(−KP(cid:49)t)Xt−1+ηt,
The Nelson–Siegel parameter restrictions on the canonical where(cid:49)t isthetimebetweenobservations.TheAFNSmeasure-
affineAFmodelgreatlyfacilitateestimation.13Theyallowaclosed- mentequationis14
y =A+BX +ε .
t t t
10 Withoutlossofgenerality,wewilltakeittobeuppertriangularinwhatfollows. In both the DNS and AFNS environments, the assumed error
11 Notethat,giventhiscanonicalrepresentation,thereisnolossofgeneralityin structureis
fixingtheAFNSmodelmeanundertheQ-measureat0andleavingthemeanunder η  [   ]
theP-measure,θP,tobeestimated. εt ∼N 0 , Q 0 ,
12 For both specifications, there is a further modest restriction described in t 0 0 H
AppendixC:κ2n,,3Q musthavetheoppositesignofκ2n,,2Q andκ3n,,3Q,butitsabsolute
numericalsizecanvaryfreely.
13 NotethatintheAFNSmodel,theconnectionbetweentheP-dynamicsand
the yield function is explicitly tied to the yield-adjustment term through the 14 NotethatthematrixBisidenticalintheDNSandAFNSmodels(compare
specificationofthevolatilitymatrix,whileinthecanonicalrepresentationitis Eqs.(8)and(10)).TheonlydifferenceistheadditionofthevectorAcontainingthe
blurredbyaninterplaybetweenthespecificationsofδnandKQ. yield-adjustmenttermsintheAFNSmodels.
1 n
10 J.H.E.Christensenetal./JournalofEconometrics164(2011)4–20
wherethematrixHisdiagonal,andthematrixQ isdiagonalinthe Table2
independent-factorandnon-diagonalinthecorrelated-factorcase. Estimatedindependent-factorDNSmodel.ThetoppanelcontainstheestimatedA
matrixandµvector.Thebottompanelcontainstheestimatedqmatrix.Standard
IntheAFNScase,moreover,Q hasspecialstructure, errorsappearinparentheses.Theestimatedλis0.06040(0.00100)formaturities
∫ (cid:49)t measuredinmonths.Themaximizedloglikelihoodis16,332.94.
Q = e−KPsΣΣ′e−(KP)′sds. AMatrix Mean
0 A·,1 A·,2 A·,3 µ
In addition, thetransition and measurementerrors are assumed
A1,· 0.9827 0 0 0.0696
orthogonaltotheinitialstate. (0.0128) (0.0137)
Now we consider Kalman filtering, which we use to evaluate A2,· 0 0.9778 0 −0.0249
(0.0166) (0.0151)
the likelihood functions of the DNS and AFNS models. We
initialize the filter at the unconditional mean and variance of A3,· 0 0 0.9189 −0.0108
(0.0284) (0.0079)
the state variables under the P-measure.15 For the DNS models
wehaveX =µandΣ =V,whereVsolvesV =AVA′+Q.Forthe qMatrix
0 0
AFNSmodelswehaveX =θP andΣ =∞e−KPsΣΣ′e−(KP)′sds, q·,1 q·,2 q·,3
0 0 0
which we calculate using the analytical solutions provided in q1,· 0.0025 0 0
FisherandGilles(1996). (0.0002)
DenotetheinformationavailableattimetbyY =(y ,y ,..., q2,· 0 0.0033 0
t 1 2 (0.0002)
yt), and denote model parameters by ψ. Consider period t − 1 q3,· 0 0 0.0075
andsupposethatthestateupdateXt−1anditsmeansquareerror (0.0004)
matrixΣt−1havebeenobtained.Thepredictionstepis
Xt|t−1 =EP[Xt|Yt−1]=ΦtX,0(ψ)+ΦtX,1(ψ)Xt−1, TEsatbimlea3tedcorrelated-factorDNSmodel.ThetoppanelcontainstheestimatedA
Σt|t−1 =ΦtX,1(ψ)Σt−1ΦtX,1(ψ)′+Qt(ψ), merarotrrisxaapnpdeaµrvinecptaorre.nTthheebseost.toTmhepeasntiemlcaotendtaλinisst0h.0e6e2s4ti8m(a0t.e0d01q0m9)aftorirxm.Sattaunrditaireds
wherefortheDNSmodelswehaveΦX,0 = (I −A)µ,ΦX,1 = A, measuredinmonths.Themaximizedloglikelihoodis16,415.36.
