Multi-Factor Cox-Ingersoll-Ross Models of the Term Structure: Estimates and Tests From a Kalman Filter Model. Ren-Raw Chen* and Louis Scott**

Transcription

1 Muli-Facor Cox-Ingersoll-Ross Models of he Term Srucure: Esimaes and Tess From a Kalman Filer Model By Ren-Raw Chen* and Louis Sco** July 993 Revised July 995 Revised Ocober 998 Revised February 00 Revised Sepember 00 * Associae Professor of Finance, Rugers Universiy. ** Execuive Direcor, Morgan Sanley & Co. Ren-Raw Chen Louis Sco Rugers Business School Morgan Sanley & Co. 94 Rockafeller Rd., Room 5 Cabo Square, Canary Wharf Piscaaway, NJ London E4 4QA U.S.A. Unied Kingdom rchen@rci.rugers.edu Louis.Sco@morgansanley.com

2 Muli-Facor Cox-Ingersoll-Ross Models of he Term Srucure: Esimaes and Tess From a Kalman Filer Model REN-RAW CHEN School of Business, Rugers Universiy, Piscaaway, NJ LOUIS SCOTT Morgan Sanley & Co., London E4 4QA Absrac This paper presens a mehod for esimaing muli-facor versions of he Cox, Ingersoll, Ross (985b) model of he erm srucure of ineres raes. The fixed parameers in one, wo, and hree facor models are esimaed by applying an approximae maximum likelihood esimaor in a sae-space model using daa for he U.S. reasury marke. A nonlinear Kalman filer is used o esimae he unobservable facors. Muli-facor models are necessary o characerize he changing shape of he yield curve over ime, and he saisical ess suppor he case for wo and hree facor models. A hree facor model would be able o incorporae random variaion in shor erm ineres raes, long erm raes, and ineres rae volailiy. Key Words: ineres raes, erm srucure, Kalman filer. Inroducion The Cox, Ingersoll, Ross (985b) model is an equilibrium asse pricing model for he erm srucure of ineres raes. The model provides soluions for bond prices and a complee characerizaion of he erm srucure which incorporaes risk premiums and expecaions for fuure ineres raes. The model is frequenly presened as a one facor model, bu in secions 6 and 7 of heir paper, Cox, Ingersoll, and Ross, hereafer CIR, show how o incorporae muliple facors and how o exend he model o value nominal bonds and nominal claims. The model is imporan for several reasons: i provides a link beween ineremporal asse pricing heory and he erm srucure of ineres raes, preserves he requiremen ha ineres raes remain nonnegaive, and produces relaively simple closed form soluions for bond prices. The model is also useful as a ool for valuing ineres rae derivaive securiies. In his paper, we esimae muli-facor versions of he CIR model by using a sae space model in which esimaes of he unobservable sae variables are generaed by a Kalman filer. One, wo, and hree facor models are esimaed, and several ess are performed o deermine wheher hese models can accuraely capure he variabiliy of he erm srucure over ime. The esimaion echnique is differen from mehods previously used o esimae CIR models, and he economeric model is able o capure several imporan feaures of he erm srucure. The Kalman filer model does no require he addiional resricive assumpions associaed wih previous work based on maximum likelihood esimaion. 3 Our resuls suppor he case for muli-facor models and he empirical ess idenify several advanages associaed wih a hree facor model. The hree facor model ha we esimae represens an exension of he wo facor model analyzed by Longsaff and

3 Schwarz (99). In ha model, Longsaff and Schwarz inerpre he wo facors as he shor erm ineres rae and ineres rae volailiy. An alernaive inerpreaion is one in which he wo facors are he shor erm rae and a long erm rae, which is similar in spiri o he work of Brennan and Schwarz (979). The wo facor model ha we esimae seems o fi more closely o his laer inerpreaion. If he shor erm rae, a long erm rae, and ineres rae volailiy are hree disinc, imporan influences for he erm srucure, hen a hree facor model is necessary. Using facor analysis applied o a sample of bond reurns, Lierman and Scheinkman (99) have found ha hree facors are necessary o characerize empirically he ineremporal variaion of he erm srucure: he general level of ineres raes, he slope of he yield curve, and curvaure, which is associaed wih volailiy. 4 The hree facor model ha we esimae capures hese hree feaures of he erm srucure wihin a model ha can be used for asse pricing. We direcly esimae he fixed parameers of he CIR model, and we effecively impose all of he resricions implied by he model. These fixed parameers deermine he cross correlaions and he dynamic behavior of bond raes. In addiion, he economeric model produces esimaes of he parameer combinaions ha are relevan for asse pricing. Finally, i is necessary o observe ha he CIR model does no saisfy all of he normaliy assumpions required for saisical consisency in he maximum likelihood esimaion of a saespace model. As we explain in secion, he Kalman filer for esimaing he unobservable sae variables requires modificaion. As a resul, he filer for he CIR model is no linear and may have a bias even hough i is a minimum mean squared error esimaor. The innovaions in he CIR model have noncenral χ disribuions, in conras o he normal disribuion ha is assumed for maximum likelihood esimaion of Kalman filer models. Maximum likelihood esimaion under he assumpion of normaliy is ofen applied in cases where he fundamenal innovaions are no normally disribued. When hese esimaors are consisen, hey are classified as quasi maximum likelihood esimaors. Consisency for hese esimaors can be verified by seing up he firs order condiions for he maximizaion, ln L β 0, and checking he large sample properies. In he case of quasi maximum likelihood esimaors ha impose normaliy assumpions, saisical consisency depends on correcly modeling he firs and second momens. In he applicaion of he CIR model here, he firs and second momens are modeled correcly, bu saisically consisency canno be esablished because he Kalman filer produces esimaes of he unobservable sae variables ha may be condiionally biased. The Kalman filer esimaes are, however, minimum mean squared error esimaes and are uncondiionally unbiased. We examine he seriousness of his poenial bias by performing a Mone Carlo analysis of he approximae maximum likelihood esimaor. This analysis reveals significan biases for he parameers ha deermine he ime series properies of ineres raes, bu he biases for he parameer combinaions ha are relevan for asse pricing are found o be eiher small or insignifican. The paper is organized as follows. In secion we presen a muli-facor model for pricing nominal bonds which follows from secions 6 and 7 of CIR, and we show ha he model can be se up in discree ime as a sae space model, which is esimaed by approximae maximum likelihood. Esimaes of he unobservable sae variables are compued wih a nonlinear Kalman filer. In

