An Admissible Macro-Finance Model of the US Treasury Market.

Transcription

1 An Admissible Macro-Finance Model of he US Treasury Marke. Peer Spencer* Universiy of York, U.K. This paper develops a macro-finance model of he yield curve and uses his o explain he behavior of he US Treasury marke. Unlike previous macro-finance models which assume a homoscedasic error process and suppose ha he one-period reurn is direcly observable, I develop a general affine model which relaxes hese assumpions. My empirical specificaion uses a single condiioning facor and is hus he macro-finance analogue of he EA (N) specificaion of he mainsream finance lieraure. This model provides a decisive rejecion of he sandard EA 0 (N) macro-finance specificaion. The resuling specificaion provides a flexible 0-facor explanaion of he behavior of he US yield curve, keying i in o he behavior of he macroeconomy. (JEL: C3, C3, E30, E44, E5) I. Inroducion Macro-finance models use boh observable macroeconomic and unobservable laen variables o model he macroeconomy and bond marke, in conras o he convenional approach which only uses laen variables. Like he convenional approach, i describes yields as linear funcions of hese driving variables in a way ha removes arbirage opporuniies. This new approach allows he parameers of he model o be informed by boh macroeconomic and yield daa. I generaes models ha are easier o inerpre and undersand since hey are based upon * Deparmen of Economics and Relaed Sudies; Universiy of York, York Y00 5DD, UK, , ps35@york.ac.uk. I am graeful o Karim Abadir, Francesco Bravo, Iryna Kaminska, Zhuoshi Liu, Renaas Kyzys, Peer N Smih, Andy Tremayne, Mike Wickens and Tao Wu and paricipans in he Mulinaional Finance and Economeric Sociey European meeings 006 for helpful commens. An edior and a referee of his Journal also provided helpful suggesions. I am paricularly graeful for he edior s suggesion ha he spo rae migh be considered a linear combinaion of all facors, no jus he T-bill rae. (Mulinaional Finance Journal, 009, vol. 3, no. /, pp. 38) Mulinaional Finance Sociey, a nonprofi corporaion. All righs reserved.

2 Mulinaional Finance Journal sandard macroeconomic srucures. However, he curren macro-finance specificaion suffers from a number of drawbacks compared o he convenional one. In paricular i assumes ha he volailiy srucure is consan, while he convenional lieraure finds ha square roo volailiy is significan. Also, macro-finance modelers assume ha he ineres rae ha is relevan for he yield srucure can be idenified and observed wihou error, while he convenional finance model esimaes his as a linear combinaion of he laen facors. In his paper I develop a model of he US economy and Treasury bond marke which relaxes hese assumpions, bridging he gap beween he macro-finance and convenional models of he erm srucure. Since hese various models are linear in variables, his means ha if here are N underlying driving variables, hey can be represened by N bond yields or macroeconomic variables or boh. For example he convenional yield facor approach jus uses linear combinaions of N bond yields as facors assuming ha hese are observed wihou measuremen error. I has been exensively used for esing affine specificaions (Brown and Schaefer (994), Duffie and Kan (996), Dai and Singleon (00)). Macro-finance models on he oher hand are based on he cenral bank model (CBM) developed by Svensson (999); Smes (999) and ohers. This represens he behavior of he macroeconomy in erms of hree variables: inflaion (π ), he gap beween oupu and is inflaion-neural level (g ) and a policy ineres rae like he Fed Funds rae (r ). This provides a basic dynamic descripion of an economy in which he cenral bank implicily arges inflaion using a Taylor rule, which deermines he policy rae in erms of inflaion and he oupu gap. Early macro-finance papers (Ang and Piazzesi (003)) revealed ha he CBM provides a good descripion of he behavior of shor erm yields bu ha a laen variable known as he financial facor has o be added o explain long erm yields. Consequenly, my model employs he hree macroeconomic variables of he CBM wih wo ses of lags, ogeher wih a single laen variable represening he financial facor. This gives a oal of N = 0 sae variables. The financial facor is backed ou from he yield model as a linear combinaion of he 9 observable variables and he 5 year yield. I depar from he macro-finance lieraure in assuming ha boh he mean values and variances of he sysem are linear in he financial facor. This means ha he yield curve is deermined by he square roo volailiy model of Cox e al (985). To handle quarerly economic daa I employ he discree ime version of his model developed by Sun

3 Macro-Finance Model of he US Treasury Marke. 3 (99). In order o ensure ha he variance srucure remains non-negaive, I also employ admissibiliy resricions similar o hose proposed for he coninuous ime model by Dai and Singleon (00). This is he analogue of he (EA ) yield facor model developed by Dai and Singleon (00) and Dai and Singleon (00), which as hey say: builds upon a branch of he finance lieraure ha posis a shor-rae process wih a single sochasic cenral endency and volailiy. Despie he exensive use of sochasic volailiy models in heoreical and empirical finance papers and he evidence of heeroscedasiciy in macroeconomic and asse price daa his is he firs macro-finance model wih his feaure. Finally I follow he convenional finance approach in assuming ha he one-period yield or spo rae relevan o he erm srucure ( y, ) is a linear combinaion of he sae variables and no necessarily equal o he policy ineres rae (r ) generaed by he macro model. These innovaions significanly improve he explanaory power of he macro-finance model and provide furher insighs ino he working of he US economy and bond marke. The paper is se ou along he following lines. The nex secion develops a Vecor Auo-Regression (VAR) model of he economy and secion III shows how his can be used o derive an affine erm srucure under he no-arbirage assumpion. Secion IV hen compares he performance of my models agains he sandard macro-finance model and discusses he implicaions for he economy and bond marke. Secion V offers a brief conclusion. II. The Macroeconomic Framework My model represens he behavior of he macroeconomy in erms of he annual CPI inflaion rae (π ), oupu gap (g ) and he 3 monh Treasury Bill rae (r ). These form he vecor z = {π, g, r } of macroeconomic variables. This T-bill rae is chosen as he policy rae in preference o alernaives like he Federal Funds and Euro-dollar raes since i has a 3 monh mauriy and is defaul free, likely o make i more relevan o a quarerly model of he Treasury marke. In addiion, x, represens he financial facor. This is assumed o follow he firs order auoregressive process:

