Forecasting Interest Rates: An Application of the Stochastic Unit Root and Stochastic Cointegration Frameworks *

Transcription

1 Forecasing Ineres Raes: An Applicaion of he Sochasic Uni Roo and Sochasic Coinegraion Frameworks * Rober Sollis Universiy of Newcasle upon Tyne Absrac This paper invesigaes forecasing U.S. Treasury bond and dollar Eurocurrency raes using he sochasic uni roo (STUR) model of Leybourne, McCabe and Tremayne (1996), and he sochasic coinegraion (SC) model of Harris, McCabe and Leybourne (00, 006). Boh models have imevarying parameer represenaions and are concepually aracive for modelling ineres raes as boh allow for condiional heeroscedasiciy. I find ha for many of he series considered STUR and SC models generae saisically significan gains in ou-of-sample forecasing accuracy relaive o simple orhodox models. The resuls obained highligh he usefulness of hese exensions and raise some issues for fuure research. JEL classificaions: C, C53, G1 Keywords: Ineres rae forecasing; Sochasic uni roo; Sochasic coinegraion * Sollis (corresponding auhor): Universiy of Newcasle upon Tyne Business School, Ridley Building, 3 rd Floor (Office 14), Newcasle upon Tyne, NE1 7RU, Unied Kingdom (Rober.Sollis@ncl.ac.uk).

2 1. Inroducion Mos of he exising lieraure on ineres rae modelling can be caegorized as falling ino one of hree broad groups: (i) heoreical and/or empirical work involving no-arbirage ineres rae models, (ii) heoreical and/or empirical work involving equilibrium coninuous ime ineres rae models and (iii) empirical work using ad-hoc economeric models. The no-arbirage approach o ineres rae modelling involves mahemaical models for ineres raes ha are consisen wih he erm srucure, such ha arbirage agains he curren erm srucure is impossible. Imporan examples are he models of Ho and Lee (1986), Black and Karansinski (1991) and Heah e al. (199). Equilibrium coninuous ime ineres rae models have heir foundaions in equilibrium economic models. Typically an appropriae funcional form for he shor rae ha saisfies cerain condiions (such as non-negaiviy) is derived firs in coninuous ime. The dynamics of he whole yield curve are hen assumed o be driven by he shor rae as described by his funcional form. Thus in conras o no-arbirage models, which use he erm srucure as an inpu, equilibrium coninuous ime models produce he erm srucure as an oupu. Examples are he models of Vasicek (1977), Cox e al. (1985), and Chan e al. (199, CKLS). Boh ypes of model can be used for pricing ineres rae derivaives and for empirical work such as forecasing. For ineres rae modelling and forecasing, which is he focus of his paper, an alernaive approach o using a model moivaed by heoreical argumens (i.e., a no-arbirage or coninuous ime model) is o employ an ad hoc economeric model chosen because of is abiliy o capure cerain feaures of hisorical daa. For a recen example of his approach see Gospodinov (005), who uses a discree ime auoregressive specificaion able o model condiional heeroscedasiciy and hreshold nonlineariy in hisorical daa. The empirical conribuion here falls ino his caegory. The applicaion of convenional uni roo ess will end o lead o he conclusion ha U.S. ineres raes conain a single uni roo (see, e.g., Campbell and Shiller, 1987; Mishkin, 199), and in ou-of-sample ineres rae forecasing compeiions models ha allow for mean-reversion ypically do no improve on uni roo models such as he random walk (see Diebold and Li, 006). However despie heir relaively good forecasing performance, convenional uni roo models such as he random walk suffer from descripive failure in he sense ha hey do no allow for condiional 1

3 heeroscedasiciy. Condiional heeroscedasiciy, and in paricular level-dependen condiional heeroscedasiciy (LDCH) is a commonly observed feaure of hisorical daa on ineres raes. There is an ineresing conradicion in ha discreisized versions of coninuous ime ineres rae models, which do generally allow for LDCH bu which assume mean-reversion, end no o be any beer for forecasing (paricularly shor-horizon forecasing) han he simple random walk (see, e.g., Duffee, 00). 1 In ligh of his, a specificaion ha allows for LDCH, bu which is based on he assumpion ha ineres raes are sochasically rending migh prove o be a useful forecasing alernaive o convenional uni roo models and coninuous ime ineres rae models. I empirically invesigae forecasing monhly U.S. ineres raes over shor horizons using wo aypical economeric models ha allow for forms of LDCH bu are based on he assumpion ha he daa is sochasically rending. One is a ime-varying parameer (TVP) univariae model he sochasic uni roo (STUR) model developed by Leybourne, McCabe and Tremayne (1996, LMT), Leybourne, McCabe and Mills (1996, LMM) and Granger and Swanson (1997); he oher is a sochasic coinegraion (SC) model developed by Harris e al. (00, 006), which has a TVP unobserved componens represenaion. Since boh models have TVP represenaions hey can each be inerpreed as a form of srucural break model. In a STUR model, on average he daa is inegraed of order one, I (1), bu deviaions above and below his cenral case are also allowed (i.e., he dominan roo of he auoregressive (AR) lagpolynomial migh someimes be less han one and someimes be greaer han one). The deviaions from he cenral I (1) case are modelled as a sochasic process. I is sraighforward o show ha STUR processes are non-saionary in variance, conain LDCH, and ha hey canno be differenced o saionariy. Yoon (006) discusses several oher ineresing saisical feaures of STUR processes, including he link beween STURs and long memory. The SC model of Harris e al. (00, 006) is an exension of he sandard Engle-Granger coinegraion framework (Engle and Granger, 1987; EG) ha allows for he regressand and regressors 1 Diebold and Li (006) find for long-horizon forecasing ha models based on exensions of he Nelson and Siegel (1987) exponenial componens framework are capable of generaing superior forecass compared o a random walk.