4 secions 3 and 4, we presen he esimaes for one, wo, and hree facor models, and we perform a variey of ess on he models. In secion 5, we presen he Mone Carlo analysis of he approximae maximum likelihood esimaor.. The Sae Space Model for Parameer Esimaion. The CIR Model of he Term Srucure The model for he analysis is he nominal pricing model in equaions (57) - (60) of CIR (985b). The insananeous nominal ineres rae is assumed o be he sum of K sae variables, r K y and he sae variable are assumed o be independen and generaed as square roo diffusion processes: dy κ ( θ y ) d + σ y dz, for,... K., The soluion for he nominal price a ime of a nominally risk-free bond ha pays $ a ime s is deermined as follows: { B y B y } N ( s) A (, s ) AK (,s ) exp K K. where A (, s) and B (, s ) have he form ha is given in CIR: A γ e ( κ + λ γ ) ( s) ( γ e, s ) γ ( s ) γ ( s + ( κ ) ( ) + λ + γ e ) κ θ σ γ ( s ) ( ( e ) B,s ) γ ( s) γ ( s γ e + ( κ ) ( ) + λ + γ e ) and γ ( κ + λ ) + σ. Each sae variable has a risk premium, λ y, and each λ is reaed as a fixed parameer. 5 The coninuously compounded yield for a discoun bond is defined as follows: R ( s) ln N ( s) s 3

5 which is a linear funcion of he unobservable sae variables. Given a se of yields on K discoun bonds, one can concepually inver o infer values for he sae variables. 6. The Sae Space Model for Esimaion To esimae he CIR model, we use he sae space model, which is described in Engle and Wason (98) and Wason and Engle (983). 7 Because he unobservable sae variables are disribued condiionally as noncenral χ variaes, adusmens mus be made o he Kalman filer. If we consider observaions for he sae variables and bond raes sampled a discree ime inervals, he coninuous ime model can be expressed as follows: y a + Φ y + v R A + B y, where y, v, and a are K vecors, R and A are M vecors, Φ is a K K diagonal marix, and B is an M K marix. y conains he unobservable sae variables and R conains he coninuously compounded yields for various discoun bonds, R ( s i ), i,..., M. A, B, a, and Φ are funcions of he fixed parameers in he sochasic processes for he sae variables. The individual elemens of y and R are as follows: y θ ( e κ ) + e κ y, + v,,..., K R ( s ) i K k ln Ak (, si) s i + K k Bk (, si) s i y k, i,..., M where is he size of he ime inerval in he discree sample. The equaions for he sae variables follow direcly from he noncenral χ disribuion. The expecaion for y condiional on informaion a - is κ e κ θ ) + e y. (, The error erm v represens he unanicipaed change in value of zero and a condiional variance equal o y and i has a condiional expeced κ ( θ ( e + e y ) κ e κ σ ) κ,-. There is no serial correlaion in v, bu here is serial dependence in he variance. The model described up o his poin is an exac discree ime represenaion of he CIR model, wihou any of 4

6 he ypical approximaions ha are applied in deriving discree ime represenaions of coninuous ime models. This model can be expressed in he sae space form by adding error erms o he equaions for he observable bond raes: y R a A + Φ y + B y + + ε v, () where ε ( ε,..., ε M ). Each error erm is a measuremen error, or noise erm, ha is inroduced o allow for small errors and imperfecions in he observed bond raes. Bond raes are ypically compued from averages of bid and ask prices, and in many samples, he raes for long erm discoun bonds mus be compued from various coupon bond issues. 8 We assume ha here is no serial correlaion and no cross correlaion in hese measuremen errors for he bond raes. This simple srucure for he measuremen errors is imposed so ha he serial correlaion and he cross correlaion in bond raes is aribued o he variaion of he unobservable sae variables. Wih hese assumpions, he covariance marix for he error erms in () can be wrien as follows: E v ε v ε Q 0 0 U, where Q is a diagonal marix wih he condiional variances of he sae variables on he diagonal, and U is a diagonal marix wih he variances of he measuremen errors on he diagonal..3 The Kalman Filer The fixed parameers of a sae space model are ypically esimaed by he mehod of maximum likelihood using he Kalman filer o compue esimaes of he unobservable sae variables. The model in () fis ino he framework of he sae space model described in equaions () - (3) in Wason and Engle (983), bu he innovaions for he sae variables in he CIR model are no normally disribued. If he innovaions in a sae space model are no normally disribued, he sandard linear Kalman filer is no longer condiionally unbiased as an esimaor of he unobservable sae variables. 9 The fixed parameers in a sae space model are ypically esimaed by using he Kalman filer o compue innovaions in he unobservable sae variables and maximizing a likelihood funcion ha imposes normal disribuions for all of he innovaions. There are models in which he normaliy assumpions can be relaxed and he maximum likelihood esimaor is sill consisen. These esimaors are known as quasi maximum likelihood esimaors. In his applicaion for he CIR model, he quasi maximum likelihood esimaion is no consisen because here is a bias in he Kalman filer. To develop a consisen quasi maximum likelihood esimaor, one mus develop an unbiased esimaor for he unobservable sae variables. To clarify some of hese issues, we begin wih a review of he linear model. The Kalman filer is an algorihm for compuing esimaes of he sae variables a each ime period during he sample. 0 The innovaions for he observed bond raes are defined as: 5