4 4 Mulinaional Finance Journal x = θ + ξx + w, +,, + () w, + where is an equaion residual or error defined in he nex subsecion and z is driven by he L h order difference sysem: L + = κ + φ0, + φ l l + l + =, + z x z w () w, where is an error vecor. These are decomposed ino componens ha are relaed o and an orhogonal componen η : w, w = Hw + Gη, +, + + (3) x = x,, z ; w =,,, and combining () and (), o give an L h order difference sysem for n sochasic variables: This sysem is consolidaed by defining { } { w, w } ; υ = { w, η } ; x = xˆ + w (4) and: where: L + = + Γ l = l + l x ˆ K x θ 0 w ; ;,3 =Γ υ K = Γ= ; κ HG ξ Γ = ; Γ = ; l =,..., L.,3,3 l 0 Φ0 Φ 3, Φl In his paper, a ha over any variable like ˆ indicaes is condiional expecaion in he previous period. The yield model employs he sae space form, obained by arranging his as a firs order difference sysem describing he dynamics of he sae vecor (see appendix ): x +. In his paper, Diag{y} represens a marix wih he vecor y in he diagonal and zeros elsewhere. 0 a is he (a ) zero vecor; a is he (a ) summaion vecor; 0 a,b he (a b) zero marix; I a he a ideniy marix. and I a,b an a marix wih ones in he firs b elemens of he leading diagonal and zeros elsewhere.

5 Macro-Finance Model of he US Treasury Marke. 5 X = Θ+ΦΧ + W (5) = { } { } where X x,, z,..., z L is he sae vecor, W = C. w,0, N n and Θ, Φ & C are defined in appendix. X has dimension N = +3L = 0. Similarly, wriing X = { x,, X, } and pariioning W, Θ, Φ, C conformably (see appendix ), (5) becomes: x, + θ ξ 0 x N, w, + x = + + Θ Φ Φ x W, +,, + (6) The macroeconomic daa were provided by Daasream and are shown in figure. π is he annual CPI inflaion rae and r he 3 monh Treasury Bill rae. The oupu gap series g is he quarerly OECD measure, derived from a Hodrick-Presco filer. The yield daa were aken from McCulloch and Kwon (99), updaed by he New York Federal Reserve Bank. These have been exensively used in he empirical lieraure on he yield curve. The 5 year yield (he longes for which a coninuous series is available) is used in equaion (6) below o idenify he financial facor. The,,3,5,7, and 0 year yields are modelled as dependen variables. The macroeconomic daa dicaed a quarerly ime frame ( , a oal of 70 periods). The quarerly yield daa are shown in figure. The 5 year yield is shown a he back of he figure, while he shorer mauriy yields are shown a he fron. A. The Sochasic Srucure The sandard macro-finance model assumes ha he volailiy srucure is homoscedasic and Gaussian: W+ N( 0 N, Ω). However, convenional finance models usually assume ha volailiy is sochasic. In he affine model developed by Duffie and Kan (996) and Dai and Singleon (00), condiional heeroscedasiciy in he errors is driven by square roo processes in he sae variables. In he admissible version of his specificaion developed by Dai and Singleon (00),. I am graeful o Tony Rodrigues of he New York Fed for supplying a copy of his yield daase.

6 6 Mulinaional Finance Journal 0 % 5 0 Ineres rae 5 CPI inflaion Oupu gap -0 FIGURE. Macroeconomic variables Noe: CPI Inflaion and 3 monh T-bill ineres rae are from Daasream. Oupu gap is from OECD. regulariy or admissibiliy condiions are imposed o ensure ha he variance srucure remains non-negaive definie. Variaions in he risk premia depend enirely upon variaions in volailiy in hese models. In he Essenially Affine model of Duffee (00) sae variables can affec risk premia hrough he price of risk as well as hrough volailiy. In he noaion of Dai and Singleon (00) an admissible essenially affine model wih N sae variables and m independen square roo facors condiioning volailiy is classed as EA m (N). Thus he sandard macrofinance model (which is essenially affine and homoscedasic) is denoed EA 0 (N). There may in general be several sochasic volailiy erms, bu in his paper I assume ha m = : sochasic volailiy is condiioned by a single variable x, ha follows a discree ime process represened by () wih: w u x, + =, + δ0 + δ, ; (7) where u,+ is a sandardized normal i.i.d. error erm. Seing δ = 0; δ 0 > 0 gives my Vasicek (977) equivalen EA 0 model. Seing δ > 0; δ 0