4 in he long-run equaion o be heeroscedasically inegraed (HI). The concep of SC is weaker han convenional coinegraion as i requires only ha I (1) behaviour is absen from he residuals of he long-run model, raher han requiring he presence of I (0) saionariy. HI processes have a sochasic rend componen and a nonlinear heeroscedasic componen which can be wrien algebraically as an I (1) process muliplied by an I(0) process. They conain a form of nonsaionary heeroscedasiciy. In he SC model he disequilibrium is also allowed o be nonsaionary heeroscedasic. The SC model is relaed o he univariae STUR model in he sense ha in STUR processes he condiional variance is level-dependen, while in HI processes he condiional variance can be level-dependen, and as will be show below, boh HI and SC models have TVP represenaions. I should be noed ha he uncondiional variance of STUR and HI processes are boh ime-dependen bu in a differen way. In HI processes his dependence is linear (as in he random walk) while in he STUR process i is exponenial. For his reason HI is concepually more aracive han STUR, alhough boh models can produce series wih similar characerisics. Noe ha forecass from STUR models can be compued using he Kalman filer. I forecas from SC models using simple vecor error correcion (VECM) forms ha employ he SC disequilibrium (raher han he Engle-Granger disequilibrium). An added complexiy when using SC models is ha when underaking regressions involving HI variables, convenional ordinary leas squares (OLS) loses is consisency propery in some scenarios. Harris e al. (00, HMLa) prove ha if he regressand is HI bu he regressors are I (1), OLS esimaes of he long-run regression parameers are consisen a he 1/ T rae (alhough ineresingly no super-consisen). However HMLa also prove ha if regressors are HI, hen OLS esimaes of he long-run regression parameers are inconsisen. HMLa develop a consisen mehod of esimaion for his case an asympoic insrumenal variables (AIV) esimaor, which is employed here. To disinguish HI from I (1), and SC from convenional coinegraion, Harris e al. (006, HMLb) develop a series of hypohesis ess. 3 Hansen (199) also makes his poin. 3 As an empirical applicaion of heir ess, HMLb invesigae coinegraion beween ineres raes of differen mauriies for a number of counries including he U.S. For he U.S. hey use a differen daa se o ha employed here and do no consider forecasing for any of he counries. 3

5 . Sochasic Uni Roos, Heeroscedasic Inegraion and Sochasic Coinegraion.1. Sochasic Uni Roos The seminal papers on STURs are LMT, LMM and Granger and Swanson (1997). LMT develop and invesigae a score es for he presence of a STUR. Granger and Swanson (1997) focus on modelling and forecasing STURs in a selecion of U.S. macroeconomic ime series. Furher ess for STURs are developed by LMM and applied o financial ime series. Taylor and van Dijk (00) presen exensive Mone Carlo evidence on he finie-sample properies of STUR ess. Empirical applicaions of he STUR model include Wu and Chen (1997), Sollis e al. (000) and Yoon (005). Yoon (006) provides a useful summary of some of he saisical properies of STUR processes. I follow he approach of LMT and define a firs-order STUR process as y (1 ρ) y 1 ε = + +, (1) where ρ and ε are sequences of zero mean, independenly and idenically disribued (iid) random variables, independen of each oher and wih variances he uncondiional mean, variance and covariance of his process are ω and σ, respecively. Assuming y 0 = 0, E( y ) = 0; () 1 j V( y ) = σ (1 + ω ), j= 0 = σ {1 (1 + ω ) } /{1 (1 + ω )}, if ω 0, = σ, if ω = 0 ; (3) s 1 s j cov( y, y s) = ρσ (1 + ω ), j= 0 s s = σ ρ {1 (1 + ω ) }/{1 (1 + ω )}, if ω 0, s = σ ρ ( s), if ω = 0. (4) When ω 0 he nonsaionary behaviour of he variance of y clearly differs from he fixed uni roo case. Raher han increasing linearly wih, he variance of he STUR process (1) increases 4

6 exponenially wih. Furhermore i can be shown ha unlike an I (1) process, he STUR process canno be differenced o saionariy. Rewriing (1) as y y 1 ωζy 1 ε = + +, (5) where ζ is iid(0,1), hen he uncondiional mean, variance and covariance of y are E( y ) = 0, (6) V y = σ + ω 1, (7) ( ) (1 ) cov( y, ) = 0. (8) y s Clearly, is nonsaionary in variance. The condiional mean and variance of are y y E ( y ) = y E (1 + ρ ) + E ( ε ), (9) V 1( y) y 1V 1(1 ρ) V 1( ε) cov 1{(1 ρ) ε} y 1 = , (10) where E (1 ) ρ =, E ( ) 0 1 ε =, V (1 ) 1 ρ ω + =, V ( ) 1 ε σ =, and cov {(1 ρ ) ε } ξω 1 + =, and he condiional mean and variance of y follow sraighforwardly. Thus i follows from (10) ha y (and consequenly y ) are boh condiionally heeroscedasic, where he condiional heeroscedasiciy is level-dependen. Noe ha he firs-order STUR model can be easily exended o include furher dynamics and deerminisic componens by wriing (1) as y = (1 + ρ ) y + ε, (11) 1 where p * j j= 1 y = y λ α y j, (1) and λ conains any deerminisic componens in y such as consans and rends. The Granger and Swanson (1997) and LMT approach o modelling a STUR differ. More specifically, in heir work on STURs Granger and Swanson (1997) use he following model: y ay 1 ε = +, (13) where ε is zero mean, iid wih variance σ ε, and where a = exp( α ), (14) 5

7 wih α being a saionary Gaussian series wih mean m and variance σ α. By expanding he exponenial funcion (14) in a Taylor series wih zero as he poin of expansion, Granger and Swanson s formulaion can be relaed o he STUR model of LMT. Consider he firs-order expansion, α exp( α )~exp(0) + exp(0), (15) 1! so ha (13) becomes, o his order of approximaion, y = (1 + α) y + ε. (16) If α is zero-mean iid hen (16) is he firs-order STUR model given by equaion (1) in LMT and equaion (1) above. To illusrae he difference beween STUR and convenional uni roo processes, in Figure 1 wo simulaed series of 300 observaions from (1) are ploed. The same pseudo-random residuals are used o generae boh series, however he firs series assumes ω = 0 ; he second series assumes ω = Thus, he firs series is a random walk (RW1) (since ω = 0 ) while he second series conains a STUR. FIGURE 1 ABOUT HERE There is a marked difference in volailiy beween he wo series wih he higher volailiy of he STUR process paricularly noiceable for he higher values owards he end of he sample.. Tesing for a STUR To es for a STUR LMT develop a score es. Consider he following exension of he firsorder STUR specificaion ha allows for addiional dynamics p ε y φ y = ρ ( y φ y ) +, (17) i i 1 i i 1 i= 1 i= 1 p 6