7 ˆ ( A + B a + Φ yˆ )) u, R ( where y is an esimae of y based on u and y. For he iniial esimae ŷ 0, one can use he uncondiional means for he sae variables. The innovaions for he sae variables, given he previous esimaes, are defined as: ˆ y Φ ˆ a y η. The Kalman filer is a linear model for compuing esimaes of he sae variables: y ˆ ˆ + D u, () a + Φ y where beween D is an y and M K marix of coefficiens which are se o minimize he mean squared error ŷ. If he innovaions are normally disribued, his esimaor is also he expecaion condiional on he curren and pas values of he observed variables. In he esimaion of he unobservable sae variables, he fixed parameers of he model are presumed o be known. The esimaor is formed by solving he following minimizaion: min E ( y yˆ ) ( y yˆ ) yˆ K E ( y yˆ ) Because y yˆ η D u, he minimizaion can be resaed as min E ( η D u ) ( η D u ). D Here, he expecaion is condiional on he observaions available a ime ( R, y, R, ). The Kalman filer uses a leas squares proecion of η on u o esimae he coefficiens in D, which deermine he curren innovaions for he sae variables. The firs order condiions for his minimizaion problem are E The covariance marices are defined as follows: [( D u ) u ] E ( η u ) D E ( u u ) 0 η. E (η u ) Σˆ B and E ( u u ) H B Σˆ B U, + ˆ where E ( η ) ˆ η Σ and Σˆ is deermined recursively 6

8 Σ Σˆ Σˆ B H B Σˆ Σˆ Φ Σ Φ + Qˆ. In he CIR model, he diagonal elemens of Q are linear funcions of y, and Qˆ is formed by replacing y in Q wih y ˆ. The soluion ha minimizes he mean squared errors is D ˆ Σ B H and he esimaes are compued as follows: yˆ a + Φ yˆ ˆ H u. + Σ B To sar his algorihm, one ses ŷ 0 equal o he uncondiional mean for y and Σ 0 equal o he uncondiional variance, and he calculaions are done recursively. This filer is he sandard Kalman filer wih one imporan difference: H depends on y ˆ, which depends on observaions hrough ime - ( R, R,...). The model also has one more imporan difference because here is an exra resricion on he sae variables, y 0. If he Kalman filer produces a negaive esimae for y, one can generae a beer esimae, in he sense of minimizing he mean squared error, by seing ŷ equal o zero. One can add his nonnegaiviy consrain o he minimizaion of he mean squared errors and use he Kuhn-Tucker condiions. min E ( η D D u ) ( η D u ) s.. y ˆ a + Φ yˆ + D u 0. The firs order condiions are now modified as follows: Σˆ B D H 0 as yˆ 0. If ŷ > 0, he corresponding equaion holds as an equaliy. The soluion o hese firs order condiions can be found by firs compuing he linear soluion D ˆ Σ B H. If an elemen of ŷ is negaive, se ha esimae equal o zero and drop he corresponding row from he sysem of equaions. The resul is ˆ * * Σ B D H 0, where ˆ * Σ B H ˆ * Σ is ( K ) K and he row dimension of * D is K -. The resuling soluion, * D, produces he same esimaes for he nonnegaive elemens of ŷ found in he original soluion. The ne resul is he linear esimaor from he sandard Kalman filer, wih any negaive esimaes replaced wih zeros. We refer o his Kalman filer as a quasi linear Kalman filer, bu i 7

9 8 is nonlinear. The quasi linear Kalman filer minimizes he mean squared errors subec o he resricion ha he esimaes mus be nonnegaive. Even hough we have reained he linear srucure of he sandard Kalman filer, he resuling filer is nonlinear in wo respecs: he nonnegaiviy resricion and he dependence of he coefficiens in D on ˆ y. Because he esimaor is no sricly linear, i is no a bes linear esimaor ha minimizes mean squared error, and i is possible ha here are oher nonlinear esimaors ha produce smaller mean squared errors. This esimaor, like he sandard Kalman filer, is compued by invering a marix and performing several marix muliplicaions, and i does no require an ieraive soluion o a se of nonlinear equaions..4 The Approximae Maximum Likelihood Esimaor The maximum likelihood esimaor is obained by maximizing he following log-likelihood funcion: ( ) + T T u H u H L L ln ln max β, (3) where β is a vecor conaining all of he fixed parameers o be esimaed. In our applicaion of he sae space model, u is no normally disribued, bu his approximae maximum likelihood esimaor, based on he normaliy assumpion, is a mehod of momens esimaor. The esimaes are found by solving he likelihood equaions, he firs order condiions for he maximizaion problem in (3): 0 r ln + T u H H H u H H u H u L β β β β, for,, N. Seing hese derivaives equal o zero is equivalen o seing 0 r r + T T u u H H H H H T u H u T β β β, for,, N. Here we have used he resul ha r (AB) r (BA). In each one of he firs order condiions, he approximae maximum likelihood esimaor effecively ses he sum of wo sample momens equal o zero. The parial derivaives in he equaions are funcions of he fixed parameers and pas values of he random variables, R, R,, ec. Consisency could be esablished if he following resuls were o hold: 0...),, ( ) ˆ ( R R u E y u E and H R R u u E y u u E...),, ( ) ˆ (.