7 Macro-Finance Model of he US Treasury Marke % Mauriy FIGURE. US Treasury discoun yields Noe: US Treasury discoun bond equivalen yield daa compliled by McCulloch and Kwon (990) updaed by he NY Fed. = 0 gives he discree ime Cox e al (985) equivalen specificaion of Sun (99) and Campbell e al (996). In his model, volailiy disappears as x, falls o zero and wih θ > 0, mean reversion helps o ensure ha x, + > 0. This means ha he probabiliy of a negaive value of variance erm is very small and goes o zero in he coninuous ime limi. If δ 0 0 he model generalizes o ha of Duffie and Kan (996). In his model volailiy disappears as δ 0 +δ x falls o zero and x falls o x min = ( ) δ 0 /δ. Provided ha θ > ( ξ)x min hen mean reversion means ha he probabiliy of a negaive value of variance erm (i.e. he probabiliy ha x falls below x min ) is very small. This is he basic specificaion of my EA model. In EA, his financial facor also condiions he volailiy of he oher variables. Subsiuing η = su, ino (3), where u, is a vecor of sandard normal normal i.i.d. error erms: where: w = Hw + Gsu, +, +, + [ w ] Eu 0;, +, + = 3 (8)

8 8 Mulinaional Finance Journal s = Diag ( δ0 + δx, ),...,( δ40 + δ4x, ). and δ i0, δ i $ 0, i =, 3, 4 and he componens of normal variables. I follows from (4) ha: u, + are sandardized υ + N( 0, Δ ); w + N( 0, ΓΔ Γ ); n n (9) where: Δ = Δ 0 + Δ x, ; Δ j = Diag{δ j,..., δ 4j }; j = 0,. The sochasic srucure for (6) is described in appendix. To make he EA model admissible in he sense of Dai and Singleon (00), he esimaion program checks ha: δ / δ > δ / δ, =,3,4 i0 0 i i (0) ensuring ha δ i0 + δ i x min > 0. This keeps he elemens of s and hence he variance srucure non-negaive. One implicaion of his resricion is ha δ i ; i =, 3, 4 mus go o zero wih δ making he srucure enirely homoscedasic. III. The Bond Pricing Framework The aim of his paper is o use his framework o model he macroeconomy and he yield curve joinly. The macro model is defined under he sae probabiliy measure P, bu asses are priced under he risk neural measure. This adjuss he sae probabiliies in such a way ha all asses have he same expeced reurn. A. The Pricing Kernel Discoun bond prices are obained using he pricing kernel (Campbell e al (996), Cochrane (00)): { } [ ] P = exp y E P τ,, τ, + [ ] = E M P ; τ =,..., M. + τ, + ()

9 Macro-Finance Model of he US Treasury Marke. 9 where P τ, is he -period price and denoes he condiional expecaion under he risk neural measure. M + is he nominal Sochasic Discoun Facor (SDF) which changes he probabiliy measure from P o and applies he ime discoun using: y,. This is known as he spo rae : he one-period yield relevan o he erm srucure and is assumed o be a linear combinaion of he sae variables: E y jx J X., =, +, () where J is a 9 vecor. Following he convenional finance approach, hese can consis of 9 freely esimaed weighs. In he sandard macrofinance model hese weighs are resriced o pick ou he curren value of he policy ineres rae r from he sae vecor ( y, = r = J X, ) by seing he hird elemen of J o uniy and he oher weighs o zero. Tess of his resricion are repored in he nex secion. The logarihm of he SDF is m+ = ( ω + y, + λ, w, + + λ, υ, + ), where λ, is a scalar and λ, a 3 vecor of coefficiens relaed o he prices of risk associaed wih shocks o z +. For he yield model o be affine hese coefficiens mus also be affine in he sae variables. So for example, he variable λ, shows he price of risk associaed wih he financial facor, which plays an imporan role in his analysis and is specified as: λ λ λ., = 0 + x, (3) If his is zero, hen a porfolio ha is consruced so ha i is only exposed o shocks in x, has a zero risk premium and is expeced o earn he spo rae. If i is consan λ, = λ0, hen variaions in his risk premium depend only upon variaions in volailiy, such as hose induced by x, in EA. This parameer plays he key role in ha model. If λ is also non-zero hen his facor can influence he risk premia horough variaions in he price of risk, even if volailiy is fixed as in EA 0, so λ plays he key role in ha model. Appendix shows how he prices of risk associaed wih he oher variables are adjused, following Duffee (00).

10 0 Mulinaional Finance Journal B. Affine Yield Models Appendix shows ha hese specificaions generae an exponenial affine bond price (affine yield) model: P, = exp [ γ Ψ X] ; τ =,..., M. τ τ τ (4) where Ψ τ is pariioned conformably wih (6) as Ψ τ = { ψ, τ, Ψ, τ}. Taking logs, reversing sign and dividing by mauriy τ gives he discoun yield: y τ, = pτ, / τ (5) = a + β X + e τ τ τ, L τ β τ,0, β l = τ, l + l τ, = a + x + z + e where: a τ = γ τ / τ; β τ = Ψ τ / τ; and e τ, is an i.i.d. error. The slope coefficiens of he yield sysem β τ are known as facor loadings. The sandard assumpion is ha e τ, represens measuremen error which is homoscedasic and orhogonal o he errors W in he macroeconomic sysem (5). Following he yield facor approach, I assume ha he 5 year (60 quarer) mauriy yield y60, is measured wihou error: y60, = a60 + L β60,0x, + β 60,. This allows he financial facor o be backed l lz = + l ou from he sysem as: L ( β l = + ) x = β y a z, 60,0 60, 60 60, l l. (6) Sacking (5) for he,,3,5, 7 and 0 year mauriies ha are modelled gives a mulivariae sysem for y = y, y, y, y, y, y, : { } 4, 8,, 0, 8, 40, L = + β0, + β l = l + l + y a x z e; (7) e N( 0, P) ; P = Diag{ ρ, ρ,..., ρ }. 6