8 where i is assumed ha he lag polynomial in he φ parameers has roos ouside he uni circle, and ε σ and ~ iid (0, ) ρ ω. This represenaion can also be wrien as ~ iid (1, ) p+ 1 y = δ y + ε, (18) i i i= 1 where δ1 = ( ρ + φ1), δi = ( φi ρφ i 1), i=,..., p, (19) and δ p+ 1, = ρφ p. (0) Boh an I (1) null hypohesis and STUR alernaive hypohesis can hen be specified in erms of H : ω = 0, H : ω > ω as Under he null hypohesis ω = 0, ρ = 1 for all, and i can be seen ha he coefficiens of equaion (18) sum o uniy and hus y is an AR ( p + 1) process wih a fixed uni roo, whereas if ω > 0, y is an AR ( p + 1) process wih a STUR. On specifying ha y is condiionally Gaussian, LMT find from he log of he likelihood funcion he firs-order condiion, 4 L(.) / ω = σ ( ε ) ( ε σ ) ο (1). (1) ω = 0 T 1 + = p+ j= 1 Following simplificaion, he es saisic can be wrien as T 1 j p ˆ 3/ 1 Z = T ˆ σ ˆ κ ( ˆ ε ) ( ˆ ε ˆ σ ), () T = p+ 3 j= p+ j where κ is he variance of ε. Consisen esimaors of he nuisance parameers can be obained from he leas squares regression: y β γ φi y i i= 1 p = ε. (3) To obain criical values for heir es, LMT simulae he null disribuion hrough Mone Carlo mehods. Due o invariance properies of he es saisic under he null hypohesis, he mos parsimonious I (1) represenaion is used when consrucing he null disribuion a random walk wih sandard normal errors. 7

9 Noe ha here are some similariies beween a STUR process and models of condiional heeroscedasiciy, since if y is an inegraed generalised auoregressive condiionally heeroscedasic process (IGARCH), hen Z ˆT will rejec he I (1) null in subsanial frequencies. However, if y is a GARCH process, i urns ou ha here are insubsanial rejecion frequencies (see Wu and Chen, 1997). Therefore he Z ˆT es saisic can be used as a way of selecing beween GARCH and IGARCH models. Wu and Chen (1997) for example inerpre he Z ˆT es saisic as he Wald es o examine an IGARCH(1,1) null agains a GARCH(1,1) alernaive, finding for heir exchange rae daa ha he IGARCH(1,1) null is rejeced for four of heir six series. Noe ha analyically IGARCH and STUR processes are no he same. In a STUR process i is he lagged level of he series ha drives any condiional heeroscedasiciy, whereas in he IGARCH specificaion i is he lagged incremen ha drives any condiional heeroscedasiciy. For he score es oulined above, LMT define ρ as a sequence of iid random variables. However for some economic ime series inuiion suggess ha movemens away from a cenral I (1) model (i.e., he deviaions of he roo from uniy) may well be auocorrelaed. LMM develop score ess for he case in which under he alernaive hypohesis he deviaions of he roo around uniy follow a saionary AR(1) process, and he case in which under he alernaive hypohesis he deviaions of he roo around uniy follow a random walk (hereafer his es saisic will be represened by E ˆT ). While he es for an AR roo differs from he es for a random walk roo, LMM show ha for an AR roo, he relevan es saisic is idenical o Z ˆT. Thus LMM appeal o he asympoic resuls of LMT o esablish he asympoic disribuion of an operaional version of heir es for an AR roo, bu hey require new heorems o derive he asympoic disribuion of an operaional version of heir es for a random walk roo. The power and size properies of boh of hese es saisics are invesigaed in LMM. 8

10 .3. Heeroscedasic Inegraion, Sochasic Coinegraion and Heeroscedasic Coinegraion SC involves modelling he relaionships beween HI series. Here SC is used o model he relaionships beween ineres raes of differen mauriies. SC is srongly relaed o he concep of heeroscedasic coinegraion (HC) firs proposed by Hansen (199). In defining HC, Hansen (199) uses he following simple model y β0 β1x w = + +, (4) w = σ u, (5) where x is an I (1) process (or vecor of I (1) processes), σ is an I (1) process and u is an I(0) process. w is no I (1), and does no sochasically rend (hence he series comove), bu w is nonsaionary (Hansen, 199, uses he erm bi-inegraed process). Specifically, w is nonsaionary in uncondiional variance in he same way ha x is, being a linear funcion of, bu i behaves differenly o x because of he saionary elemen u. Since he sandard coinegraion heory assumes asympoic saionariy of he coinegraing residual, he model given by (4) and (5) is clearly no covered by his heory. Hansen (199) develops asympoic heory for his paricular case. He shows ha OLS is consisen a he usual rae (i.e., ) bu no super-consisen, and ha in pracice he normal disribuion offers a reasonable approximaion of he rue disribuion for parameer inference. To clarify he link beween HC and TVP models, assume for simpliciy ha he daa generaion process (DGP) is a TVP model: y βx u 1/ T = +, (6) where β and u are I(0), and x is I (1), bu ha a consan parameer model is esimaed. Since he parameer variaion is ignored, i follows sraighforwardly ha he consan parameer model becomes a model in which he residuals conain an I(0) elemen (being a funcion of he ignored parameer variaion in β ) muliplied by an I (1) elemen ( x ). To define HI and SC consider he following sochasic TVP unobserved componens model for a vecor ime series z proposed by HMLa: z = µ + Πw + ε, (7) 9