and Q = Q, and for the AFNS modtels we have ΦX,0 t= (I − AMatrix Mean
exp(−tKP(cid:49)t))θP,ΦX,1 = exp(−KP(cid:49)t),andQ = (cid:49)tte−KPsΣΣ′ A·,1 A·,2 A·,3 µ
e−(KP)′sds,where(cid:49)ttisthetimebetweenobsertvation0s. A1,· (00..90817645) (00..00015803) −(00..00105977) (00..00174253)
In the time t update step, Xt|t−1 is improved by using the A2,· 0.0066 0.9332 0.0819 −0.0294
additionalinformationcontainedinY .Wehave (0.0228) (0.0229) (0.0202) (0.0159)
t A3,· 0.0152 0.0401 0.9011 −0.0120
Xt =E[Xt|Yt]=Xt|t−1+Σt|t−1B(ψ)′Ft−1vt, (0.0526) (0.0418) (0.0377) (0.0126)
Σt =Σt|t−1−Σt|t−1B(ψ)′Ft−1B(ψ)Σt|t−1, qMatrix
q·,1 q·,2 q·,3
where q1,· 0.0025 0 0
(0.0001)
vt =yt −E[yt|Yt−1]=yt −A(ψ)−B(ψ)Xt|t−1, q2,· −0.0022 0.0023 0
Ft =cov(vt)=B(ψ)Σt|t−1B(ψ)′+H(ψ), q3,· (00..00002083) 0(0.0.0000061) 0.0066
H(ψ)=diag(σε2(τ1),...,σε2(τN)). (0.0007) (0.0006) (0.0004)
At this point, the Kalman filter has delivered all ingredients 4.2. DNSmodelestimation
needed to evaluate the Gaussian log likelihood, the prediction-
errordecompositionofwhichis Independent-factor DNS estimates appear in Table 2, and
correlated-factorDNSestimatesappearinTable3.Inbothmodels
logl(y ,...,y ;ψ)
1 T the level factor is the most persistent, and the curvature factor
=−T −N2 log(2π)− 21log|Ft|− 12vt′Ft−1vt, iosffl-edaisatgopnearslisetleenmt.eInnttohfetchoerreeslatitmeda-tfeadctoArmDNatSrimxoisdeslt,aotinsltyicoalnlye
t=1 significant.16
Volatilityparametersaremosteasilycomparedbyconverting
where N is the number of observed yields. We numerically
ψ from Cholesky factors to conditional covariance matrices. For
maximizethelikelihoodwithrespectto usingtheNelder–Mead
independent-factorDNSwehave
simplexalgorithm.Uponconvergence,weobtainstandarderrors
fromtheestimatedcovariancematrix, QDNS = qq′
indep
Ω(ψ)= 1 1 −T ∂lo∂gψlt(ψ)∂lo∂gψlt(ψ)′−1, = 6.17×010−6 1.11×010−5 00 , (12)
T T t=1 0 0 5.58×10−5
whereψdenotestheestimatedmodelparameters. andforcorrelated-factorDNSwehave
QDNS =qq′
corr
15 WeensurecovariancestationarityundertheP-measureintheDNScaseby
restrictingtheeigenvaluesofAtobelessthan1,andintheAFNScasebyrestricting 16 Interestingly,thesignificantparameterisASt,Ct−1,whichisthekeynon-zero
therealcomponentofeacheigenvalueofKPtobepositive. off-diagonalelementrequiredinProposition1fortheAFNSspecification.
J.H.E.Christensenetal./JournalofEconometrics164(2011)4–20 11
Table4
Summarystatisticsforin-samplemodelfit.Residualmeansandrootmeansquarederrorsforsixteenmaturities.Maturitiesareinmonths;meansandRMSEsareinbasis
points.