11 Macro-Finance Model of he US Treasury Marke. This defines he yields in erms of he curren sae vecor. The condiional expecaion for he nex period can be wrien using (4) as: y = yˆ + u (8) L β + + l = l + l where: yˆ = a+ B xˆ + z u = BΓ υ + e ; B= [ ββ] Since he macro errors are heeroscedasic in EA, so is u : ( 0, Γ( Δ +Δ ) ΓΒ ) u+ N k B 0 x, (9) C. Yield Model Coefficiens The coefficiens of (4) are derived in appendix. These coefficien sysems are recursive in mauriy. They are also recursive in he sense ha Ψ,τ does no depend upon ψ, τ (or γ τ ). This sub-srucure is sandard and common o boh models, as is he recursion for he inercep erm. The only difference beween he EA 0 and EA models is found in he wo recursion relaionships for he firs slope coefficien. These are encompassed by he model: ψ = ( ξ + j ) ψ + ψ Φ ψ Σ Φ, τ, τ, τ, τ, τ δ ( ) ψ, τ + ψ, τ C τ =,..., M. (0) where: Σ i = CDC i ; Di = Diag{ { δi,..., δ4i},0 N 4} ; i= 0, The EA 0 model simplifies his by seing δ (and hence G ) o zero: ( j ) ψ, = ξ + ;,...,60. ψ, + ψ, Φ τ = τ τ τ ()

12 Mulinaional Finance Journal 0 where: ξ = ξ = ξ δ0λ; Φ =Φ ϒ. ϒ is a 9 vecor in which he firs hree elemens are free parameers and he res are zero. 0 If ξ + j =, hen () has a uni roo and i can be shown ha in he limi he forward rae falls wihou bound as shown in appendix, a problem originally poined ou by Campbell e al (996). The EA model specifies he firs coefficien of (0) as: ξ = ξ = ξ δλ0. I reains he quadraic erms, which show he Jensen effecs implied by he square roo volailiy specificaion. As Campbell e al (996) noe, his means ha ψ,τ and hence he forward rae have well defined asympoes even if ξ + j. This makes i more suiable for use wih daa ses such as he one used in his research ha exhibi uni or near-uni roos. The oher slope coefficiens are common o boh models and follow he sandard recursion: ( ) Ψ = Φ Ψ + J, τ, τ ( ( ) ) ( ) ( ) τ = I Φ I Φ J () where: Φ =Φ Λ. I assume ha he roos of his sysem are sable under, so his has he asympoe: ( ( ) ) I Ψ = lim Ψ = Φ J. * τ, τ (3) The inerceps follow anoher sandard recursion: Δ γτ = γτ γτ =Ψ, τ Θ + ψ, τ θ Ψ, τ Σ0Ψ, τ (4) [ C ] δ0 ψ, τ + ψ, τ ; τ =,..., M. where: θ = θ δ0λ0; Θ =Θ F;. In EA here is a resricion across ξ = ξ δλ0 and θ = θ δ0λ0 because λ0 hen

13 Macro-Finance Model of he US Treasury Marke. 3 deermines boh coefficiens given ξ, θ, δ, δ 0. M enforces his resricion. The encompassing model M relaxes his resricion and defines θ as a free parameer defined independenly of he oher parameers. IV. Model Esimaion and Evaluaion The macro (5) and yield (8) models are esimaed joinly by maximum likelihood. Appendix 3 derives he likelihood funcion and describes he numerical opimisaion procedure. Table provides some basic summary saisics for hese daa. Preliminary work designed o esimae he dimensionaliy of he model esimaed OLS regression equaions for he inflaion rae (π), he oupu gap (g); and he 3-monh Treasury bill discoun rae (r) using y60, as a proxy for x,. This sysem was esimaed for L =, 3, 4 and 5 lags, wih boh homoscedasic and heeroscedasic error srucures and he resuls suggesed he use of a hree-lag model. This gives a vecor X of en sae variables (i.e. x, and curren and wo lagged values of z ). I began by esimaing he sandard macro-finance model EA 0 (0). This is homoscedasic and idenifies he one period yield wih he T-bill rae: y. Wih his dynamic srucure i has 6 parameers 3, = r. The empirical version of his model is called M0 and has a loglikelihood value of Model M is he empirical version of he equivalen EA (0) specificaion. This uses anoher 4 parameers (for Δ (4)) bu saves one degree of freedom by using he resricion λ = 0. I has a loglikelihood value of Model M relaxes his resricion and hus encompasses boh M0 and M. I employs a oal of 66 parameers and has a loglikelihood value of 64.3 A sandard likelihood raio es of M0 agains M gives a χ (4) value of 0., providing a decisive rejecion of ha model (he probabiliy of observing his by chance is almos zero). However, M is accepable (χ () =.; p = 0.07). Finally, I esed he sandard macro-finance resricion: y, = r by reaing he 0 coefficiens of J as parameers o be esimaed. This gave a significan increase in fi in all hese models. However, only wo of hese 3. These are: θ; ξ; κ(3); Φ 0 (3); Φ (9); Φ (9); Φ 3 (9); G(3); H(3); λ 0 ; λ ; Λ 0 (3); Λ (3); Λ (9) and Δ 0 (4).