11 w = w + η, (8) 1 Π = Π+ V, (9) where z, w, ε, η and µ are m 1 vecors, and Π, Π and V are m m marices. The disurbances ε, η and V are mean zero I(0) processes. w is a vecor I (1) process. µ is a vecor of consans. Noe ha sricly (7) (9) is a represenaion raher han an esimable model, since he only observed variables are hose in z. (7) (9) implies ha he individual elemens of z have he following represenaion: i i i i z = e Πw + eε + evw, (30) where e is an m 1 vecor wih 1 in is i h posiion and 0 elsewhere. Thus he sochasic parameer i represenaion (7) (9) gives rise o a fixed parameer represenaion wih an addiional heeroscedasic disurbance erm evw. i The definiion of HI follows sraighforwardly from (30). If ei Π 0 hen HMLa define z i as being sochasically inegraed. If in addiion, E( evve ) > 0 hen is heeroscedasically i i z i inegraed (HI), while if ei Π 0, E ( evve i i) = 0 hen z i is simply I (1). Thus sochasic inegraion ness convenional inegraion and HI. Noe ha he heeroscedasiciy in (30) is driven by a funcion of he level of he process (specifically he I (1) componen of he process). HMLb exend his represenaion so ha he heeroscedasiciy can be driven by a funcion of any developed by HMLb are applicable o eiher case. I (1) process. The ess To illusrae he difference beween HI and convenional uni roo processes, in Figure wo simulaed series of 300 observaions are ploed. The firs is a random walk (RW) while he second is an HI series simulaed using (30) where RW is used for w. FIGURE ABOUT HERE As wih he STUR process in Figure 1, he HI process shows an increased level of volailiy relaive o he I (1) process, which is paricularly noiceable for he higher values owards he end of he sample. 10

12 To define SC le c be a non-zero m 1 vecor and consider he following represenaion: cz = cµ + cπw + ( cvw + cε ). (31) Assuming ha z is generaed according o (7) (9), if c Π = 0 hen he variables in are said o z be sochasically coinegraed (SC), oherwise hey are no coinegraed. The concep is inuiively simple; in he case of sandard coinegraion a linear combinaion of variables removes he sochasic rend; in he case of SC a linear combinaion of variables removes he sochasic rend bu he resuling process may display nonsaionary heeroscedasiciy (i.e., he variance is driven by an I (1) process). Noe ha he concep of SC encompasses boh HC and sandard coinegraion. If c Π = 0, E ( cvvc ) = 0 and V 0, hen he variables are inegraed and coinegraed in he sandard EG = sense. If c Π = 0, E ( cvvc ) > 0, he variables in z are heeroscedasically coinegraed (HC). HMLa invesigae he esimaion of SC models and uncover an ineresing asymmery ha requires he developmen of a new esimaor. For esimaion, HMLa reformulae (7) (9) ino a regression model, where z is pariioned ino a scalar y and an ( m 1) 1 vecor x ( z = ( y, x ) ), y = α + x β + u, (3) where u = c Πw + c'vw + c' ε, and c = (1, β ')'. Thus he error erm is composed of a sochasic rend c Πw, a heeroscedasic elemen c'vw and a saionary erm c'ε, and HMLa show ha for he OLS esimaor ˆβ, x = µ x + Π xw + ε x, (33) Πx = Πx + V x. (34) ˆ d β β f( W( s), Σ, Σ, Π ), (35) xv xx x where Σ = cov( vec ( Π ), Vc ), Σ = var( vec( Π )) and W( s) is an m -dimensional Brownian xv x xx x moion. I hen follows (see HMLa) ha if Σ 0 xv, ˆβ will be inconsisen and ˆα divergen. This resul is a consequence of he fac ha if Σ 0, hen x will be correlaed wih u. Noe ha if xv Vx = 0 hen x is I (1), in which case if y is HI he model (3) collapses o he HC model of 1/ Hansen (199) and herefore ˆβ is consisen a he usual rae T rae (bu no super-consisen). If 11

13 V x = 0 and y is I (1) hen (3) collapses o he EG model and ˆβ is super-consisen a rae T. Thus, an asymmery exiss in he sense ha ˆβ is consisen if he regressand is HI bu he regressors are I (1), bu if he regressors are HI and he regressand is I (1) or HI, hen OLS is inconsisen. To summarise, HMLa show ha under SC he OLS esimae of he long-run slope parameer is inconsisen if x is HI, and he OLS esimae of he consan is divergen. Forunaely HMLa develop a consisen mehod of esimaion, uilizing an insrumenal variables (IV) esimaor. Their IV 1/ esimaor employs (1, x ) wih k > 0 as an insrumen where k = O( T ). Noe ha if k were k fixed hen he esimaor (defined below) would be inconsisen. By allowing k as T he esimaor is consisen. Hence HMLa call he esimaor an asympoic IV (AIV) esimaor. The AIV esimaor of ( α, β ) is 1 T T T k ˆ k y α k x = k+ 1 = k+ 1. (36) ˆ = T T T β k x k x kx x ky = k+ 1 = k+ 1 = k+ 1 Using bespoke limi heory, HMLa prove consisency of he AIV esimaor and asympoic normaliy subjec o some addiional exogeneiy resricions. HMLb exend his mehodology furher by developing a series of hypohesis ess o deermine he presence of HI and SC, and o disinguish beween sandard coinegraion and HC. There are hree ess in all: S is he es of a fixed uni roo versus HI; S is a es of he null of SC ˆhi ˆnc (saionary or heeroscedasic); Sˆhc ess he null of saionary coinegraion agains HC. All of he ess are asympoically normally disribued. Deails of he ess will no be presened here and he reader is referred o HMLb, which conains relevan proofs and simulaion experimens illusraing he good finie-sample properies of he es saisics. 1

