State-Space Estimation of Multi-Factor Models of the Term Structure: An Application to Government of Jamaica Bonds. Abstract

Transcription

1 State-Space Estmaton of Mult-Factor Models of the Term Structure: An Applcaton to Government of Jamaca Bonds R. Bran Langrn Abstract Ths paper estmates mult-factor versons of the Vascek (977) and the Cox, Ingersoll and Ross (CIR; 985) models of the term structure of nterest rates usng zero-coupon Government of Jamaca bond prces. Statstcal tests confrm that the two-factor CIR-model best accounts for the dynamcs of the term structure. The emprcal analyss revealed that the level of the short rate exhbts strong and smooth mean reverson and the exstence of a large and sgnfcant rsk premum that ncreases wth tme to maturty. Based on estmated factor loadngs, the unobserved short rate has a sgnfcant mpact on the short end of the yeld curve but a relatvely mnmal mpact on the long end. JEL classfcaton: C, E4, G Keywords: affne term structure, state-space models, Kalman flter Fnancal Stablty Department, Bank of Jamaca, Nethersole Place, P.O. Box 6, Kngston, Jamaca, W.I. Tel.: (876) Fax: (876) Emal: bran.langrn@boj.org.jm

2 .0 Introducton The econometrc estmaton of the term structure of nterest rates has receved tremendous attenton from fnancal- and macro-economsts, partcularly n the context of bond prcng. Based on the Expectatons Theory of the term structure, the yelds on long-term bonds are the expected value of rsk-adjusted average future short-term yelds. Hence, measurement of the term structure of nterest rates allows for the extracton of nformaton on nvestors expectatons about future nterest rates. Term structure measurement models have a range of applcatons. Specfcally, nterpretng the emprcal propertes of bond yeld dynamcs that are provded by term structure measurement models s mportant for a number of purposes that nclude: Influencng aggregate demand through monetary polcy. The short rate s the fundamental polcy nstrument of the central bank. That s, central banks may shft the short end of the yeld curve when adjustng ther polcy stance. However, movement n long-term rates have a greater nfluence on aggregate demand. Thus, knowledge of yeld curve dynamcs provdes nformaton to the central bank on how ther nterest rate decsons wll mpact the future path of the economy. Rsk management through the prcng and hedgng of nterest rate-contngent clams ncludng caps, floors and swaptons. Further, value-at-rsk estmates for fxed ncome portfolos can be obtaned through smulatng paths for the term structure. 4 Publc debt management through bond ssues. 5 Knowledge of the dynamc propertes of the yeld curve provdes nformaton on the mpact of fscal polcy on nvestor rsk preferences See, for example, Babbs and Nowman (999), Da and Sngleton (000) and Pearson and Sun (994). See, for example, Ang and Pazzes (00), Debold, Redebusch and Aruoba (00), Fendel (004), Hördahl, Trstan and Vestn (00), Pazzes (00) and Rudebusch and Wu (00). See, for example, Amn and Morton (994), Buhler et al (999), Dressen, Klaasen and Melenberg (00), Canabarro (995), Chernov and Ghysels (000), Jagannathan et al (000) and Longstaff et al (00). 4 Value-at-Rsk s defned as the maxmum potental loss on a portfolo for a gven horzon and probablty. 5 See, for example, Da and Phlppon (004).

3 and future yeld expectatons of bonds across maturtes. Fscal authortes can use ths nformaton when decdng the length of tenors n ther fnancng decsons. There has been enormous growth snce the 990s n the soveregn bond markets for emergng economes, ncludng for Jamaca. Ths has led to the ncreased mportance of obtanng nformaton concernng the term structure of emergng countres soveregn bond yelds n order to predct the tmng of possble adverse credt events n these economes. Whereas a number of studes exst that examne the term structure of specfc emergng market soveregn bond yelds, no known study exsts for the Jamacan case. Ths paper estmates the two most popular versons of affne dffuson term structure models usng zero-coupon Government of Jamaca (GOJ) soveregn bonds for the perod 4 September 004 to 8 July 006. Specfcally, mult-factor versons of the Vascek (977) and the Cox, Ingersoll and Ross (CIR; 985) models of the nomnal nterest rate term structure are estmated usng a state-space approach. Ths approach smultaneously ntegrates tme seres and cross-sectonal GOJ soveregn yelds to generate the unobservable state varables usng a Kalman flter. The objectve of ths exercse s to examne the usefulness of popular term structure models n explanng the yeld curve dynamcs n Jamaca n order to derve nformaton on nvestor expectatons as well as to accurately prce GOJ bonds and hedgng nstruments. The next secton focuses on the theoretcal formulaton of the Vascek and CIR mult-factor affne models. The state space representaton of the Vascek and CIR term structure models and the Kalman flter algorthm are presented n Secton. One to three factor versons of these models are used to explan the dynamcs of the term structure of GOJ bonds for the perod 4 September 004 to 8 July 006. The data descrpton and emprcal results are reported n Secton 4. Secton 5 provdes a bref concluson.