14 4 Mulinaional Finance Journal TABLE. Daa Summary Saisics: y 60 g π y y 4 y 8 y y 0 y 8 y 40 Mean Sd Skew Kur Auo KPSS ADF Noe: Inflaion (π) and ineres rae are from Daasream. Oupu gap (g) is from OECD. Yield daa are US Treasury discoun bond equivalen daa compliled by McCulloch and Kwon (990) updaed by he New York Federal Reserve Bank. Mean denoes sample arihmeic mean expressed as percenage p.a.; Sd. sandard deviaion and Auo. he firs order quarerly auocorrelaion coefficien. Skew. & Kur. are sandard measures of skewness (he hird momen) and kurosis (he fourh momen). KP SS is he Kwiaowski e al (99) saisic esing he null hypohesis of level saionariy. The 0% and 5% significance levels are and respecively. ADF is he Adjused Dickey-Fuller saisic esing he null hypohesis of non-saionariy. The 0% and 5% significance levels are.575 and.877 respecivly.

15 Macro-Finance Model of he US Treasury Marke. 5 TABLE. Coefficiens of deerminaion (R ), x π g y y 4 y 8 y y 0 y 8 y 40 M M M

16 6 Mulinaional Finance Journal TABLE 3. Dynamic model srucures (asympoic -values in parenheses.) Parameer M0 M3 Parameer M0 M3 Parameer M0 M3 Φ 0 κ θ (.7) (.05) φ 0, κ ξ (4.03) (4.03) (.4) (.4) (5.) (50.00) φ 0, κ j (3.3) (3.33) (7.00) (6.69) ( ) (6.) φ 0, κ j r 0.99 (6.79) (7.00) (7.03) (7.44) ( ) (5.) Φ Φ Φ φ, φ, φ 3, (6.0) (6.88) (0.90) (0.86) (.) (.08) φ, φ, φ 3, (3.7) (3.00) (0.36) (0.06) (.67) (.84) φ, φ, φ 3, (0.40) (0.30) (.67) (.90) (.48) (.) φ, φ, φ 3, (.96) (.75) (0.69) (0.60) (0.57) (0.37) φ, φ, φ 3, (0.36) (3.83) (.03) (.44) (8.89) (8.90) φ, φ, φ 3, (.03) (.09) (0.7) (0.09) (.38) (.40) ( Coninued )

17 Macro-Finance Model of he US Treasury Marke. 7 TABLE 3. (Coninued) Parameer M0 M3 Parameer M0 M3 Parameer M0 M3 φ, φ, φ, (.00) (.08) (.96) (.90) (0.60) (0.6) φ, φ, φ, (.00) (.05) (4.04) (4.9) (4.89) (6.0) φ, φ, φ, (7.7) (7.58) (6.83) (6.99) (.93) (.06)

18 8 Mulinaional Finance Journal parameers were saisically significan, hose aached o he financial facor ( J x ) and he T-bill rae (J r ). Adding hese o model M gives my preferred specificaion M3. This has 67 parameers and a loglikelihood of 648.3, revealing a significan improvemen upon M (χ () = 5.0; p 0). These ess srongly suppor he sochasic volailiy hypohesis: inroducing he 4 condiioning parameers of Δ dramaically increases he likelihood. This parallels he resuls of Duffee (00) and ohers using he convenional yield-facor model. This modificaion has wo effecs (a) i inroduces quadraic Jensen erms ino he yield coefficien associaed wih he financial facor (0) and (b) i allows for condiional volailiy in he daa. Theoreically, hese wo effecs are inexricably linked because he yield srucure depends upon he sochasic srucure. However, i is possible o analyze hemseparaely. Table repors he R saisics associaed wih he 0 equaions of M, M and M3, showing ha hese models have similar predicion errors despie he quesionable mahemaical properies of he M0 yield specificaion. These are only marginally higher in M3 han in M0. This is perhaps no surprising since hese models are all linear in variables and he macro-dynamic srucures are idenical. Their esimaed coefficiens, (boh he macro parameers and he facor loadings) are numerically very similar. In fac, he main reason why he likelihood of he EA models are so much higher is because of effec (b) hey allow for condiional volailiy in he daa, damping he effec ha large residuals have on he likelihood value as explained in he nex secion. This resul again parallels ha of Duffee (00) for he convenional model. In oher words, he informaion ha he esimaion procedure uses o pin down he Δ parameers comes largely from he heeroscedasic behavior of he daa raher han he behavior of he yield curve iself. I now look a he empirical resuls in deail. The parameers of M0 and M3 are se ou in Tables 3, 4 and 5. These are generally well deermined, alhough as we would expec in VAR-ype analysis, some of he off-diagonal dynamic coefficiens are insignifican. As in previous sudies of essenially affine yield srucures, many of he risk-adjusmen parameers are poorly deermined. Since he macro parameers and facor loadings for he alernaive specificaions are similar, I focus on he EA model and in paricular he insighs i yields ino he sochasic volailiy srucure.