14 3. Daa, Uni Roo Tess and STUR Resuls 3.1. Daa Two daa ses are employed. The firs consiss of monhly observaions on zero-coupon Treasury bond yields (inerpolaed from coupon bonds) over 195: :1. This daa has been used in previous empirical research on ineres rae and erm srucure modelling (see, e.g., Duffee, 00), and is an exension of a daa se employed by Bliss (1997). The mauriies are 3, 6, 1, 4, 60, and 10 monhs. The second daa se consiss of monhly observaions on U.S. dollar Eurocurrency raes over 1975:01 005:1. The mauriies are 1, 3, 6, and 1 monhs. Eurocurrency raes have been sudied in a number of previous empirical sudies using convenional coinegraion echniques; see, e.g., Gerlach and Smes (1997) and Dominguez and Novales (00). 4 Seleced Treasury bond and Eurocurrency series are ploed in Figure 3 and Figure 4. FIGURE 3 ABOUT HERE FIGURE 4 ABOUT HERE 3.. Uni Roo Tes Resuls Table 1 gives he resuls from he applicaion of a convenional augmened Dickey-Fuller (ADF) es, τ ˆµ, (Dickey and Fuller, 1979), a KPSS es K ˆ s (Kwiakowski e al., 199), and he Z ˆT es of LMT. 5 As is ypically found when convenional uni roo ess are applied o U.S. ineres raes, boh τ ˆµ and K ˆ s suppor he hypohesis ha all of he series conain a uni roo. Ineresingly, however, Z ˆT generaes very srong rejecions of a fixed uni roo in favour of he sochasic uni roo alernaive for all of he series. 4 See Duffee (00) for he source of he Treasury bond daa. The Eurocurrency daa is from he FT daabase. 5 For he ADF es he lag lengh is deermined using he general-o-specific approach. Noe ha for breviy I do no presen resuls for he LMM es EˆT, since he random walk roo model is inappropriae for hese series wih he esimaed roo being no very persisen in all cases (see, e.g., Table ). 13

15 TABLE 1 ABOUT HERE 3.3. Esimaed STUR Models and Forecass As an example, for each of he ineres rae series Table gives he esimaed parameers obained for firs-order STUR models wih no deerminisics: y = (1 + ρ) y 1 + ε. The Kalman filer and maximum likelihood echniques are used for esimaion (see, e.g., Hamilon, 1994, Ch. 13). I is assumed ha he roo follows an AR(1) process ( ρ = θρ 1 + η ). The hyperparameers consis of he variance of he STUR model residuals ( σ ), he variance of he sochasic parameer ρ ( ω ), and he AR(1) slope parameer aached o ρ (θ ). TABLE ABOUT HERE For boh he Treasury bond and Eurocurrency daa, for all mauriies ˆ σ is larger han ˆω, and he slope parameer ˆ θ is quie low indicaing ha he deviaions of he model above and below he cenral I (1) case are no very persisen. As an example, in Figure 5 he 3-monh Treasury bond yield is ploed along wih he esimaed roo for his series (unsmoohed). Clearly here is a subsanial amoun of roo volailiy, paricularly around he early 1980s, coinciding wih he changes in U.S. moneary policy. FIGURE 5 ABOUT HERE The forecasing resuls for he Treasury bond series are given in Tables 3 and 4. Table 3 gives he mean square forecas error (MSFE) values for 1-, -, and 3-sep-ahead forecass from a random walk (RW) model, a convenional AR(1) model, a TVP AR(1) model (TV-AR), a firs-order STUR model wih no deerminisic componens (STURa) and a firs-order STUR model allowing for a 14

16 consan erm (STURb). 6 The forecasing period is 1990: :1 and he forecass are compued recursively. Noe ha for he purposes of exposiion, he figures in he ables are he rue MSFE values muliplied by 100. TABLE 3 ABOUT HERE I is clear from his able ha compared o he oher specificaions employed (RW, AR, TV- AR), reducions in MSFE are available by using he STUR models, paricularly for 1-sep-head forecasing. For he - and 3-sep-ahead forecass he STUR models generae lower MSFE values for he shorer mauriy raes. The STURa model is generally preferred o he STURb model, alhough noe ha because he model flucuaes beween I (0), I (1) and explosive, he consan erm does no have he same inerpreaion as in convenional fixed-parameer models. TABLE 4 ABOUT HERE Table 4 conains he resuls from a series of hypohesis ess applied o he 1-sep-ahead forecas errors from he STURa, RW and AR models. The following ess are repored for comparing he RW vs. he STURa model and he AR vs. he STURa model: he Diebold and Mariano (1995, DM) and Harvey e al. (1997, MDM) ess of equal forecasing accuracy, and he Harvey e al. (1998, HLN) es of forecas encompassing (he null hypohesis of he HLN es is ha he forecass from he STURa model are encompassed by hose from he compeing model). Using he relevan criical values, rejecions are obained from a leas one of he ess for all of he series oher han he 10-year rae. The majoriy of he rejecions are a he 1% level, wih he sronges rejecions being for he 6- monh and 1-year raes. These resuls are generally consisen wih he esimaed parameer values given in Table. In paricular he absence of rejecions for he 10-year rae is unsurprising given ha 6 I do no compare wih coninuous ime ineres rae models, leaving his for fuure research. Noe however ha previous empirical work (see e.g. Duffee, 00) has found ha for U.S. daa, simple univariae models such as he random walk end o generae forecass wih lower MSFE values han coninuous ime ineres rae models such as he CKLS model. 15

17 ˆ ρ is almos whie-noise (i.e., unpredicable) for his series (he slope coefficien in Table, ˆ θ, is no saisically significanly differen from zero). TABLE 5 ABOUT HERE TABLE 6 ABOUT HERE The MSFE values for he Eurocurrency rae forecass are given in Table 5. The forecasing period is 000:01 005:1 and he forecass are also compued recursively. In his case here is even sronger evidence supporing he STURa and STURb models. For all mauriies eiher he STURa and STURb models generae lower MSFE values han any of he compeing specificaions. The equal forecasing accuracy and encompassing ess for he 1-sep-ahead forecass from he STURa, RW and AR models are given in Table 6. When compared o he RW and AR models, virually all of he differences in MSFE are saisically significan on he basis of he DM and MDM ess. Similarly he null of forecasing encompassing is rejeced for nearly all of he mauriies. Noe ha for boh he Treasury bond and Eurocurrency daa furher sub-sample forecasing experimens were underaken. For breviy he resuls are omied, bu similar improvemens in MSFE were found for he subsamples considered HI, SC and HC Tes Resuls Table 7 conains resuls from applying he HMLb ess for HI, SC and HC o boh of he daa ses, where he long-run models are bivariae regressions of he long rae (10 monhs for he Treasury bond daa and 1 monhs for he Eurocurrency daa) on shor raes (here shor rae is aken o mean each of he oher mauriies). Also given are he esimaed slope parameers associaed wih he AIV esimaor and OLS for comparison. 7 This is paricularly rue for sub-samples where he esimaion period includes he early 1980s. 16