Maturity DNSindep-factor DNScorr-factor AFNSindep-factor AFNScorr-factor
Mean RMSE Mean RMSE Mean RMSE Mean RMSE
3 −1.64 12.26 −1.84 11.96 −2.85 18.53 −2.49 11.55
6 −0.24 1.09 −0.29 1.34 −1.19 7.12 −0.03 0.64
9 −0.54 7.13 −0.51 6.92 −1.24 3.45 −0.33 6.91
12 4.04 11.19 4.11 10.86 3.58 9.60 3.72 10.14
18 7.22 10.76 7.28 10.42 7.14 10.43 5.53 8.33
24 1.18 5.83 1.19 5.29 1.37 5.94 −1.18 4.37
36 −0.07 1.51 −0.19 2.09 0.30 1.98 −1.10 3.16
48 −0.67 3.92 −0.85 4.03 −0.40 3.72 0.93 4.13
60 −5.33 7.13 −5.51 7.31 −5.27 6.82 −2.01 5.22
84 −1.22 4.25 −1.30 4.25 −1.50 4.29 0.89 3.83
96 1.31 2.10 1.29 2.02 1.03 2.11 1.05 1.83
108 0.03 2.94 0.07 3.11 −0.11 3.02 −3.23 5.26
120 −5.11 8.51 −5.01 8.53 −4.95 8.23 −11.65 14.00
180 24.11 29.44 24.40 29.66 27.87 32.66 3.85 16.53
240 25.61 34.99 26.00 35.33 35.96 42.61 4.32 23.97
360 −29.62 37.61 −29.12 37.18 1.37 22.03 −0.81 23.04
 6.03×10−6 −5.47×10−6 6.76×10−6  term structure model is very difficult and time-consuming and
=−5.47×10−6 1.01×10−5 −4.73×10−6. (13) effectivelypreventsthekindofrepetitivere-estimationrequired
6.76×10−6 −4.73×10−6 5.09×10−5 in a comprehensive simulation study or out-of-sample forecast
exercise, which we pursue with the AFNS model in the next
The variances of shocks to each state variable are similar across section.17 By comparison, the estimation of the AFNS model is
the independent- and correlated-factor DNS models, with level straightforwardandrobustinlargepartbecausetheroleofeach
factor shocks the least volatile and curvature factor shocks the latent factor is not left unidentified as in the maximally flexible
mostvolatile.Thecovarianceestimatesobtainedinthecorrelated- A (3) model. Even though the factors are latent in the AFNS
factorDNSmodeltranslateintoacorrelationof−0.701forshocks m0odel,withtheNelson–Siegelfactorloadingstructure,theycanbe
tothelevelandslopefactors,acorrelationof0.385forshockstothe clearlyidentifiedaslevel,slope,andcurvature.Thisidentification
levelandcurvaturefactors,andacorrelationof−0.208forshocks eliminates the troublesome local maxima reported by Kim and
totheslopeandcurvaturefactors. Orphanides(2005),i.e.,maximawithlikelihoodvaluesveryclose
Theindependent-andcorrelated-factorDNSmodelsarenested, totheglobalmaximumbutwithverydifferentinterpretationsof
so we can test the independent-factor restrictions using a thethreefactorsandtheirdynamics.18
standard likelihood-ratio (LR) test. Under the null hypothesis of The estimated independent-factor AFNS model is reported in
independent-factor DNS, LR = 2[logL(θcorr) − logL(θindep)] ∼ Table5.Althoughtheindependent-factorDNSandAFNSmodels
χ2(9).WeobtainLR = 164.8,withassociatedp-valuelessthan arenon-nested,theycontainthesamenumberofparameters,so
0.0001, so we would formally reject the restrictions imposed theirlikelihoodscanbecompareddirectly.Thelowerloglikelihood
in the independent-factor DNS model. This rejection reflects an value obtained for the AFNS model (16,280 vs. 16,332) suggests
elevatednegativecorrelationbetweentheshockstotheleveland weakerin-sampleperformance,whichappearsconsistentwiththe
slope factors and a significant positive correlation through the RMSEsinTable4.