4 .0 Equlbrum Multfactor Affne Models of the Term Structure The Vascek (977) and CIR (985) models fall n the class known as equlbrum models of the term structure and are the two most popular versons of affne dffuson term structure models. These studes represent specal cases of ths class of models: the Gaussan case (Vascek ) and the non-gaussan case (CIR). Both models rely on specfc assumptons about the stochastc nature of state varables to obtan nformaton on the dynamc evoluton of the term structure wthn an economc envronment. The dstnct features of these models are that the market prce of rsk s dentfed ether exogenously or endogenously and the nstantaneous short rate s explctly specfed as a functon of unobserved state varables. 6 The man dfference between these models s that the short rate n the CIR model s specfed as a square root process that s proportonal to the level of the nterest rate, unlke the Vascek model whch assumes a constant varance. Ths feature prevents the occurrence of negatve rates under certan restrctons. 7 Sngle-factor term-structure models descrbe the dynamcs of the nstantaneous short rate. Hence, these models can only account for parallel shfts n the yeld curve. In practce, however, other factors may nfluence dfferent sectons of the yeld curve allowng for varous shapes such as twsts and nverse humps. Alternatvely, the flexblty nherent n mult-factor term-structure models allows for a wder range of possble yeld curve shapes. Three-factor term-structure models are usually estmated n practce to explan the dynamcs of the term structure of nterest rates. The specfcaton of three factors rely on the semnal study by Ltterman and Schenkman (99), based on standard prncple component analyss, whch found that three factors correspondng to the level, curvature and slope of the yeld curve explaned the term structure of 6 The market prce of rsk, otherwse called the Sharpe rato, refers to the expected standardsed excess rate of return above the rsk free rate from a specfc zero-coupon bond. 7 See Subrahmanyam (996) for an extensve dscusson on the Vascek and CIR models as well as other semnal term structure models.

5 US Treasury bond yelds n the 980s. However, many studes have found that the ncluson of addtonal factors does not ncrease the performance of term structure models. 8 Consstent wth ths fndng, the Ltterman and Schenkman (99) study determned that almost 90.0 per cent of the varaton n US Treasury yelds was drven by the varaton n the frst factor. Multfactor affne models of the term structure represent the yelds of securtes as affne functons of a vector of K unobservable state varables or factors, X = ( X, X, K, X K ), whch s governed by the followng multdmensonal dffuson process 9 The nstantaneous short rate s gven as () = μ () + σ. () d X t X t dt X t dw t K K K K K K () β β (). () r t X t = + = 0 The factors, X () t, are assumed to be ndependently generated by the Ornsten-Uhlenbeck (O-U) process n the Vascek (Gaussan) case represented as,,, () dx t = κ θ X t dt + σ dw t = K K () and the square-root process n the CIR (non-gaussan) case represented as () (),,, () = κ θ () + σ = K (4) dx t X t dt X t dw t K where κ, θ andσ are the speed of mean reverson, long-term mean and volatlty parameters, respectvely, and W () t denote ndependent Wener processes under the rsk-neutral prcng measure, Φ. The nomnal prcng formula for a pure dscount bond wth a face value of $ maturng at T s 8 See, for example, Chatterjee (005). 9 n A functon F : R R s affne f there exsts some coeffcents T n F X = a+ b X X R., a R and n b R such that 4

6 where B ( T ) and K K = exp = (5) PT A T B T X t = A T, n the Vascek model have the followng forms exp κ = ( ) ( κ T) (6) B T A T = exp κ λσ σ ( B ( T) T) κ θ κ σ B ( T) 4κ (7) and where A ( T ) and B T, n the CIR model have the followng forms A T B T γ exp ( κ + λ + γ) T ( T) + ( + + ) ( T) = γ exp γ κ λ γ ( exp γ ) ( exp( γ T )) ( T) + ( + + ) ( T) = γ exp γ κ λ γ exp γ κθ σ (8) (9) and γ = κ + λ + σ. The rsk premum for each state varable s λ X where the fxed parameter λ s the market prce of rsk for the correspondng state varable and s negatvely related wth the rsk premum. The prcng formula for a coupon bond wth a face value of $ maturng at T wth m m coupons, C, to be pad at T s Ψ ( T) = C P( T), wth an mpled yeld to maturty obtaned by = m solvng Ψ ( T) = C exp( T ). However, ϕ ϕ ( X, T ) would not be normally dstrbuted gven ts = nonlnear relatonshp wth X () t. 5