18 TABLE 7 ABOUT HERE Consider firs he es of I (1) versus HI, S. For he Treasury bond daa here are no rejecions of he ˆhi null of I (1) oher han for he 10 monhs mauriy suggesing ha while here can be HC, here will be no gains from using he AIV esimaor for regression models wih he 10 monhs rae as he regressand and one of he oher raes as regressor. However, if he 10 monhs rae is used as regressor wih one of he oher raes as regressand, hen OLS will be inconsisen and he AIV esimaor preferred. For he Eurocurrency daa here are srong rejecions for all of he mauriies, suggesing ha for his sample OLS will always be inconsisen for regressions involving differen mauriies. The oher resuls in Table 7 concern he bivariae regressions of he long rae on shor raes. ˆ c The Shin (1994) es K of he null of convenional coinegraion is compued, along wih S and Sˆhc he HMLb ess of he null of SC (encompassing convenional coinegraion and HC) agains no coinegraion, and convenional coinegraion versus HC, respecively. ˆnc For boh of he daa ses Kˆ c rejecs convenional coinegraion a he 5% level for all mauriies. However, for boh of he daa ses here are no rejecions of SC from he Sˆnc es. Thus Kˆ c finds no evidence of coinegraion, while Sˆnc overurns his resul. For he Treasury bond daa he null of convenional coinegraion is no rejeced by Sˆhc for any of he mauriies. However, for he Eurocurrency daa se Sˆhc finds srong evidence of HC for all mauriies. Noe ha for boh daa ses in all cases he esimaed slope parameer is close o uniy, consisen wih he Expecaions Hypohesis (EH). Ineresingly, for he Eurocurrency daa he AIV esimaed slope parameer is closer o uniy han he OLS esimaed parameer. Hence when AIV is used i generaes more suppor for he EH. In his case OLS is an inconsisen mehod of esimaion due o he finding ha all of he series are HI. 17

19 5. SC Forecasing Resuls To exploi he finding of HC beween Eurocurrency raes for forecasing, a series of simple vecor error correcion models (VECMs) using he SC disequilibrium (SC-VECMs) are employed o recursively compue 1-sep-ahead forecass of ineres rae changes over 000:01 005:1. For breviy, only he Eurocurrency daa is used for his forecasing exercise. Convenional VECMs (i.e., using he convenional EG disequilibrium) are used o generae forecass over he same period for comparison. Four differen model specificaions are used, including models imposing a slope of uniy and a zero inercep as suggesed by he EH (under risk neuraliy). The model definiions are given below Table 8. For breviy we focus on 1-sep-ahead forecass only. I should be borne in mind ha VECMs ypically do no ou-perform univariae echniques for shor-horizon ineres rae forecasing (see, e.g., Diebold and Li, 006). The parameers of boh he SC-VECM and convenional VECM models are esimaed by OLS. I is imporan o sress ha since under HC he SC disequilibrium will be nonsaionary heeroscedasic (alhough no sochasically rending; see Hansen, 199; HMLa; HMLb), and since he firs-differenced series in he VECMs are firs-differences of HI processes, i is no clear wheher OLS will generae consisen (or asympoically normal) esimaes of he parameers in hese VECMs. Noe also ha he appropriaeness of a fixed parameer shor-run funcional form migh be quesionable given he nonlinear model governing long-run behaviour. Therefore hese shor-run models should be viewed merely as linear approximaions of he unknown rue shor-run DGPs. The research by Hansen (199), HMLa and HMLb focuses only on long-run regressions of HI series, and hey do no explore he properies of firs-differenced HI series or VECMs involving he SC disequilibrium in fac, as of ye here has been no published research on VECMs involving HC series. Hence hese resuls should be seen as a firs sep in invesigaing he pracical benefis for VECM modelling of allowing for nonsaionary heeroscedasiciy in he long-run relaionship. TABLE 8 ABOUT HERE TABLE 9 ABOUT HERE 18

20 The MSFE values are given in Table 8 and equal forecasing accuracy and encompassing es resuls are given in Table 9. I can be seen in Table 8 ha for boh long rae and shor rae forecass he SC-VECMs generae lower MSFE values han he convenional VECMs for all mauriies. Noe in paricular ha while he long rae forecass generally have larger MSFE values han he univariae models in Table 5, for he 3-monh and 6-monh shor rae forecass he SC-VECMs generae lower MSFE values han he univariae models considered. The resuls in Table 9 indicae ha he differences in forecasing accuracy beween he SC- VECMs and he convenional VECMs are saisically significan for all mauriies, and he null of forecasing encompassing is in all cases rejeced by he HLN es. Ineresingly, in Table 8 he model ha generaes he lowes MSFE values for he long rae forecass is Model 4, which assumes a slope coefficien of uniy and a zero inercep in he long-run model. However, for he shor rae forecass he SC-VECM Model 3 generaes he lowes MSFE values. 6. Conclusions The performance of he univariae STUR model of LMT and he SC model of HMLa and HMLb is invesigaed for forecasing monhly U.S. ineres raes. These models are concepually aracive for forecasing ineres raes since boh allow for forms of level-dependen condiional heeroscedasiciy, which is a paricular feaure of hisorical daa on ineres raes, bu hey mainain a link o he convenional uni roo and coinegraion paradigms. For ineres rae modelling hese approaches bridge he gap beween discreisized coninuous ime ineres rae models (e.g., he CKLS model), which generally allow for level-dependen heeroscedasiciy bu which ypically are poor a forecasing due o he assumpion of mean reversion, and orhodox uni roo and coinegraion models which do no in heir sandard forms explicily incorporae heeroscedasiciy bu which allow for sochasic rends. I have found ha he STUR model generaes saisically significan improvemens in forecasing accuracy relaive o some simple orhodox univariae models for pos-war daa on U.S. Treasury bond yields and U.S. dollar Eurocurrency raes. Evidence of SC (and HC) beween ineres 19