mean-reversion matrix between changes in the slope factor and Although the two independent-factor models differ statisti-
deviationsofthecurvaturefactorfromitsmean. cally, they are quite similar economically, as can be seen in two
Crucially, however, the extra parameters in the correlated- ways.First,wecomparemean-reversionmatrices,covariancema-
factor model, although statistically significant, appear econom- trices,andmeanvectors.Tocomparetheindependent-factorAFNS
ically unimportant. That is, the increased flexibility of the mean-reversion matrix to that of the independent-factor DNS
correlated-factor DNS model provides little advantage in fitting model,wetranslatethecontinuous-timematrixinTable5intothe
observedyields,asdocumentedinTable4,whichreportsmeans one-monthconditionalmean-reversionmatrix,
and root mean squared errors (RMSEs) for model residuals. The
RMSEdifferencesappearnegligible(typicallylessthanonehalfof   0.993 0 0 
1
onebasispoint),maturity-by-maturity,andnoconsistentadvan- exp −KP = 0 0.983 0 . (14)
tageacrossmaturitiesaccruestothecorrelated-factormodel.In- 12 0 0 0.902
terestingly,bothmodelshavedifficultyfittingyieldsbeyondthe
ten-year maturity, which suggests that a maturity-dependent Similarly, we convert the volatility matrix into a one-month
yield-adjustmenttermcouldimprovefit.Wenowexaminetheem- conditionalcovariancematrix
piricalperformanceofAFNSmodels,whichincorporateprecisely
∫ 1
suchyieldadjustments. QAFNS = 12 e−KPsΣΣ′e−(KP)′sds
indep
0
4.3. AFNSmodelestimation
Thus far we have examined just one simple model (DNS),
comparing fit in the independent- and correlated-factor cases. 17 Forexample,Rudebuschetal.(2006)reportdifficultyreplicatingthepublished
estimates of a no-arbitrage model even though they use identical data and
NowwebringAFNSintothemix,andthingsgetmoreinteresting.
estimationprograms.
In particular, we can compare independent- and correlated- 18 Otherstrategiestofacilitateestimationincludeaddingsurveyinformation(Kim
factorcases,withandwithoutimpositionofabsenceofarbitrage.
andOrphanides,2005)orassumingthatthelatentyieldcurvefactorsareobservable
( )
As many have noted, estimation of the canonical affine A0 3 (AngandPiazzesi,2003).
12 J.H.E.Christensenetal./JournalofEconometrics164(2011)4–20
Table5 0
Estimatedindependent-factorAFNSmodel.ThetoppanelcontainstheestimatedKP
matrixandθPvector.ThebottompanelcontainstheestimatedΣmatrix.Standard
errorsappearinparentheses.Theestimatedλis0.5975(0.0115)formaturities
measuredinyears.Themaximizedloglikelihoodis16,279.92. nts -20
KPmatrix Mean poi
K·P,1 K·P,2 K·P,3 θP n basis -40
KKK123PPP,,,··· 0(000.0.0861165) (0000.2.1171840) 001.2330 −−((00000.....00000120717891232309)) adjustment term i -60
(0.4240) (0.0061) d-
el
Σmatrix Yi -80
Σ Σ Σ AFNS indep.-factor
·,1 ·,2 ·,3
AFNS corr.-factor
Σ1,· 0.0051 0 0 -100
(0.0001)
Σ2,· 0 0.0110 0 0 5 10 15 20 25 30
(0.0006)
Σ3,· 0 0 0.0264 Maturity in years
(0.0014)
Fig.1. Yield-adjustmenttermsforAFNSmodels.
2.15×10−6 0 0 
=  0 9.94×10−6 0 . (15) withtheclearresultthatcorrelated-factorDNSisdominatedby
0 0 5.26×10−5 correlated-factorAFNS.
Combining the model comparison results above with those
Inspectionrevealsthatthemean-reversionmatrixandcovariance reportedearlierinSection4.2,correlated-factorAFNSemergesas
matrix (and also the factor mean vector) are similar across the the clear in-sample favorite among all the various combinations
independent-factorDNSandAFNSmodels. of independent-factor, correlated-factor, DNS and AFNS models.