10 The firs condiion is no saisfied because he quasi linear Kalman filer for he CIR model does no necessarily equal he condiional expecaion for he sae variables. If he Kalman filer is condiionally unbiased, hen boh condiions hold and he esimaor is saisically consisen. In quasi maximum likelihood esimaion, he covariance marix for he parameer esimaes mus be adused and he likelihood raio saisics for model resricions do no have asympoic χ disribuions. As shown in Whie (98), he covariance marix for T ( ˆ β β) converges o ln L ( ) ln ( ) ln ( ) ln ( ) β L β L β L β E E E. β β β β β β (4) If he innovaions are normally disribued as in he sandard model, hen ln L ( β ) ln L( β ) ln L( β ) E E, β β β β and he covariance marix becomes he familiar inverse of he informaion marix. These saisical properies do no necessarily carry over for he approximae maximum likelihood esimaor, bu we use equaion (4) o compue sandard errors for he parameers esimaes. In secion 3, we proceed as if he biases are no serious. In secion 5, we examine he magniudes of he biases. To find he approximae maximum likelihood esimaor, we use a modified mehod of scoring which is described in Bernd, Hall, Hall, and Hausman (974) and Engle and Wason (98). The covariance marix for β is compued by replacing he expecaions in (4) wih sample momens. The firs derivaives and he expecaion of he second derivaive marix are given in Engle and Wason..5 Comparison wih Oher Esimaion Techniques in he Lieraure Several approaches have been used in he previous research on he empirical esimaion of CIR models. Gibbons and Ramaswamy (993) and Heson (989) have used uncondiional sample momens in a generalized mehod of momens (GMM) framework o esimae and es differen versions of he model. GMM esimaors of his form are ypically less efficien han alernaive esimaors and he resuling esimaes for he model parameers have relaively large sandard errors. This approach does have some advanages: specific disribuional assumpions are no required, and as Gibbons and Ramaswamy have noed, he GMM esimaors allow for measuremen errors in he bond raes. Longsaff and Schwarz (99), in heir esimaion of a wo facor model, use one monh T-Bill raes as a proxy for he insananeous ineres rae and esimaes of ineres rae volailiy generaed from a GARCH model. 3 Their parameers are esimaed by regressing bond yield changes on he changes in hese wo esimaed facors. This approach depends on he assumpion ha he esimaes for he wo facors do no conain measuremen error. A hird approach is maximum likelihood esimaion, which has been used in Chen and Sco (993) and Pearson and Sun (994). The likelihood funcion for he observed bond raes is developed from he condiional densiy funcions for he sae variables, which have noncenral χ disribuions. Because here are ypically more bond raes han unobservable facors or sae 9

11 variables, Chen and Sco inroduce measuremen errors for seleced bond raes. To obain racable likelihood funcions, hey assume ha some of he bond raes are measured wihou error. This approach o maximum likelihood is no racable if all of he bond raes are measured wih some error, and i requires a specific disribuion for he measuremen errors. Duffie and Singleon (997) have applied his maximum likelihood esimaor o raes in he ineres rae swap marke. Pearson and Sun have circumvened his problem by assuming no measuremen errors and by resricing he number of cross secions for he bond raes o be equal o he number of facors. In he esimaor developed in his secion, we allow for measuremen errors on all of he bond raes and he likelihood funcion remains relaively simple and racable. This esimaor differs from he previous maximum likelihood esimaors because we do no use he non-cenral χ densiy funcion and we are no required o impose a specific assumpion for he disribuion of he measuremen errors. The esimaor uses he srucure imposed by he CIR model on he firs and second momens of he condiional disribuions, and i imposes more of he model srucure han he previously implemened GMM esimaors. The negaive feaure of he esimaor developed in his secion is he poenial large sample bias. The magniudes of he biases are examined in secion 5. More recenly, a simulaed mehod of momens (SMM) esimaor has been developed by Dai and Singleon (000) for exponenial affine erm srucure models ha include he CIR model as a special case. Their SMM esimaor is saisically consisen, bu he esimaor is compuaionally slow as i requires ieraive soluions coupled wih Mone Carlo simulaions. Two recen papers have used Kalman filers o esimae muli-facor erm srucure models. Babbs and Nowman (999) use he sandard Kalman filer, sae-space model o esimae a generalized Vasicek model, in which all of he innovaions are normally disribued. They use weekly daa on 8 mauriies over he period 987 o 996 o esimae,, and 3 facor models. Their zero coupon raes are exraced from quoes for U.S. LIBOR and U.S. swap raes, and he longes mauriy is 0 years. Geyer and Pichler (999) have esimaed muli-facor CIR models using he esimaion echnique developed in his secion. They use monhly daa for 6 mauriies in he U.S. Treasury marke over he period 964 o 993, and hey esimae models wih up o 5 facors. The longes mauriy in heir sudy is 5 years. In he nex secion, we use 4 mauriies from he U.S. Treasury marke o esimae,, and 3 facor models, and he longes mauriy varies from 5 o 30 years. We emphasize he use of long mauriies because we feel ha he lengh of he mauriies used in he analysis will have a significan impac on he esimaion of he mean reversion parameers. 3. Esimaion of One, Two, and Three Facor Models 3. Daa We use wo daa ses for he esimaion of he CIR model. The firs daa se includes yields for discoun bonds calculaed by McCulloch, and presened in Shiller and McCulloch (990). We use he raes from he able for he zero coupon yield curves for 3 monhs, 6 monhs, 5 years, and he longes mauriy available (0-5 years). The raes are annualized and saed on a coninuously compounded basis and represen raes for discoun bonds. These raes have been compued from monh end prices in he Treasury bond marke, and we use he daa for he period 960 o 987. This monhly daa se serves as a longer sample for he esimaion of he parameers in he ineres raes processes. Ineres rae volailiy appears o have changed over he las 30 years, and Treasury bond 0