7 .0 The State-Space Approach to Estmate Mult-Factor Term Structure Models A state-space approach s adopted n ths paper to estmate the unknown parameters and extract the unobservable state varables. A state-space representaton s a dynamc system that comprses measurement equatons, whch condton observed varables on unobserved or state varables, as well as transton equatons, whch descrbe the path of the state varables. Ths system may be expressed n a form that may be examned usng the Kalman flter whch orgnates from the engneerng control lterature. 0 The Kalman flter s an algorthm for sequentally updatng a lnear projecton for the system usng nformaton from the observed varables. The exact statespace representaton for a mult-factor model wth state vector X ( t ) s based on the assumpton that X (0), X(), K, X( t ) s a Markov process wth X(0) ~ δ (0)[ X(0) ] and X () t X ( t ) ~ δ X () t X ( t ) where δ (0) [ X (0) ] and δ X () t X ( t ) represent the densty of the ntal state vector and the transton densty, respectvely.. The CIR (non-gaussan) model Consder the followng CIR square-root process for the spot nterest rate () () () = κ θ () + σ (0) dr t r t dt r t dw t The change n the nstantaneous short rate has a mean-revertng drft as well as a varance whch s proportonal to the level of the short rate. The affne drft ( t) = κ( θ r( t) ) r () t > θ ( () t < θ () 0 μ ensures that f r ) then dr t < ( dr() t > 0) should hold under the assumptonκ > 0. The Feller (95) condton κθ < σ ensures that the process has a reflectng boundary at r( t) = 0 so that the condtonal varance σ r( t) does not collapse to zero. Ths condton does not allow the process to be nonstatonary (.e., κ = 0 ). 0 See Duan and Smonato (995), Babbs and Nowman (999), Chen and Scott (00), De Jong (998), Geyer and Pchler (999) and Lund (997) for applcatons of the Kalman flter to term structure models. See Hamlton (994). 6

8 The soluton for nomnal prce of a pure dscount bond wth a face value of $ maturng at T s PT AT exp( BT rt) = () where A( T ) and BT are matrces wth ndvdual elements depcted by equatons (8) and (9), respectvely. The ndvdual elements of X ( T ) and Y( T ) are X t = θ exp κ Δ t + exp κ Δt X t + η t, () η t Ω t- ~ Ν 0, t ; =, K, K or, n the K = -factor case () () () () ( ( Δt) ) ( κ ) exp( κ Δ ) X t θ exp κ exp κ Δt 0 0 X t η t X t = θ exp Δ t + 0 exp ( κδt) 0 X( t ) + η() t X t θ t 0 0 exp( κδt) X( t ) η() t and K ln A t, sj B t, sj X t Y ( sj) = +, j =,, M = s t s t K () j The lmt of the yeld to maturty, or the long-term yeld, as the tme to maturty gets longer s lm log Y = P T T = κθ ( κ + λ + γ ). T j () The unobservable state varables for the CIR model are dstrbuted condtonally as non-central χ varates. In order to estmate the unobservable state varables, the exact transton densty s substtuted by a normal densty X () t X ( t ) ~ N μ ( t), Σ ( t). The matrces for the condtonal mean and condtonal varance of X () t for the CIR model are determned such that they are equal to the frst two moments of the exact transton densty wth elements defned as () t = exp( Δ t) + exp( Δt) ( ) μ θ κ κ Y t (4) 7

9 and the matrx, Σ() t, has K dagonal elements ( κ t) exp Δ () t = θσ exp( κ t) exp( κ t) Y( t ) κ Δ + Δ (5). The Vascek (Gaussan) model Consder the followng Vascek spot nterest rate O-U process () = κ θ () + σ (0 ) dr t r t dt dw t and κ > 0 s requred for the process to be statonary. The soluton for nomnal prce of a pure dscount bond wth a face value of $ maturng at T s P T A T exp( B T r t ) = ( ) where BT and A( T ) are matrces wth ndvdual elements depcted by equatons (6) and (7), respectvely. The ndvdual elements of X ( T ) and Y( T ) are the same as equatons () and () for the CIR model. The matrces for the condtonal mean and condtonal varance of X ( t ) for the Vascek model are and () t = exp( Δ t) + exp( Δt) Y ( t ) (4 ) μ θ κ κ exp () t = σ κ Δ κ ( κ t ) (5 ) See Dullmann and Wndfuhr (000). 8

10 . The Kalman Flter The contnuously compounded yeld to maturty on a pure dscount bond s Y T K ln A( T) B( T) X ( t) (6) = + = T whch affne n the unobserved vector of state varables X ( t ). In order to estmate the system, t T s assumed that yelds for the N maturtes are observed wth errors of unknown magntudes. Hence, equaton (6) may be expressed as Y T K ln A ( β; T) B( β; T) X ( t) ε () t (7) = + + = T T where ( h β = θ κ σ λ ) s a vector of unknown parameters and ε ( t) has zero mean and varance, H () t, but not necessarly normally dstrbuted. Equaton (7), whch s the measurement equaton of the state-space model, s expressed n stacked form as ( () ) () ( ) N N () () Y X t ; β, T ln A β; T T T B β; T ε t Y( X t ; β, T) ln ( A β; T ) T T B β; T ε t X() t = + +, M M M K M Y( X() t ; β, T ) ln ( ( ; )) ( TN) B( ; TN) N () t N A β TN T β ε N N K N () Ν H() t where ε t ~ 0, (8) () H t () h t 0 L 0 0 h t L 0 () = M M O L hn () t. The transton equaton of the state-space model over the tme nterval Δt of the dscrete sample may be expressed as ( β ) ( () β ) ω ( X ( t+ ) =Γ X ( t ) ;, Δ t +Σ X t ;, Δ t t+ ) (9) 9