21 raes of differen mauriies is found for he Eurocurrency daa, and employing he AIV esimaor of HMLa is found o improve forecass of his daa from VECMs. I is imporan o menion some caveas. The lieraure on SC is relaively new and is complicaed by saisical asymmeries and he need for new mehods of esimaion and bespoke limi heory o deal wih HI series. Furhermore, while Hansen (199), HMLa and HMLb have developed he heory for analysis of HI daa in levels, here has been no heoreical analysis of he properies of parameer esimaes for VECMs involving he firs-differences of HI series. Noe also ha he SC framework does no currenly allow for muliple coinegraing vecors o be esimaed simulaneously and hen idenified as in he Johansen (1988) framework, and herefore for he momen i is limied o applicaions where his is no a prioriy. However, he resuls obained here highligh he usefulness of hese exensions, paricularly for ineres rae modelling and forecasing where he evidence from uni roo ess suggess ha he daa conains sochasic rends, bu where convenional uni roo and coinegraion models are no sufficienly flexible o capure heeroscedasiciy in he daa. Much of he recen research on forecasing under srucural breaks ends o make he implici assumpion ha he breaks are occasional. The effec of occasional srucural change on he forecasing performance of an economeric model can ofen be miigaed by including dummy variables or by using sub-samples of he full sample of daa. Whils for many macroeconomic and financial variables he assumpion of a small number of discree breaks may be a reasonable one, for oher variables i migh be more appropriae o model breaks as a sochasic process, effecively allowing for breaks a every single ime observaion. Boh he STUR and SC models uilized here allow for his ype of srucural change and hey appear o perform well for ineres rae forecasing relaive o simple fixed parameer models. Comparing he forecasing performance of STUR and SC models wih occasional-break models is an ineresing opic lef for fuure research. 0

22 References Black, F. and Karansinski, P. (1991). Bond and opion pricing when shor raes are lognormal. Financial Analyss Journal (July-Augus), Bliss, R.R. (1997). Tesing erm srucure esimaion mehods. Advances in Fuures and Opions Research 9, Campbell, J.Y. and Shiller, R. (1987). Coinegraion and ess of presen value models. Journal of Poliical Economy 95, Chan, K.C., Karolyi, G.A., Longsaff, F.A., and Sanders, A.B. (199). An empirical comparison of alernaive models of he shor-erm ineres rae. Journal of Finance 47, Cox, J.C., Ingersoll, E., and Ross, S.A. (1985). A heory of he erm srucure of ineres raes. Economerica 53, Dickey, D.A. and Fuller, W.A. (1979). Disribuion of he esimaors for auoregressive ime series wih a uni roo. Journal of he American Saisical Associaion 74, Diebold, F. and Li, C. (006). Forecasing he erm srucure of governmen bond yields. Journal of Economerics 130, Diebold, F.X. and Mariano, R.S. (1995). Comparing predicive accuracy. Journal of Business and Economic Saisics 13, Dominguez, E. and Novales, A. (00) Can forward raes be used o improve ineres rae forecass? Applied Financial Economics 1, Duffee, G. (00). Term premia and ineres rae forecass in affine models. Journal of Finance 57, Engle, R.F. and Granger, C.W.J. (1987). Co-inegraion and error correcion: represenaion, esimaion and esing, Economerica 55, Gerlach, S. and Smes, F. (1997). The erm srucure of Euro-raes: some evidence in suppor of he expecaions hypohesis. Journal of Inernaional Money and Finance 16, Gospodinov, N. (005). Tesing for hreshold nonlineariy in shor-erm ineres raes. Journal of Financial Economerics 3,

23 Granger, C.W.J. and Swanson, N.R. (1997). An inroducion o sochasic uni roo processes. Journal of Economerics 80, Hamilon, J.D. (1994). Time Series Analysis., Princeon, New Jersey: Princeon Universiy Press. Hansen, B.E. (199). Heeroskedasic coinegraion. Journal of Economerics 54, Harris, D., McCabe, B.P.M., and Leybourne, S.J. (00). Sochasic coinegraion: esimaion and inference. Journal of Economerics 111, Harris, D., McCabe, B.P.M., and Leybourne, S.J. (006). A residual-based es for sochasic coinegraion. Economeric Theory, Harvey, D.I., Leybourne, S.J., and Newbold, P. (1997). Tesing he equaliy of predicion mean squared errors. Inernaional Journal of Forecasing 13, Harvey, D.I., Leybourne, S.J., and Newbold, P. (1998). Tess for forecas encompassing. Journal of Business and Economic Saisics 16, Heah, D.R., Jarrow, R., and Moron, A. (199). Bond pricing and he erm srucure of ineres raes. Economerica 60, Ho, T.S.Y., and Lee, S.B. (1986). Term srucure movemens and pricing ineres-rae coningen claims. Journal of Finance 41, Johansen, S. (1988). Saisical analysis of coinegraing vecors. Journal of Economic Dynamics and Conrol 1, Kwiakowski, D., Phillips, P.C.B., Schmid, P., and Shin, Y. (199). Tesing he null of saionariy agains he alernaive of a uni roo: how sure are we ha economic ime series have a uni roo? Journal of Economerics 54, Leybourne, S.J., McCabe, B.P.M., and Mills, T.C. (1996). Randomized uni roo processes for modelling and forecasing financial ime series: heory and applicaions. Journal of Forecasing 15, Leybourne, S.J., McCabe, B.P.M., and Tremayne, A.R. (1996). Can economic ime series be differenced o saionariy? Journal of Business and Economic Saisics 14, Mishkin, F. (199). Is he Fisher effec for real? A re-examinaion of he relaionship beween inflaion and ineres raes. Journal of Moneary Economics 30,

24 Nelson, C. R. and Siegel, A. F. (1987). Parsimonious modelling of yield curve. Journal of Business 60, Shin, Y. (1994). A residual based es of he null of coinegraion agains he alernaive of no coinegraion. Economeric Theory 10, Sollis, R., Leybourne, S.J., and Newbold, P. (000). Sochasic uni roos modelling of sock marke indices. Applied Financial Economics 10, Taylor, A.M.R. and van Dijk, D. (00). Can ess for sochasic uni roos provide useful pormaneau ess for persisence? Oxford Bullein of Economics and Saisics 64, Vasicek, O. (1977). An equilibrium characerizaion of he erm srucure. Journal of Financial Economics 5, Wu, J.-L. and Chen, S.-L. (1997). Can nominal exchange raes be differenced o saionariy? Economics Leers 55, Yoon, G. (005). Sochasic uni roos in he Capial Asse Pricing Model. Bullein of Economic Research 57, Yoon, G. (006). A noe on some properies of STUR processes. Oxford Bullein of Economics and Saisics 68,

25 Table 1: Uni roo es resuls Mauriy τˆ µ A. Treasury bond yields Kˆ s 3 monhs monhs monhs monhs monhs monhs B. Eurocurrency raes 1 monh monhs monhs monhs ZT Noes: ˆµ τ is he ADF es. ˆ s K is he KPSS saionariy es. ˆT Z is he LMT es (fixed uni roo vs. STUR). Bold indicaes a rejecion a he 1% significance level.