12 prices are also available on a daily or weekly basis. The second daa se consiss of bond prices on Thursdays from January 980 o December 988. Prices for 3 week and 6 week T-bills, 5 year Treasury noes, and he longes mauriy noncallable bonds available have been colleced from he Wall Sree Journal. For his period, here are 470 weekly observaions. This paricular se of observaions was seleced for several reasons. T-bills maure on Thursdays and here are only a few holidays ha fall on Thursday during he sample period. 4 By using weekly daa insead of monhly daa, we have a much larger sample size. The four differen mauriies were chosen so ha differen poins along he yield curve could be used o esimae he facors and he parameers in he processes ha deermine he facors. The wo Treasury bills are discoun bonds, bu he wo longer erm bonds are coupon bonds. The yields for 5 year discoun bonds and long erm discoun bonds have been approximaed from he coupon bonds by assuming wo forward raes: one o apply from 6 monhs o 5 years and one from 5 years o 30 years. The longes mauriies on noncallable bonds were 5 years a he beginning of he 980's and 30 years a he end of he sample period. 3. Empirical Resuls The resuls of he esimaion are presened in Tables I and II. Table I conains he esimaes for he monhly daa se, , and Table II conains he esimaes for he weekly daa se, Time is measured in years so ha all of he parameer values are expressed on an annual basis. All of he variance parameers are saisically significan, bu he resuls are mixed for he individual esimaes of he κ, θ, and λ parameers. Mos of he esimaes for he risk premiums are negaive and approximaely half are saisically significan. Only one of he risk premium esimaes is posiive and i is no significan. The κ and θ esimaes are saisically significan for he firs facor, bu hey are generally insignifican for he second and hird facors in he muli-facor models. The log-likelihood values increase dramaically as he number of facors is increased. The sandard deviaions for he measuremen errors naurally decrease as he number of facors is increased. In mos of he cases, he sandard deviaions for he measuremen errors on he 6 monh T- Bill rae go o zero. This is a common phenomenon in facor analysis, and he esimaes for he variances, and he sandard deviaions, are consrained o be nonnegaive. In he weekly daa se, he variance for he measuremen error on he long erm bond rae also goes o zero in he hree facor model. The esimaed sandard deviaions for he measuremen errors are much larger for he one facor model. In he monhly daa se, hese sandard deviaions are 33 basis poins for he 3 monh T-Bill rae, 0 basis poins for he 5 year bond rae, and 3 basis poins for he long erm bond rae. In he weekly daa se, he sandard deviaions are 40 basis poins for he 3 monh T-Bill rae, 04 basis poins for he 5 year bond rae, and basis poins for he long erm bond rae. The measuremen errors for he 5 year bond rae and he long erm bond rae are quie large in he one facor model. The larges sandard deviaions for measuremen errors in he muli-facor models are 30 o 37 basis poins, and mos of he esimaes are much smaller. For example, in he hree facor model esimaed from he weekly daa se, hese sandard deviaions are 3 basis poins for he 3 monh T-Bill rae, 0 for he 6 monh T-Bill rae, 7 basis poins for he 5 year bond rae, and 0 for he long erm bond rae. These resuls indicae ha mos of he variaion in he observed bond raes is explained by he common facors in he muli-facor models. The esimaes for he κ parameers are close o zero for he exra facors in he muli-facor models, and almos all of he esimaes are smaller han heir sandard errors. The only excepion is κ in he wo facor model esimaed from he weekly daa, bu he esimae,.085, is relaively

13 small. The sochasic processes for he sae variables resemble firs order auoregressions if hey are sampled a discree ime inervals, and he κ parameers deermine he rae of mean reversion. The coefficien on y in he auoregression is equal o exp( κ ), where is he size of he ime inerval over which he daa are sampled. If he κ parameer is close o zero, hen he auoregression coefficien is close o one, which is equivalen o having a roo close o he uni circle in he ime series represenaion. If κ is zero, he sae variable is a random walk and he bond raes are no saionary ime series. Cooley, LeRoy, and Parke (99) have argued from a heoreical perspecive ha ineres raes should be saionary ime series. 5 If κ 0 for a square roo process in he CIR model, he facor behaves like a pure random walk bu zero becomes an absorbing barrier for he process; if he process his zero, i disappears. If he κ parameer is small so ha κθ < σ, he process can hi zero, bu zero serves as a reflecing barrier so ha he process coninues. The small κ esimaes indicae ha some of he facors in he muli-facor models do exhibi characerisics similar o random walks, and hese facors are he ones ha explain he variaion of he long erm bond raes. The rae of mean reversion for each facor can be measured by compuing mean half lives. 6 In he wo facor model esimaed from he weekly daa se, he mean half lives are.95 years for he firs facor and 3.7 years for he second facor. In he hree facor model esimaed from he weekly daa se, he mean half lives for he hree facors are.48 years, 40.9 years, and 9.7 years. The facors wih long mean half lives are he ones ha deermine he variaion of he longer erm bond raes. The esimaes for he mean half lives are much shorer in he sudies by Babbs and Nowman(999) and Geyer and Pichler (999). We aribue he difference o our inclusion of 30 year mauriies. The esimaor developed in secion is based on he assumpion ha he bond raes and he sae variables are saionary ime series, bu he Kalman filer can be applied o nonsaionary ime series. 7 If he series are no saionary, one uses a saring value for y 0 which is reaed as a fixed parameer and Σ 0 is se equal o zero. We have reaed he bond raes as saionary ime series and he uncondiional means, θ, are used for he iniial esimae ŷ 0, and he uncondiional variances and covariances are used for Σ 0. If κ is close o zero, he corresponding variance in Σ 0 is large so ha he esimae, ŷ, for he firs period is allowed o have a large deviaion from he uncondiional mean. The Kalman filer uses condiional variances for he subsequen observaions in he sample. The large variance associaed wih a small κ parameer affecs he firs observaion only. We have run he models in Tables I and II wih he nonsaionary Kalman filer and he changes in he parameer esimaes are very small. An analysis of he facor loadings can be used o deermine he naure of he facors calculaed by he Kalman filer. In his model, he facor loadings are he coefficiens in he marix B defined in equaion (). In Figures -4, we presen graphs of hese coefficiens across differen mauriies in he wo and hree facor models. The coefficiens for he wo and hree facor models compued from he esimaes in he sample are presened in Figures and. The coefficiens compued from he esimaes for he weekly daa se, , are in Figures 3 and 4. The paerns are similar for boh ses of graphs. In he wo facor model, he coefficiens on he firs facor decrease quickly as ime o mauriy increases. The coefficiens for he second facor are approximaely one for all mauriies. The firs facor has a srong influence on shor erm raes, bu a diminished effec on long erm raes, and his facor deermines he slope of he erm srucure. The second facor affecs all raes and deermines he general level of ineres raes. We find ha esimaes of he firs facor are highly correlaed wih he slope of he erm srucure, specifically he