11 () () ω ( + ) K where Γ X () t ; β, Δ t = E X t+δ t X t, Σ X () t ; β, Δ t = Var X t+δ t X t and t s a error vector wth zero mean and unt varance. The Kalman flter provdes an optmal soluton to predctng, updatng and evaluatng the lkelhood functon for Gaussan state-space models. For the non-gaussan case, the Kalman flter may be used to extract approxmate frst and second moments of the model. In these models the Kalman flter s quas-optmal and may be used to construct an approxmate quaslkelhood functon. Defne the mean state matrx as and the state covarance matrx as ( ) () ( X () t β t) a( β t) b( β t) Xˆ ( Γ ˆ ;, Δ =, Δ +, Δ t ) (0) P t t Var X t t + = + Ω and P() t = Var X () t Ω () t () where X () t E X () t () t = Ω, Ω() t represents the nformaton avalable at tme and are K and K K matrces, respectvely. t a( ) and b( ) Equatons (8) and (9) descrbe the state space representaton. The Kalman flter provdes optmal estmates, Xˆ ( t+ ), of the state varables gven nformaton at tme t +. The condtonal mean and varance of Xˆ ( t+ ) may be expressed as and { } ( β) ( β) X ˆ t+ t = E t X t+ = a + b X ˆ t () P t t E t X t X t t X t X t t ( + ) = () ( + ) ˆ ( + ) ( + ) ˆ ( + ) () 0

12 Gven Σ( X () t ; β, Δ t) s affne n X () t and () ( () ) Cov X t, Σ X t ; β, Δ t ω ( t + ) Ω() t = 0 and usng the law of terated expectatons ˆ P t+ t = b β, Δt P t b β, Δ t +Σ X t ; β, Δ t. (4) Equatons () and (4) are referred to as the predcton step. The second step n calculatng the Kalman flter nvolves updatng the estmaton from the predcton step gven the arrval of new nformaton based on actual observatons, Y() t. Hence, the optmal estmates of the state vector and state covarance matrx are gven by and where ˆ ( ) ν Xˆ t+ = X t+ t + K t+ t+ (5) ( ) ( ) P t+ = P t+ t K t+ B t+ P t+ t (6) ( t ) Y( t ) Y( t ) ν + = + +t (7) ( ) () ˆ ( ) Y t+ t = B t X t+ t + A t (8) K t+ = P t+ t B t+ F t+ (9) ( ) F( t+ ) = B( t+ ) P( t+ t) B( t+ ) + H( t+ ) (0) Equatons (5) and (6) are referred to as the update step and equatons (7) to (0) are the observaton estmaton error, transton estmaton, Kalman gan and covarance matrx of R( t t + ), respectvely. For the Kalman flter to provde an optmal estmaton of Xˆ ( t+ ), the followng condton must hold ( ˆ ) Cov X t X t, Y s + + = 0; s =, K, t +. ()

13 The log-lkelhood functon may be expressed as log LY,, Y N; log T N log F t t F t t, N N ( () K β) = π ν + ( + ) ( + ) ν( + ) () = = wth the nverse and determnant of F ( t+ ) expressed as F t H t H t B t P t t B t H t B t B t H t + = + + ( + ) ( + ) + ( + ) ( + ) ( + ) ( + ) ( + ) F t Ht * Pt t * Pt t Bt Ht Bt. ( + ) = ( + ) ( + ) ( + ) + ( + ) ( + ) ( + ), () In the Gaussan case, the condtonal mean and varance of the system s correctly specfed. Thus, the measurement and transton equatons and the Kalman flter recurson can be used to conduct predcton-error decomposton n the evaluaton of the exact lkelhood functon. However, n the non-gaussan case, the lnear Kalman flter does not produce Xˆ ( t+ ) but rather X ( t+ ), the lnear projecton of X ( t+ ) on the lnear sub-space generated by the observed yelds. Ths lnearly optmal approxmaton yelds a quas-lkelhood functon. As dscussed n Bollerslev and Wooldrdge (99), the hyperparameter vector ˆ β ( T ) that maxmses the quaslkelhood functon n the non-gaussan case s approxmately consstent and asymptotcally normal. Alternatvely, n the Gaussan case, the quas-lkelhood functon turns nto the exact lkelhood functon gven normally dstrbuted measurement errors. The asymptotc dstrbuton of β = ( θ κ σ λ h) where s ˆ ( ) ( ˆ β β 0 ~ 0, ˆ ˆ ) (4) T T N F T G T F T T = ˆ ( β T t = ) (5) Fˆ ( T ) f T ; Y t, T ˆ ( ˆ ln l β T ; Y t, t) ln l( β ( T) ; Y( t), t T ) (6) t = GT ˆ = T ˆ β ˆ β