Second,thesimilarityoftheindependent-factorDNSandAFNS Presumably,thisisduetothegreaterflexibilityofthecorrelated-
modelscanbeseenbynotingthattheymakeidenticalassumptions factorAFNSyieldadjustment.Wereporttheestimatedcorrelated-
about the P-dynamics and therefore differ only by the yield- factor AFNS model in Table 6, from which we can infer the
adjustment term, which is quite rigid in the independent-factor estimatedyieldadjustment.Inpopulation,theadjustmentis
case.Inparticular,theindependent-factorAFNSyieldadjustment

is A(t,T) (T −t)2 1 1 1−e−λ(T−t)
− =−σ2 −(σ2 +σ2) −
−A(t,T) =−σ121 1 ∫ TB1(s,T)2ds− σ222 1 T −t 11 6 21 22 2λ2 λ3 T −t
T −∫t T 2 T −t σt2 1 ∫ T 2 T −t + 1 1−e−2λ(T−t)−(σ2 +σ2 +σ2)
× B2(s,T)2ds− 33 B3(s,T)2ds 4λ3 T −t 31 32 33
2 T −t
t t
 
=−σ121(T −6t)2 −σ222 21λ2 − λ131−Te−−λ(tT−t) × 21λ2 + λ12e−λ(T−t)− 41λ(T −t)e−2λ(T−t)
+41λ31−Te−−2λt(T−t)−σ32321λ2 + λ12e−λ(T−t) −43λ2e−2λ(T−t)− λ231−Te−−λ(tT−t) + 85λ31−Te−−2λt(T−t)
−41λ(T −t)e−2λ(T−t)− 43λ2e−2λ(T−t) −σ11σ21[21λ(T −t)+ λ12e−λ(T−t)− λ131−Te−−λ(tT−t)]
−λ231−Te−−λ(tT−t) + 85λ31−Te−−2λt(T−t), −σ11σ31λ32e−λ(T−t)+ 21λ(T −t)+ λ1(T −t)e−λ(T−t)

andisplottedinFig.1.Itiseverywherenegative,monotonically 3 1−e−λ(T−t)
increasinginabsolutevalue,andverysmooth.Presumablyamore − −(σ σ +σ σ )
λ3 T −t 21 31 22 32
flexible yield-adjustment term is needed to achieve substantial
improvement in fit. The correlated-factor AFNS model, to which 
wenowturn,achievesthis. × 1 + 1 e−λ(T−t)− 1 e−2λ(T−t)
Webeginwithtwomodelcomparisonsthatinvolvecorrelated- λ2 λ2 2λ2
factor AFNS. First consider independent- vs. correlated-factor 
AFNS. The models are nested, so under the null hypothesis of − 3 1−e−λ(T−t) + 3 1−e−2λ(T−t) .
independent-factor AFNS, LR = 2[logL(θ ) − logL(θ )] ∼ λ3 T −t 4λ3 T −t
corr indep
χ2(9).WeobtainLR = 428.7,withassociatedp-valuelessthan
0.0001, so independent-factor AFNS is dominated by correlated- Replacing population parameters with estimates delivers the
factorAFNS.Second,considercorrelated-factorDNSvs.correlated- correspondingestimatedyieldadjustment,whichweplotinFig.1.
factorAFNS.Themodelsarenon-nestedbutcontainequalnumbers Itisindeedmoreflexible,withaninterestinghumpinthefifteen-
of parameters, so we compare their log likelihoods directly, to twenty-year maturity range, which improves the fit of those
J.H.E.Christensenetal./JournalofEconometrics164(2011)4–20 13
Table6 mean-reversionmatrix
Estimatedcorrelated-factorAFNSmodel.ThetoppanelcontainstheestimatedKP
matrixandθPvector.ThebottompanelcontainstheestimatedΣmatrix.Standard  1  0.917 −0.107 0.122
errorsappearinparentheses.Theestimatedλis0.8244(0.0122)formaturities exp −KP = 0.0390 0.981 0.0112 . (16)
measuredinyears.Themaximizedloglikelihoodis16,494.29. 12 0.456 0.769 0.0667
KPmatrix Mean
Evidently,thelevelfactorbecomeslesspersistentoncetheflexible
K·P,1 K·P,2 K·P,3 θP correlated-factorAFNSyieldadjustmentisincorporated,because
K1P,· 5.2740 9.0130 −10.7100 0.0794 thelevelfactorismorefreetoworkwithslopeandcurvatureto
(1.3100) (1.4200) (1.4800) (0.0084) improvefitatshortermaturities,giventhattheyieldadjustment
K2P,· −0.2848 0.5730 −0.5528 −0.0396 ismosthelpfulatlongmaturities.