14 shor erm rae minus he long erm rae, and esimaes of he second facor are highly correlaed wih he long erm ineres rae. In he hree facor model, he coefficiens for he firs facor are similar o hose for he firs facor in he wo facor model. The coefficiens decrease sharply as ime o mauriy increases, and his facor deermines he slope of he erm srucure. The coefficiens for he second facor decrease slowly as ime o mauriy increases, and he coefficiens for he hird facor increase wih ime o mauriy and hen level off beween 0 and 30 years. The second and hird facors deermine he general level of ineres raes and he relaionship beween medium and long erm raes. We find ha he sum of he second and hird facors is highly correlaed wih medium and long erm raes. In hese model esimaes, he second facor has a higher volailiy and he hird facor has a lower volailiy. The ineracion of hese wo facors deermines he curvaure of he erm srucure and he volailiy of bond raes. In all of hese models, he relevan combinaions of parameers for valuing bonds and ineres rae derivaive asses are ( κ + λ ), κ θ, and σ. The esimaes for hese parameer combinaions in he muli-facor models are presened wih heir asympoic sandard errors in Table III. Mos of hese parameer combinaions are saisically significan in he sense ha he esimaes are large relaive o heir sandard errors, bu several of he esimaes for κ θ are close o zero and are smaller han heir sandard errors. All of he esimaes for ( κ + λ ) and σ are several imes greaer han heir sandard errors. Mos of he parameer combinaions ha are relevan for asse pricing are esimaed wih a high degree of precision. We urn now o saisical ess of he differen models. Because he hree models are nesed, one could use he likelihood raio saisic for hypohesis esing, bu his saisic does no have he sandard χ disribuion when he innovaions are no normally disribued. The asympoic disribuion for he likelihood raio saisic would be a weighed sum of χ disribuions, as described in Vuong (989). We have already noed ha he approximae maximum likelihood esimaor is poenially biased and he saisical resuls for quasi maximum likelihood esimaors may no apply. Comparisons of log likelihood funcions across he models do serve as indicaors of he model performance in fiing he daa. The values for he log likelihood funcion in he monhly daa se are 5,88.77 for he one facor model, 6,730. for he wo facor model, and 6, for he hree facor model. In he weekly daa se, hese values are for he one facor model, for he wo facor model, and 0,44.4 for he hree facor model. The likelihood raio saisics for ess of he one facor model versus he wo facor model are,803 in he monhly daa and 3,007 in he weekly daa. The likelihood raio saisics for ess of he wo facor model versus a hree facor model are 449 in he monhly daa and 83 in he weekly daa. The values for hese likelihood raio saisics are exremely large and would indicae reecion of he null hypoheses a low significance levels if one could apply eiher he sandard χ disribuion or he resuls for weighed sums of χ disribuions in Vuong. There is also a subsanial reducion in he sandard deviaion of he measuremen error for he 5 year bond rae as we move from he wo facor model o he hree facor model. In he monhly daa se, his sandard deviaion is reduced from 37 basis poins o 6 basis poins. In he weekly daa se, i is reduced from 34 basis poins o 7 basis poins. There are reducions in he sandard deviaions for he oher measuremen errors in he hree facor model, bu he improvemens are smaller. During he period 979-8, here was a shif in Federal Reserve policy oward an emphasis on growh raes of he money supply, and ineres rae volailiy increased. In Tables IV and V, we presen esimaes of he CIR model wih his period removed from he wo samples. The resuls are 3

15 similar o hose repored in Tables I and II, excep for he esimaes of he volailiy parameers. The esimaes for he σ parameers are smaller and in some cases he esimaes are much smaller. For example, in he weekly daa se he σ esimaes for he wo facor model decrease from.6885 and o.0855 and when he period is removed. The sandard deviaions for he measuremen errors are also smaller when his period is no included in he samples. The oher aspecs of he resuls remain he same. In he wo and hree facor models, here are facors wih slow mean reversion (κ esimaes close o zero). Mos of he risk premium esimaes are negaive, and he log likelihood funcion increases significanly as he number of facors is increased. 4. Can he CIR Model Explain he Term Srucure of Ineres Raes Over Time? In his secion we examine he abiliy of he CIR models o fi acual bond prices and he differen poins along he yield curve. For seleced days from 980 hrough 99, we have colleced all of he available bond prices for he U.S. Treasury marke. The flower bonds, he callable bonds, and he coupon issues wih less han a year o mauriy have been excluded. The specific daes are given in he noe o Table VI. Eigh daes from June 989 o December 99 fall ouside of he esimaion period. On each day, we compue prices and yields using he hree CIR models wih he parameer esimaes from Table II for he weekly daa se, which covers he period The esimaes for he sae variables are compued from he Kalman filer. The same values for he fixed parameers have been used in he pos sample period, To calculae he esimaes for he sae variables during he pos sample period, we use weekly observaions on he same bond raes from 989 o 99. We hen compare he prices from he hree CIR models wih acual bond prices, and we compare he yields-o-mauriy compued from he CIR model prices wih yields-o-mauriy compued from acual bond prices. To measure he fi for he hree models, we compue roo mean square errors for he errors in prices and yields. Boh absolue pricing errors and percenage pricing errors are compued. The calculaions are summarized in Table VI. The roo mean squared errors are similar for he absolue pricing errors and he percenage pricing errors. The relaive ranking of he hree models is he same across all of he measures of model fi: he hree facor model performs marginally beer han he wo facor model, and boh muli-facor models perform much beer han he one facor model. For he sample of,304 bonds, he muli-facor models have roo mean squared errors which are much smaller han he roo mean squared errors for he one facor model. The muli-facor models coninue o ouperform he one facor model in he pos sample period. During he sample period, he roo mean squared error for prices wih he hree facor model is 35% lower han he roo mean squared error for prices from he wo facor model. In erms of yields, he roo mean squared error for he hree facor model is 9% lower. The hree facor model ouperforms he wo facor model by a small margin in he pos sample observaions from 989 o 99. Wha is he naure of he pricing errors of hese models? In Figure 5, we presen graphs of he yield curve for seleced days on which we have complee ses of observaions for he Treasury marke. 8 In he graphs we plo yield-o-mauriy versus duraion, a common measure of mauriy for bonds. The acual yields are ploed as aserisks agains curves for he hree CIR models. The one-facor model performs poorly and on many days here are large misses for he long end of he yield curve. In all of he graphs, he wo facor model is able o fi boh he shor end of he yield 4