14 and ψ t ( ˆ; (), ) () ψ t t β ( () () = Ψ + Ψ Ψ ) () () Ψ() Ψ() t f Y t t t t t β β β β T ( β ) ( L ; Y( T), T = = l β; Y( t), T t ) (7) (8) where ψ and Ψ are the condtonal mean and varance functons from the lnear Kalman flter. 4.0 Estmaton Results of Mult-factor Models Term structure models were orgnally estmated wth ether tme seres bond yelds or a crosssecton of bond yeld over dfferent maturtes. The tme seres approach ncorporates the ntertemporal dynamcs of the term structure but not cross-secton nformaton. 4 However, to ensure the model s arbtrage free, a range of maturtes should be ncluded n the estmaton. The cross-secton approach whch uses bond yeld data across maturtes at a pont n tme has the drawback that the parameters can be unstable over dfferent ponts n tme. 5 Hence, the ncorporaton of both tme seres and cross-secton data n emprcal tests of the term structure allows for the proper use of nformaton from both dmensons n order to obtan more accurate parameter estmates. 6 Nevertheless, a man drawback of tme seres/cross-secton models of the term structure s that f the number of maturtes s larger than the number of factors, the model wll be under dentfed. In order to crcumvent ths problem, ths paper follow the approach of most term structure models that rely on panel data whch add Gaussan measurement errors when estmatng the relatonshp between the maturty yelds and the unobserved state factors to obtan See Duan and Smonato (998). 4 Examples of recent term structure models that rely on tme seres data nclude: Anderson and Lund (997), Brenner, Harjes and Kroner (996), Broze, Scallet and Zakoan (995) and Chan, Karoly, Longstaff and Sanders (99). 5 Examples of recent term structure models that rely on cross-secton data nclude: Brown and Dybvg (986), Brown and Schaefer (994) and De Munnk and Shotman (994). 6 Examples of recent term structure models that ncorporate both tmes seres and cross-secton data nclude: Babbs and Nowman (999), Ball and Torous (996), Chatterjee (005), Chen and Scott (995), De Jong (000), Duan and Smonato (995), Geyer and Pchler (996), Gbbons and Ramaswamy (99), Jegadeesh and Pennacch (996), Lund (997), Pearson and Sun (994), and Pennacch (99).

15 consstent parameters. The ncluson of measurement errors s consstent wth the exstence of market regulartes such as bd-ask spreads and non-synchronous tradng. 4. Data Descrpton The data used n the emprcal study conssts of daly zero coupon GOJ domestc bond yelds from 4 September 004 to 8 July 006 obtaned from Bloomberg. In partcular, the panel data set covers 45 observatons and N=5 nterest rates. The maturtes ncluded 0.5-, 0.5-, -, -, -, 4-, 5-, 6-, 7-, 8-, 9-, 0-, 5-, 0- and 0-year tenors. Table. Summary Statstcs: GOJ Zero Coupon Bond Yelds 9/4/004-7/8/006 Maturty mth 6 mth yr yr yr 4 yr 5 yr 6 yr 7 yr 8 yr 9 yr 0 yr 5 yr 0 yr 0 yr Mean Medan Maxmum Mnmum Std. Dev Skewness Kurtoss The average yeld per maturty over the sample perod ndcates that, on average, the GOJ zero coupon term-structure s upward slopng (see Table ). The average spread or premum between the -month and 0-year spot rates s approxmately 500 bass ponts. Ths sgnfcant rsk premum of 8.0 per cent demanded by nvestors s lkely to be caused by unfavourable GOJ debt ratos. The volatlty of the spot rates s greatest at the 0-year maturty and lowest at the - month to 4-year maturtes. Ths s nconsstent wth expectatons of greater volatlty at the shorter maturtes whch may be due to greater uncertanty regardng the rskness of GOJ bonds. The skewness and kurtoss parameters ndcate that the dstrbutons are not normal across maturtes. 7 The skewness coeffcent of all yelds, except the 0-year yeld, s greater than zero ndcatng a lower downsde rsk relatve to the normal dstrbuton. The kurtoss values below 7 The skewness and kurtoss of the Normal dstrbuton s 0 and, respectvely. 4

16 for all yelds apart from the 0-year maturty, mples lower losses when compared to the normal dstrbuton. The zero-coupon yelds on the GOJ bonds are hghly correlated (>80.0 per cent) across all maturtes, abstractng from the 0- and 0-year maturtes whch exhbt much lower correlaton coeffcents (see Table ). For the most part, the correlatons are close to perfect between yelds on maturtes up to one year apart. As the number of years ncrease between maturtes, these par-wse correlaton coeffcents declne, suggestng the use of a mult-factor term structure model. Table. Correlaton Matrx: GOJ Zero Coupon Bond Yelds 9/4/004-7/8/006 -month 6-month -year -year -year 4-year 5-year 6-year 7-year 8-year 9-year 0-year 5-year 0-year 0-year -month month year year year year year year year year year year year year year Emprcal Results One-, two-, and three factor Vascek and CIR models are estmated to obtan the parameters estmates of λ, κ, θ and σ, the standard devaton estmates of the N measurement errors, h, as well as the values for the log-lkelhood and Akake Informaton Crteron (AIC) 8 (see Tables and 4; standard errors are shown n talcs). 8 The ntal startng values chosen for these parameters were the same across both models. Further, the parameter estmates were robust to varatons n the startng values. 5