(1.3200) (2.3200) (2.7600) (0.0200)
K3P,· −37.3100 −66.7700 80.0900 −0.0279 Theone-monthconditionalcovariancematrixis
(11.0000) (11.9000) (12.1000) (0.0193) ∫ 1
Σmatrix QcAoFrNrS = 12 e−KPsΣΣ′e−(KP)′sds
0
Σ·,1. Σ·,2 Σ·,3  7.42×10−6 −6.11×10−6 −7.62×10−6
Σ1,· (00..00010544) 0 0 =−6.11×10−6 1.07×10−5 5.89×10−7 . (17)
Σ2,· −0.0013 0.0117 0 −7.62×10−6 5.89×10−7 1.87×10−4
(0.0051) (0.0018)
Σ3,· −0.1641 −0.0590 0.0001 Theconditionalvariancesinthediagonalareaboutthesameforthe
(0.0069) (0.0106) (6.8900) levelandslopefactorsasthoseobtainedinthecorrelated-factor
DNS model, but the conditional variance for curvature is much
7.5 larger.Intermsofcovariances,thenegativecorrelationbetween
the shocks to level and slope is maintained. For the correlations
between shocks to curvature and shocks to level and slope, the
signshavechangedrelativetotheunconstrainedcorrelated-factor
7.0
DNSmodel.Thissuggeststhattheoff-diagonalelementsofΣ are
heavilyinfluencedbytherequiredshapeoftheyield-adjustment
termratherthanthedynamicsofthestatevariables.Ontheother
ent 6.5 hand,theestimatedcovariancesoftheshocksintheDNSmodels
perc arelikelytobeunbiasedastheyarevariedtoprovidethebestfit
n
d i oftheP-dynamicswithoutanyimplicationsforthecross-sectional
el 6.0
Yi fitofthemodel.
Empirical mean yields 5. Out-of-samplepredictiveperformance
5.5 Indep.-factor DNS mean yields
Corr.-factor DNS mean yields
Indep.-factor AFNS mean yields Here we investigate whether the in-sample superiority of
Corr.-factor AFNS mean yields
the correlated-factor AFNS model carries over to out-of-sample
5.0
forecastaccuracy.Wefirstdescribetherecursiveestimationand
0 5 10 15 20 25 30 predictionprocedureemployed.Second,wecompareperformance
Time to maturity in years of the four uncorrelated/correlated-factor DNS/AFNS models,
exactly as in the in-sample analysis of Section 4 except that we
Fig. 2. Mean yield curves. We show the empirical mean yield curve, and the workout-of-sampleasopposedtoin-sample.Third,wecompare
independent-andcorrelated-factorDNSandAFNSmodelmeanyieldcurves.
theout-of-samplepredictiveperformanceofAFNStothatofthe
( )
canonicalA 3 model.
long-termyieldsinparticular,althoughitalsohelpswithshorter 0
maturities.
Another way to appreciate the role of the yield-adjustment 5.1. Constructionofout-of-sampleforecasts
term is to compare the mean fitted yield curves from the
independent-andcorrelated-factorAFNSandDNSmodelstothe We construct six- and twelve-month-ahead forecasts from
samplemeanyieldcurve,whichisdoneinFig.2.Allofthemodels the four DNS and AFNS models for yields at various maturities.
match the mean yield curve well for maturities up to ten years, We estimate and forecast using an expanding sample. The first
but their behavior diverges for longer maturities. Note that the estimationsampleisJanuary1987toDecember1996;thenJanuary
DNSmodelcurveismonotonicallyincreasing,whilewiththeyield- 1987 to January 1997, and so on. The largest estimation sample
adjustment terms, the AFNS models can bend downward and for the one-month-ahead forecasts ends in November 2002 (72
achievebetterlong-maturityfit.19 forecasts in all). For the six- and twelve-month horizons, the
The enhanced flexibility produced by the correlated-factor largestsamplesendinJune2002andDecember2001(67and61
AFNSyield-adjustmenttermallowsthelevelfactortobecomeless forecasts),respectively.
persistent,asevidencedbytheestimatedone-monthconditional Underquadraticlosstheoptimalforecastissimplytherelevant
τ
conditionalexpectation.TheoptimalDNSforecastforamaturity-
yieldmadeattimetfortimet+histherefore
19 ThisresultsuggestswhytheDNSmodelisnotarbitragefree.Atverylong 1−e−λτ
maturities,onlythelevelfactorhasanyappreciableinfluenceonbondyields.To yDt+NhS,t(τ) ≡ EtP[yt+h(τ)]=EtP[Lt+h]+EtP[St+h] λτ
eliminatethearbitrageopportunityfromgoinglongonabondwithverylong
meinveaeqtnuutruaitlaiyltlyyanatndhdheaeyndigeilillndugscttruharetviroeinsmkobfuycsotsnhsvoloerpxtiientygd),aowwbhoninwcdhawrtdhisethD(aaNnsSliamgphoptdlliyeclascthiaoonrnnteoortfmsJueapntpusoreirntty’.s, +EtP[Ct+h]1−λeτ−λτ −e−λτ. (18)