16 curve and he long end, bu on a few days i is unable o fi he inermediae poins. The wo facor model capures he general slope of he yield curve, bu i occasionally misses he shape or curvaure of he yield curve. In almos all of he cases he hree facor model fis he general shape as well as he slope of he yield curve. For example, in June 984, he wo facor model misses he yields for bonds wih duraions of one o five years, whereas he hree facor model generally prices hese bonds correcly. For he pos sample daes, he yield curves for he muli-facor models are close and he acual yields fall very close o he wo curves. For many of he days in he sample, he wo facor model provides an adequae characerizaion of he acual yield curve, bu here are occasions when a hree facor model is necessary o capure he curvaure of he yield curve. The one facor model is unable o characerize he changes in he yield curve over ime and he errors of he model are economically significan. To make he one-facor model perform well over ime, one mus regularly adus he parameer values, bu such a procedure is inernally inconsisen and suggess ha some of he parameers should be reaed as sae variables. Geyer and Pichler (999) run addiional diagnosic ess o show ha he muli-facor CIR models are reeced by he erm srucure daa. 5. A Mone Carlo Analysis of he Approximae Maximum Likelihood Esimaor In secion (), we have modified he Kalman filer o impose a nonnegaiviy resricion and o accoun for he dependence of he variance of he sae variables on previous levels, and he resuling filer is no sricly linear. A a heoreical level, he approximae maximum likelihood esimaor can have large sample biases. In his secion, we simulae a wo facor model o sudy he properies of he modified Kalman filer and he approximae maximum likelihood esimaor. The wo facor model has been chosen because i capures much of he variaion of he erm srucure over ime, and he approximae maximum likelihood esimaor for he wo facor model converges a a much faser rae han he esimaor for he hree facor model. As shown in CIR (985b), if y is generaed as a square roo process, he disribuion of condiional on y s is a noncenral χ if we perform he ransformaion x c y, where y c κ ( e κ ( s) σ ). 4κ θ The random variable x is disribued as a noncenral χ wih degrees of freedom, ν, and a σ κ ( s) noncenraliy parameer, c e y. There are several mehods available for simulaing noncenral χ variaes, and we have used wo mehods described in Johnson and Koz (970, Ch. 8). They noe he following propery of his disribuion: χ δ χ δ χ, / / ν ( ) ( ) + ν where / χ ν ( δ ) is a noncenral χ wih ν degrees of freedom and noncenraliy parameer δ, and / χ ν is a cenral χ wih ν degrees of freedom. The χ ( δ ) variae can be simulaed by 5

17 / simulaing a sandard normal, Z, and performing he following ransformaion: χ ( ) ( Z + δ ). Then a second simulaion from a χ ν is added o his value o produce he simulaion. Some of he esimaes in our muli-facor models produce degrees of freedom ha are less han one and his mehod canno be used in hese cases. Anoher mehod is based on he observaion ha a noncenral χ variae is a mixure of cenral χ variaes. Firs simulae he degrees of freedom from a Poisson disribuion ha has an expeced value equal o δ and hen simulae a χ variae wih he simulaed degrees of freedom. We have used he firs mehod in hose cases where he degrees of freedom are greaer han one and he second mehod when he degrees of freedom are less han one. The random number generaors for he sandard normal, he chi-squared, and he Poisson disribuions conained in he Inernaional Mahemaical and Saisical Library (IMSL) have been used for he simulaions. For he Mone Carlo analysis, we have simulaed he model in () wih wo facors and four bond raes. The simulaions for he unobservable sae variables have been drawn from he noncenral χ disribuion as described above, and he measuremen errors have been simulaed as normal random variables wih zero means. The fixed parameer values have been se a he values given in Table 7, which are very close o he esimaes for he wo facor model in Table. The mauriies for he bond raes in he simulaions are 3 monhs, 6 monhs, 5 years, and 30 years. We checked firs he properies of he Kalman filer by running he filer wih rue values for he parameers. In each sample, we simulaed 470 weeks of observaions for he bond raes and hen calculaed he esimaes for he sae variables. Five hundred independen samples were simulaed, and he resuls are summarized in Table 7 where we repor he means and he roo mean squared errors for y yˆ. The means for boh sae variables are very close o zero, which confirms he resul ha he uncondiional expecaion of he bias in he quasi linear Kalman filer is zero. The possibiliy of a condiional bias has no been examined. The roo mean squared errors are also small, and In basis poins, he roo mean squared errors are 0 and 7, which sugges ha he esimaes from he filer are close o he rue values, and ha he condiional biases may be relaively small. We have examined he behavior of he approximae maximum likelihood (ML) esimaor in wo differen samples. The firs se of simulaions is for samples represening 0 years of monhly daa. The second se is for samples of weekly daa wih 470 weeks of observaions, which is approximaely 9 years of daa. In he iniial simulaions, we found ha here were some samples in which he esimae for σ ( ε ) was approaching zero and he esimaor did no converge. In hese cases, we have fixed σ ( ε ) a a value close o zero and we have resared he esimaor for he remaining parameers. The resuls are summarized in Panels A and B of Table 8. In boh ses of simulaions, here are clearly biases in he esimaes of he κ, θ, and λ parameers, bu no significan biases in he variance parameers, σ and σ, and no biases in he sandard deviaions for he measuremen errors. In secion (3), we noed ha he relevan parameers for asse pricing are he σ, κ + λ, and κ θ combinaions. We have also repored he simulaion resuls for κ + λ and κ θ and here are no significan biases for hese combinaions. The approximae ML esimaor produces reliable esimaes for he sandard deviaions of he measuremen errors and he parameers ha are relevan for asse pricing, bu here are significan biases in he separae esimaes of he κ, θ, and λ parameers. The κ and θ parameers deermine some of he ime series properies of he facors: he κ parameer measures he rae of mean reversion and he θ δ 6