17 The results for the Vascek model ndcate that all of the λ, θ and σ parameters are statstcally nsgnfcant at the 5.0 per cent level (see Table ). In addton, the standard errors are generally very large and n most cases ncrease sgnfcantly as the number of factors ncreases. The results are mxed for the κ parameters. The κ parameters are statstcally sgnfcant n the two- and three-factor models but not sgnfcant n the one-factor model. All of the estmated standard devaton parameters for the measurement errors are statstcally sgnfcant. The log-lkelhood values show strong ncreases as the number of factors ncrease. However, only one of the 5 estmated standard devaton parameters for the measurement errors dsplays a consstent declne as the number of factors ncrease. The smallest standard devatons for measurement equaton n the Vascek models are, and 0 bass ponts for the 5-year bond rate n the one-, two- and three-factor models, respectvely. The largest standard devatons are, 85 and 95 bass ponts for the 0-year bond rate n the one-, two- and three-factor models, respectvely. These large measurement errors suggest that the models are unable to explan a sgnfcant porton of the 0-year yeld movements. The parameter results from the CIR model estmaton produced sgnfcantly more favourable results (see Table 4). Most of the λ, κ, θ and σ parameter estmates are statstcally sgnfcant at the 5 percent level, except θ n the one-factor model and λ, θ, θ, κ and σ n the three-factor model. All of the parameter estmates are statstcally sgnfcant for the two-factor model. The estmates of the market prce of rsk parameter, λ, for the CIR models have plausble values. These estmated parameters also have large negatve values, ndcatng the exstence of large and postve rsk prema for the latent factors. 9 9 Some examples of rsk premum estmates for the level factor usng CIR models n the lterature nclude: -0. and 0.0 for the UK and German term structure over 6//99 8//04, respectvely (see Chatterjee (005); -0. and. for the US two-factor and three-factor term structure models over /8 /88 (see Chen and Scott (00). 6

18 The estmates of the rate of mean reverson parameter, κ, are also sgnfcant except for the frst factor of the three-factor model. These estmates range from 0.5 to 0.8, ndcatng that the mean half lves, or the expected tme for the short rate to return halfway to ts long-term average mean, ranges between 0.9 to.4 years. 0 Ths narrow range of mean half-lfe values mples that mean reverson for GOJ rates s relatvely fast and that the factor determnes varatons prmarly at the short end of the yeld curve. The values for the volatlty estmates, σ, are statstcally sgnfcant and small ( bass ponts for each factor), ndcatng a relatvely smooth process of mean reverson. Half of the parameter estmates for long-term average rate (asymptotc nterest rate), θ, are sgnfcant and ther values are very close to zero. However, the condton κθ < σ does not hold, ndcatng that the orgn acts as both a reflectng and absorbng barrer for the process. Ths mples that the process remans strctly postve. The correlaton coeffcent between factors one and two n the two factor model s The log-lkelhood value and AIC values mprove by. per cent when movng from the one-factor model to the two-factor model but deterorates notably (-7.6 per cent) when movng to the three-factor model. Ths s taken as evdence that the two-factor model out-performs the one- and three-factor models. 0 The half lfe s computed usng: exp( κ jt) t = ln(0.5) κ j. The lkelhood rato (LR) statstc rejects the null hypotheses that the addtonal factors are not jontly sgnfcant at the.0 per cent level. However the LR test s unrelable n ths case because t does not have the standard asymptotc χ dstrbuton when the errors are not Gaussan. 7

19 Table. Estmates from Vascek Model for GOJ Bond Yelds One Factor Model Two Factor Model Three Factor Model λ (9.078) ( ) (665.79) λ ( ) ( ) λ -.5 (86.0) θ (4.6795) (66.575) (400.4) θ ( ) (86.9) θ (.7575) κ (0.009) (0.06) (0.05) κ (0.005) (0.0079) κ.68 (0.0670) σ (0.057) (0.4045) (.097) σ (0.7) (.786) σ (4.7800) h (0.000) (0.0004) (0.000) h (0.000) (0.0004) (0.0004) h (0.0000) (0.0004) (0.0007) h (0.000) (0.000) (0.0006) h (0.000) (0.000) (0.005) h (0.000) (0.000) (0.00) h (0.000) (0.000) (0.0000) h (0.000) (0.0005) (0.000) h (0.0007) (0.000) (0.00) h (0.0005) (0.0006) (0.007) h (0.0005) (0.0005) (0.00) h (0.0005) (0.0004) (0.0006) h (0.00) (0.00) (0.006) h (0.00) (0.004) (0.0009) h (0.0059) (0.005) (0.004) LogL AIC