19 Macro-Finance Model of he US Treasury Marke. 9 TABLE 4. Variance srucures (asympoic -values in parenheses) Parameer M0 M Parameer M0 M Δ 0 Δ δ δ (8.04) (0.99) ( ) (.75) δ δ (5.07) (.99) ( ) (3.7) δ δ (5.94) (.77) ( ) (3.33) δ δ (4.67) (.) ( ) (4.9) G H g h (6.) (.88) (0.) (0.) g h (0.) (3.83) (0.) (4.35) g h (9.) (8.08) (4.) (5.93) A. The Sochasic Srucure A he core of his model here is an auoregressive sysem (5) deermining he macro-dynamics. The novely here is he inroducion of square roo volailiy effecs ino his srucure. The ime variaion in volailiy is driven by he financial facor x, inferred from (6): x, = y60, π g r ( 6.0) ( 0.5) (.38) ( 4.55) π g r π ( 0.36) (.64) (.98) ( 0.55) g r (.4) ( 3.33) This facor is dominaed by he curren value of he long bond yield, 4 4. The models of risk premia developed by Glosen e al (993) and Scruggs (998) are similar in his respec. The firs condiions volailiy on he yield gap and he second on a

20 0 Mulinaional Finance Journal TABLE 5. Risk adjusmen srucures (asympoic -values in parenheses.) Parameer M0 M3 Parameer M0 M3 Parameer M0 M3 Λ F Υ λ F υ (3.93) (5.83) (3.3) (.3) (8.4) (8.4) λ 3.54 F υ ( ) (.30) (8.87) (7.00) (.9) (.9) F υ (.77) (.77) (.) (.) Λ λ, λ, λ, (0.4) (0.4) (.4) (.4) (0.8) (0.8) λ, λ, λ, (.4) (.4) (.9) (.9) (0.0) (0.0) λ, λ, λ, (.6) (.6) (.0) (.0) (0.0) (0.0) shor erm ineres rae.

21 Macro-Finance Model of he US Treasury Marke The financial facor is backed ou from he 5 year yield and dominaes he long end of he yield curve. The implici 3 monh yield is esimaed as 0.9 imes he T-bill rae plus 0. imes his facor, and dominaes he shor end. These esimaes are from he preferred specificaion, model M3. 0 %p.a. 8 6 x 4 y r FIGURE 3. Policy rae (r), one period yield (y ) and financial facor (x ) which dominaes he behavior of he long end of he yield curve. The shor end of he curve is dominaed by he implici 3 monh yield which is esimaed as y, = 0.9r + 0.x,. The facor loadings are repored in figure 7. This shows how he loading on he T-bill rae (r ) decays and ha on he financial facor ( x, ) increases wih mauriy in M3. These hree raes are depiced in figure 3. The dynamic properies of he model are dominaed by he auoregressive coefficien associaed wih x,, which is close o uniy under boh measures. Solving he model condiional upon x, shows he seady sae effec of a permanen percenage poin increase in x, would be o raise he seady sae rae of inflaion by 0.448, he T-bill rae by 0.8 and he 5 year rae by.038 poins, implying a rise in boh he real rae of ineres and he risk premium. Consequenly i seems o reflec expecaions abou boh underlying inflaion and real ineres raes. The M3 model esimaes of he financial facor are shown in Figure 4(a). This shows he one-period ahead expecaion along wih he 95% confidence inerval compued from he one-period ahead condiional

22 Mulinaional Finance Journal 0 8 % FIGURE 4(a). Variabiliy of he financial facor (One sep ahead esimae and 95% confidence inerval) volailiy. 5 This inerval rises wih during he 970s and afer ha boh subside. This facor also drives he volailiy in he oher model variables. The one-quarer-ahead forecas values and 95% confidence inervals for he hree macro variables and he 5 and 0 year yields are shown in figures 4(b)-(e). The effec of sochasic volailiy is paricularly pronounced in he case of he T-bill rae, consisen wih he finding in univariae models (Chen and Sco (993), Ai-Sahalia (996), Sanon (997) and ohers). Is variance is low over he firs four years and las wo years of he esimaion period, consisen wih he ex pos sabiliy of ineres raes over hese periods (figure 4(d)). These flucuaions in volailiy are a very imporan facor in explaining he superior performance of he EA models. Tha is because he likelihood funcion (4) normalizes he squared predicion errors in he sum of squares by he condiional variances, as in E(() of Duffee (00). In M-M3 hese variances depend upon x, and his helps o reduce he x, 5. These inervals are compued as.96 imes he sandard deviaions implied by he square roos of he condiional variances (9) and (9).