26 Table : STUR parameer esimaes Mauriy ˆ θ A. Treasury bond yields 3 monhs 0.15 (0.057) monhs 0.03 (0.056) monhs 0.03 (0.054) monhs (0.05) monhs (0.046) monhs (0.046) B. Eurocurrency raes 1 monhs (0.051) monhs (0.051) monhs (0.051) monhs 0.18 (0.058) Noes: Parameers are esimaed using he Kalman filer and maximum likelihood. Sandard errors are in parenheses. ˆω ˆ σ

27 Table 3: Forecass of Treasury bond yields over 1990: :1, MSFE values Mauriy RW AR TV-AR STURa STURb A. 1-sep-ahead 3 monhs monhs monhs monhs monhs monhs B. -sep-ahead 3 monhs monhs monhs monhs monhs monhs C. 3-sep-ahead 3 monhs monhs monhs monhs monhs monhs Noe: Bold indicaes he minimum MSFE value across he differen models.

28 Table 4: 1-sep-ahead forecass of Treasury bond yields over 1990: :1, equal forecasing accuracy and encompassing ess Mauriy DM MDM HLN A. Random walk vs. STURa 3 monhs monhs monhs monhs monhs monhs B. AR vs. STURa 3 monhs monhs monhs monhs monhs monhs Noes: DM sands for he Diebold and Mariano (1995) es of equal forecas accuracy. MDM sands for he Harvey e al. (1997) modified DM es. HLN sands for he Harvey e al. (1998) es of forecas encompassing. Bold indicaes a rejecion a he 5% level (or sronger). Significan posiive values indicae ha he STURa model is preferred.

29 Table 5: Forecass of Eurocurrency raes over 000:01 005:1, MSFE values Mauriy RW AR TV-AR STURa STURb A. 1-sep-ahead 1 monh monhs monhs monhs B. -sep-ahead 1 monh monhs monhs monhs C. 3-sep-ahead 1 monh monhs monhs monhs Noe: Bold indicaes he minimum MSFE value across he differen models.

30 Table 6: 1-sep-ahead forecass of Eurocurrency raes over 000:01 005:1, equal forecasing accuracy and encompassing ess Mauriy DM MDM HLN A. Random walk vs. STURa 1 monh monhs monhs monhs B. AR vs. STURa 1 monh monhs monhs monhs Noes: See Table 4. Bold indicaes a rejecion a he 1% level.

31 Table 7: Tess for HI, SC and HC A. Treasury bond yields Regressor: ˆOLS β K ˆ c ˆAIV β S ˆhi 3 monhs monhs monhs monhs monhs Regressand 10 monhs NA NA NA.056 NA NA B. Eurocurrency raes Regressor: 1 monh monhs monhs Regressand: 1 monhs NA NA NA.41 NA NA S ˆnc S ˆhc ˆ c Noes: K is he Shin (1994) es of he null of coinegraion (saionary). S is he HMLb es (fixed uni roo versus HI). denoes he HMLb es of he null of SC (saionary or heeroscedasic). Sˆhc is he HMLb es of he null of saionary coinegraion agains HC. Bold indicaes a rejecion a he 5% level (or sronger). Sˆnc ˆhi

32 Table 8: 1-sep-ahead forecass of Eurocurrency raes over 000:01 005:1 from VECMs and SC- VECMs, MSFE values A. Long rae forecass Regressor: Model 1 Model Model 3 Model 4 VECM SC-VECM VECM SC-VECM 1 monh monhs monhs B. Shor rae forecass Regressand: 1 monh monhs monhs Noes: The long rae here is he 1-monh Eurocurrency rae. Regressor refers o he regressor in he long-run model. Regressand refers o he regressand in he VECM and SC-VECM models. The shaded blocks indicae he minimum MSFE values for each model. Bold denoes he minimum MSFE value across he differen models. The model definiions are given below where y1 1 denoes he long rae a ime 1 and where i and j indicae mauriy ( i = 1,3, 6,1, j = 1,3,6, i j ). For models 3 and 4 he number of lagged differences o include is deermined using he general-o-specific approach. The long-run parameers ˆα and ˆβ are esimaed by OLS for he VECMs and AIV for he SC-VECMs. Model 1: yi = γ 0 + λ( y1 1 yj 1) + η Model : y ˆ ˆ i = γ 0 + λ( y1 1 α βyj 1) +η v Model 3: y = γ 0 + γ ˆ ˆ 1 y + γ y + λ( y1 1 α βy 1) + w η i p j p q i q j p= 1 q= 1 v Model 4: y = γ 0 + γ1 y + γ y + λ( y1 1 y 1) + w η i p j p q i q j p= 1 q= 1

33 Table 9: 1-sep-ahead forecass of Eurocurrency raes over 000:01 005:1 from VECMs and SC-VECMs, equal forecasing accuracy and encompassing ess A. Long rae forecass Regressor: Model : VECM vs. SC-VECM Model 3: VECM vs. SC-VECM DM MDM HLN DM MDM HLN 1 mons monhs monhs B. Shor rae forecass Regressand: 1 monh monhs monhs Noes: See Tables 4 and 8. Bold indicaes a rejecion a he 10% level or sronger. Significan posiive values indicae ha he SC-VECM is preferred.

34 Figure 1. Simulaed RW and STUR Processes 0.5 RW1 STUR

35 0.650 Figure. Simulaed RW and HI Processes RW HI

36 Figure 3. Seleced U.S. Treasury Bond Yields monh 10-monh

37 0.0 Figure 4. Seleced U.S. Dollar Eurocurrency Raes 1-monh 1-monh

38 15 Figure 5. 3-monh Treasury Bond Yield and Esimaed STUR 3-monh Treasury bond yield ˆ ρ