20 Table 4. Estmates from CIR Model for GOJ Bond Yelds One Factor Model Two Factor Model Three Factor Model λ (0.006) (0.045) (0.4) λ (0.0074) (0.044) λ (0.0460) θ (0.0000) (<0.000) (0.00) θ (<0.000) (0.0004) θ 0.00 (0.0006) κ (0.00) (0.080) (0.455) κ (0.06) (0.049) κ (0.0557) σ (0.000) (<0.000) (<0.000) σ (<0.000) (0.000) σ (0.000) h (0.000) (0.000) (0.05) h (0.000) (0.000) (0.0) h (0.000) (0.00) (0.006) h ( ) (0.0007) (0.0007) h (0.0007) (0.0008) (-0.000) h (0.0004) (0.000) (0.000) h (0.000) (0.000) (0.000) h (0.08) (0.000) (0.000) h (0.008) (0.0004) (0.0005) h (0.007) (0.000) (0.000) h (0.00) (0.0004) (0.000) h (0.0009) (0.000) (0.0669) h (0.006) (0.008) (0.000) h (0.005) (0.008) (0.009) h (0.007) (0.00) (0.0064) LogL AIC

21 Two of the 5 estmated standard devaton parameters for the measurement errors tend to zero as the number of factors ncrease. The smallest standard devatons for measurement equaton n the CIR models are 6 and bass ponts for the 5-year bond rate n the one- and three-factor models, respectvely, and 0 bass ponts for the 0-year bond n the two-factor model. The largest standard devatons are 47, 450 and 5 bass ponts for the 0-year bond rate n the one-, two- and three-factor models, respectvely. Smlar to the Vascek models, these values are sgnfcantly larger compared to the relatvely low standard devatons for the remanng bond rates. Hence, asde from the 0-year yeld, the factors explan most of the yeld fluctuatons n the CIR models suggestng that the 0-year yeld fluctuaton s not adequately explaned by the CIR model. The tme seres evoluton of the combned factors of the two-factor CIR model are compared wth the evolutons of the -month to 0-year bond yelds (see Fgure ). The combned factors are strongly correlated wth these yelds suggestng that monetary polcy nfluences these yelds. The correlaton coeffcents between the combned factors of the two-factor CIR model and GOJ yelds range from 94.0 per cent to 00.0 per cent for the -month to 5-year yelds and 8.0 per cent for the 0-year yeld. The correlaton between the combned factors and the 0-year yeld was sgnfcantly lower wth a value of 44.0 per cent (see Fgure ). The Kalman flter one-step ahead n-sample predcted yelds and the actual yelds for the two-factor CIR model are llustrated n Fgure. There appears to be a strong postve correlaton between these predcted and actual yelds, partcularly for the 4-year to 0-year GOJ maturty yelds. 0

22 Fgure. Evoluton of Combned Factors of -Factor CIR Model and the -month to 0-year maturtes COMBINED_FACTORS THREEMONTH SIXMONTH ONEYEAR TWOYEAR THREEYEAR FOURYEAR FIVEYEAR SIXYEAR SEVENYEAR EIGHTYEAR NINEYEAR TENYEAR FIFTEENYEAR TWENTYYEAR Fgure. Evoluton of Combned Factors of -Factor CIR Model and the 0-year maturty COMBINED_FACTORS (Left Hand Scale) THIRTYYEAR 4. Factor Loadngs The factor loadngs as a functon of maturty presented n ths secton s based on the estmated parameters of the measurement equaton n the one- and two-factor CIR models (see Fgures 4 and 5). The factor loadngs are derved usng the coeffcents of BT as expressed n equaton (6). The term structure of zero yelds can be one of three possble shapes. If the short rate, r, s less than Y ( ), then shape s monotoncally ncreasng. It s monotoncally decreasng or humped when >. r Y

23 Fgure. Actual and Predcted GOJ Yelds One-step-ahead THREEMONTH One-step-ahead SIXMONTH One-step-ahead ONEYEAR One-step-ahead TWOYEAR Std. Resduals Actual Predcted Std. Resduals Actual Predcted Std. Resduals Actual Predcted Std. Resduals Actual Predcted One-step-ahead THREEYEAR One-step-ahead FOURYEAR One-step-ahead FIVEYEAR One-step-ahead SIXYEAR Std. Resduals Actual Predcted Std. Resduals Actual Predcted Std. Resduals Actual Predcted Std. Resduals Actual Predcted One-step-ahead SEVENYEAR One-step-ahead EIGHTYEAR One-step-ahead NINEYEAR One-step-ahead TENYEAR Std. Resduals Actual Predcted Std. Resduals Actual Predcted Std. Resduals Actual Predcted Std. Resduals Actual Predcted One-step-ahead FIFTEENYEAR One-step-ahead TWENTYYEAR One-step-ahead THIRTYYEAR Std. Resduals Actual Predcted Std. Resduals Actual Predcted Std. Resduals Actual Predcted