23 Macro-Finance Model of he US Treasury Marke FIGURE 4(b). Inflaion variabiliy (One sep ahead esimae and 95% confidence inerval) impac of he large errors ha end o occur when his is high. Consequenly he likelihood is much higher han for he consan variance model M0, even hough he un-normalized R and RMSEs of he macro and yield forecass of hese models are similar. This effec can be seen using a simple wo-sage OLS mehod. Tha is because he condiional means and variances are linear in he financial facor, which can be approximaed by he (lagged) 5 year yield r 60. Using his o firs explain is own condiional mean, we run he firs sage OLS regression (using my daa se): 4 y = y + w (.68) ( 5.98) y 60, 60, + 60,, + We hen use o explain he condiional variance, represened by he squared firs-sage residuals. This gives an approximaion o he volailiy equaion (7):

24 4 Mulinaional Finance Journal FIGURE 4(c). Oupu gap variabiliy (One sep ahead esimae and 95% confidence inerval) w = ( 3.68) + + y ( 5.86) 3, 60, The slope coefficien in his regression suggess ha condiional volailiy is very significan saisically. B. The Dynamic Srucure How firmly does he financial facor anchor inflaion and ineres raes? This quesion depends upon wheher hey are co-inegraed wih he non-saionary nominal facor x,. This was checked by running ADF ess on he residuals of hese equaions, which decisively rejec nonsaionariy. The macro variables adjus surprisingly quickly and smoohly o heir equilibrium values (condiional upon x, ). This is clear from he impulse responses, which show he dynamic effecs of innovaions in he macroeconomic variables on he sysem. Because hese innovaions are correlaed empirically, I use he orhogonalized

25 Macro-Finance Model of he US Treasury Marke % FIGURE 4(d). Variabiliy of T-bill rae (One sep ahead esimae and 95% confidence inerval) innovaions obained from he riangular facorizaion defined in (4). The impulse responses show he effec on he macroeconomic sysem of increasing each of hese shocks by one percenage poin for jus one period using he Wold represenaion of he sysem as described for example in Cochrane (997). This arrangemen is affeced by he ordering of he macro variables in he vecor x, making i sensible o order he variables in erms of heir likely degree of exogeneiy or sensiiviy o conemporaneous shocks. The financial facor is assumed o represen exogenous expecaional influences, so his is ordered firs in he sequence. This means ha independen shocks o inflaion, oupu and ineres raes can hen be inerpreed as sudden shocks ha are no anicipaed by he bond marke. Following Hamilon (994) inflaion is ordered before he oupu gap, on he view ha macroeconomic shocks are accommodaed iniially by oupu raher han price. Ineres raes are placed afer hese variables on he view ha moneary policy reacs relaively quickly o disurbances in oupu and prices. Thus he variable ordering is: x, ; π ; g and r. This means ha shocks o he financial facor (ν ) disurb all

26 6 Mulinaional Finance Journal FIGURE 4(e). Variabiliy of 0 year yield (One sep ahead esimae and 95% confidence inerval) four variables conemporaneously as indicaed by he firs column of he marix Γ shown in (4), independen shocks o inflaion (ν ) affec oupu and ineres raes bu no he financial facor, and so on. Figure 5 shows he resuls of his exercise. The coninuous line shows he effec of each independen shock on he T-bill rae, he broken line he effec on inflaion and he doed line he effec on oupu. Elapsed ime is measured in quarers. Panel (i) shows he effec of a shock o he financial facor (ν ). This could reflec an increase in he bond marke s expeced rae of inflaion or he underlying real rae of reurn in he economy. Oupu and he T-bill rae increase immediaely, bu inflaion does no, meaning ha real ineres raes increase iniially. The financial facor acs as a leading indicaor for inflaion, which peaks afer hree years. Panel (ii) shows he effec of an independen shock o inflaion (ν ), essenially an inflaionary impulse ha is no anicipaed by he bond marke. The iniial effec on he T-bill rae is only abou a quarer of a poin, so real ineres raes fall. However, oupu falls back, reaching a rough afer falling by 0.8% afer wo years, reflecing real balance and oher conracionary inflaionary effecs. The fall in oupu has he effec of reversing he rise in inflaion, seing up cycles in hese variables. In

27 Macro-Finance Model of he US Treasury Marke. 7 % response % response Time Time (i) Financial facor (f ) (ii) Inflaion (π) % response % response Time Time (iii) Oupu gap (g) (iv) T-bill rae (r ) Key - effecs on: inflaion CCCCCC oupu T-bill rae FIGURE 5. Model M3 macroeconomic impulse responses Noe: Each panel shows he effec of a shock o one he four orhogonal innovaions (υ ) shown in (4). These shocks increase each of he facors in urn by one percenage poin compared o is hisorical value for jus one period. Since x, has a near-uni roo, he firs shock (υ, ) has a persisen effec, while oher shocks are ransien. The coninous line shows he effec on he spo rae, he dashed line he effec on oupu and he doed line he effec on inflaion. Elapsed ime is measured in quarers. conras o he effecs shown in he firs panel, which are highly persisen, he sysem is close o is iniial level afer 0 years following his inflaionary impulse. The oher wo panels show similarly fas responses, wih qualiaive effecs in accordance wih macroeconomic heory. These responses are refleced in figure 6, which repor he resuls of an Analysis of Variance (ANOVA) exercise. These figures show he share of he oal variance aribuable o he innovaions a differen lag lenghs and are also obained using he Wold represenaion of he sysem as described in Cochrane (997). They indicae he conribuion

28 8 Mulinaional Finance Journal (i) Inflaion variance (ii) Oupu variance (iii) Variance of T-bill rae (iv) Variance of 5 year yield Key - % of variance due o orhogonal innovaions in: financial facor; inflaion; oupu; T-bill rae. FIGURE 6. Model M3 Analysis of Variance Noe: Each panel shows he conribuion o oal variance of innovaions in he orhogonal shocks represening innovaions in each of he four driving variables. Elapsed ime is measured in quarers. each innovaion would make o he volailiy of each model variable if he error process was sared in he firs period. Iniially, he variances of hese variables are srongly influenced by heir own innovaions. However he influence of he long bond innovaions builds up over ime, paricularly in he case of he T-bill rae, where his explains over half of he oal forecas variance afer 0 years. Figure 7 shows he facor loadings as a funcion of mauriy expressed in quarers. The firs panel shows he loadings on r (coninuous line) and x, (broken line). The spo rae is he link beween he macro model and he erm srucure. Recall ha in M3 i is esimaed as y, = 0.9r + 0.x,. These regression weighs deermine he firs quarer loadings on r and, while oher facors x,