18 parameer is he long run average. 6. Summary and Conclusions In his paper we have used a sae space model o esimae muli-facor versions of he CIR model of he erm srucure of ineres raes. Esimaes of he unobservable sae variables have been generaed by a nonlinear Kalman filer. We find ha muli-facor models are necessary o explain he changes over ime in he slope and shape of he yield curve. In saisical ess, he wo facor model is reeced wih he hree facor model as he alernaive hypohesis. The diagnosic ess in secion 4 sugges ha he wo facor model frequenly performs as well as he hree facor model, bu here are periods when he hree facor model is needed o capure he general shape of he yield curve. The hree facor model has he added flexibiliy necessary o explain he random variaion in shor erm ineres raes, long erm raes, and volailiy. In he muli-facor models, he variaion of long erm raes is explained by facors ha experience very slow mean reversion. This aspec of he empirical resuls is a reflecion of he near random walk behavior of long erm raes. The approximae maximum likelihood esimaor for he CIR model is one ha is poenially biased, even in large samples. The Mone Carlo simulaions confirm ha here are significan biases in some of he parameer esimaors. The significan biases occur in he esimaes of κ, θ, and λ. The κ and θ parameers, along wih he σ parameers, conrol he ime series properies of ineres raes. The κ parameers conrol he raes of mean reversion and he θ parameers conrol he long run averages. The λ parameers conrol he risk adusmens when moving from he real world disribuion o he risk neural disribuion for asse pricing. The parameer combinaions, κ + λ, κ θ, and σ, deermine he risk neural disribuion for asse pricing. In conras, here is no evidence of significan biases in hese parameer esimaes. Geyer and Pichler (999) repor ha he inclusion of more mauriies improves he precision of he parameer esimaes; he sandard errors of he esimaes decrease as more mauriies are added. The inclusion of addiional mauriies increases he number of cross secions in he sample, and his would improve he precision of he risk neural parameer combinaions. This conclusion canno be applied o he esimaion of he long run means, he mean reversion parameers, or he risk premia. Throughou he paper, we have alluded o he poenial condiional bias in he quasi linear Kalman filer as he cause of he biases in he approximae maximum likelihood esimaor. The esimaion errors in he Kalman filer appear o be small in he Mone Carlo simulaions for he CIR model, and he biases in he κ and θ parameers could be nohing more han he familiar finie sample biases found in auoregressive models. The biases in he λ parameers are he resul of he biases in he κ parameers, because he esimaes for he κ + λ combinaions do no have significan biases. Noes. Soluions for opion and fuures prices in wo facor versions of he CIR model can be found in Beaglehole and Tenney (99), Chen and Sco (99), and Longsaff and Schwarz (99). Soluions for he muli-facor model can be found in Chen and Sco (995).. Geyer and Pichler (999) use he same esimaion echnique developed here o esimae muli-facor CIR models. 3. Specifically he esimaion echniques found in Chen and Sco (993), Longsaff and Schwarz (99), and Pearson and Sun (994). For oher work on he esimaion of CIR models, see Brown and Dybvig (986), Gibbons and 7

19 Ramaswamy (993), and Sambaugh (988). 4. There is a relaionship beween volailiy and he curvaure of he yield curve, and his aspec of he erm srucure is examined in Lierman, Scheinkman, and Weiss (99). 5. This model can be derived by applying arbirage mehods or by using he uiliy based model in CIR (985b). The risk premiums are deermined endogenously in a uiliy based model by he covariabiliy of he sae variables wih marginal uiliy of wealh. The form for he risk premium used here is consisen wih a log uiliy model. 6. This inversion of bond raes o infer values for he sae variables has been used in Chen and Sco (993), Duffie and Kan (993), and Pearson and Sun (994). 7. For an applicaion of he sae space model in he finance lieraure, see Pennacchi (99). 8. A similar allowance for measuremen errors in bond raes was used by Sambaugh (988). 9. In his case, he Kalman filer is a linear minimum mean squared error esimaor and i is uncondiionally unbiased. See Harvey (99, pp. 09-3). 0. For references on Kalman filers, see Chow (975, pp ) and Harvey (99).. A minimum mean squared error esimaor can be biased.. See Whie (98) and Vuong (989). 3. They esimaed ineres rae volailiy by applying a GARCH model o one monh T-Bill raes. GARCH is an acronym for generalized auoregressive condiional heeroskedasiciy. 4. For hose weeks during which Thursday is a holiday, we use Wednesday or Friday prices. 5. Saisical ess for uni roos have very lile power and one canno disinguish empirically in a finie sample wheher a ime series has a roo on or us close o he uni circle. 6. The mean half life is he expeced ime for he process o reurn halfway o is long run average. The mean half life is defined as follows: or, where is he mean half life. 7. See Wason and Engle (983, p.387). 8. We presen graphs for 7 days. The graphs for he oher 9 days in he sample can be obained from he auhors. 8

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