24 As gven by equaton (4) the sum of all factors n a mult-factor term structure model s equal to the level of the nstantaneous short rate. The coeffcents on the factor of the one-factor model and the st factor of the two-factor model dsplay the same pattern of rapd declne as the tme to maturty ncreases, ndcatng a strong mpact for the short-term rates. Specfcally, these factor loadngs dsplay steep declnes between 0 and.5 years. The declnes become less steep as the tme to maturty ncreases to around 0 years and level off at very low levels for the remanng maturtes. These factors could represent level factors. The nd factor loadng of the twofactor model exhbts a steep ncrease for short-term rates between 0 and 5 years whch dmnshes as the tme to maturty ncreases to around 0 years and levels off for the remanng maturtes. Ths factor could represent the steepness factor correspondng to the slope of the yeld curve. Fgure 4. Factor Loadng of One-Factor CIR Model.00% 0.00% 8.00% 6.00% 4.00%.00% 0.00% See Ltterman and Schenkman (99).

25 Fgure 5. Factor Loadngs of Two-Factor CIR Model.00% 0.00% 8.00% 6.00% 4.00%.00% 0.00% maturty (years) st Factor nd Factor 5.0 Concluson In ths paper sngle- and mult-factor verson of the Vascek and CIR models of the term structure of nterest rates were estmated usng a state space formulaton. Ths approach combnes both cross-secton and tme seres nformaton based on a system of bond prce equatons to generate estmates of unobservable state varables that drve the term structure. The models are estmated for up to three factors usng a quas-maxmum-lkelhood estmator wth a Kalman flter. Ffteen bond maturtes were used comprsng the 0.5-, 0.5-, -, -, -, 4-, 5-, 6-, 7-, 8-, 9-, 0-, 5-, 0- and 0-year computed zero-coupon GOJ bond yelds coverng the perod 4 September 004 to 8 July 006 to estmate the parameters of each model. Based on the emprcal results, the Vascek models performed very poorly relatve to the CIR models. Addtonally, the results suggested that the -factor CIR model provded the best representaton of the dynamcs of the yeld curve. Based on the factor loadngs, extracted factors of the two-factor model correspond wth the general level and slope of nterest rates, 4

26 respectvely. The emprcal analyss for the -factor model revealed that the level of the short rate exhbted strong and smooth mean reverson and ndcated the exstence of a large and sgnfcant rsk premum that ncreases wth tme to maturty. The values of the parameter estmates for the long-term average rate are all vrtually zero. However, ths s probably a result of the sample perod under analyss. That s, the perod corresponds to a consstent seres of downward adjustments to Bank of Jamaca repurchase rates followng a substantal upward adjustment of over bass ponts durng an epsode of substantal foregn exchange market nstablty n 00. The strong reversal of the short rate snce 00 could explan the domnant expectatons of nvestors for consderable loosenng of monetary polcy beng reflected n the estmated long-term average rate. A summary of the key fndngs of ths study, based on sgnfcant estmates from two-factor CIR model, are: The short-rate (nfluenced by monetary polcy) exhbts rapd declne between 0 and.5 years whch become less steep as the tme to maturty ncreases to around 0 years and levels off to a very low level for the remanng bond maturtes Rsk premum parameters have large negatve values, ndcatng the exstence of a large and postve rsk prema for the level and steepness factors that ncreases wth the tme to maturty of GOJ bonds Long-run average yeld parameters reveal that nvestors were expectng lower nterest rates over the sample perod Mean reversons for the level and steepness factors that drve the dynamcs of GOJ yelds are relatvely fast and smooth ndcatng relatvely short-lves for monetary shocks The level and steepness factors explan varatons prmarly at the short end of the yeld curve 5

27 The Kalman flter one-step ahead predctng yelds appear to closely track actual GOJ bond yelds, partcularly for the 4-year to 0-year maturty yelds. Smlar to tradtonal research on the term structure, ths study examned a yelds-only latentfactor model of the dynamcs of the yeld curve. Recent studes n the lterature have focused on uncoverng the relatonshp between term structure models and specfc macroeconomc varables. Future research wll explctly ncorporate the relatonshp between term structure latent factors and macroeconomc varables of nterest n the Jamaca case. For example, based on estmated factor loadngs, ths study concluded that the unobserved short rate (related to the BOJ polcy rate) has a sgnfcant mpact on the short end of the yeld curve and a relatvely mnmal mpact on the long end. Relevant observable macroeconomc varables that could be jontly ncorporated wth latent state varables n a state-space model of the term structure nclude monetary aggregates, the expected nflaton gap, the expected output gap, foregn nterest rates, as well as the fscal defct to account for yeld movements at the long end. See, for example, Rudebusch and Wu (00) for an applcaton of a macro-fnance term structure model to US Treasury yelds. 6

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