29 Macro-Finance Model of he US Treasury Marke Financial facor, x. ( ) and T-bill rae r ( ) Oupu g ( ) and inflaion π ( ) FIGURE 7. Model M3 Facor loadings Noe: The facor loadings show he cumulaive effec (afer hree quarers) of changes in he four facors on yields a differen mauriies have a zero loading. The loadings on r hen end o decline monoonically wih mauriy, reflecing he relaively fas adjusmen process. This mean i acs like he slope facor in he convenional 3-facor model. In conras, he slow-moving naure of x, means ha is loading increases wih mauriy over mos of his range, allowing i o ac as a level facor. The nex panel shows he loadings on π (doed line) and g (broken line). The lower righ hand panel of Figure 6 decomposes he condiional forecas variance of he 5 year yield ino he separae effecs of surprises o he four orhogonal shocks defined in (4). (ANOVA figures for he 7 and 0 year yields show a similar paern.) Innovaions in he hree macro have a modes conribuion for near-erm forecass, bu are increasingly dominaed by innovaions in he long bond innovaions. This explains over 95% of he oal forecas variance 0 years ahead. V. Conclusion Condiional volailiy is a common feaure of macroeconomic and financial daa and as Duffee (00) and many ohers have shown, i is imporan o allow for his when modelling he yield curve and derivaives ha are priced off his. My specificaion exends he new macro-finance model o allow for condiional volailiy, bringing i ino

30 30 Mulinaional Finance Journal line wih he convenional finance model. I is an EA specificaion ha condiions he cenral endency and he variance srucure of he model on he financial facor, which is closely correlaed wih he long bond yield. The likelihood of he new model is much higher han ha of he exising EA 0 macro-finance specificaion, even hough he raw forecas errors of he wo models are similar. As was found o be he case in he convenional yield facor model (Duffee (00)) his is because he EA model allows for condiional volailiy in he facors driving he sysem, damping he negaive effec ha large residuals have on he likelihood value. In pracice hen, he informaion ha he esimaion procedure uses o pin down he parameers of my EA yield model comes indirecly from he behavior of he macroeconomic and laen facors raher han he behavior of he yield curve iself. My model can be seen as a modificaion of he convenional EA yield facor model of he bond marke which replaces some of he laen facors by macroeconomic variables. I shows ha he sochasic volailiy idenified by he convenional model is relaed o macroeconomic volailiy. I can use a hird-order dynamic specificaion wih large number sae variables (0) in place of he convenional firs-order sysem because is parameers are esimaed using by macroeconomic as well as yield daa. However, he behavior of he yield curve is largely dicaed by hree facors: he financial facor, he oupu gap and he T-bill rae. The model is consisen wih he radiional hree laen facor US finance specificaion in his respec, bu aligns he las wo facors wih observable variables. This research opens he way o a much richer erm srucure specificaion, incorporaing he bes feaures of he macro-finance and convenional finance models. Acceped by: Prof. R. Taffler, Gues Edior, Sepember 008 Prof. P. Theodossiou, Edior-in-Chief, Sepember 008 Appendix : The Sae-Space Represenaion of he Model Sacking (4) pus he sysem ino sae space form (5), where : Θ= { θκ },,0 ; N 4, (5)

31 Macro-Finance Model of he US Treasury Marke. 3 ξ Φ Φ... Φ Φ,3,3,3 0 l l 0N 03, I ξ Φ= 3 3 =. Φ Φ I 0 3, and where he las marix pariions Φ conformably wih (6), so ha Φ is (N ) and Φ is (N ). Similarly: 0 0 C H G 0.,3,6 0N = 3,6 = C C 06, 06,3 0 6 (6) where he las marix pariions C conformably wih (6): C is (N ) and C is (N ). The error srucure of (6) follows from (8) as: W = C w + C SU, +, +, + (7) ( 0, ) U N I, + N N,3 IN,3 = Diag,0 3 N 4 and S = Diag δ0 + δx,,..., where: { } ( ) δ δ + + condiional expecaions: ( + x ),0 ; U = { u,0 } 40 4, N 4,, N 4 { } { [ Ψ ]} = [ Ψ Ψ ] E exp U, + exp IN,3 ; { [ ]} [. This implies he (8) { } { } E exp Ψ SU, + = exp Ψ SIN,3SΨ = exp Ψ S Ψ. where:

32 3 Mulinaional Finance Journal S = D + x D 0, (9) where: Di = Diag{ { δi,..., δ4i},0 N 4} ; i= 0,. Appendix : The EA 0 and EA Specificaions This appendix derives he arbirage-free bond price sysems for he EA 0 and EA specificaions. Subsiuing (6), (34) and (7) ino (), noing ha and are independen allows his o be facorized as: w, + U, + [ ] P = E M P τ, + τ, + { [ ω y, γτ ψ, τ ( θ ξx, ), τ ( x, = Ψ Θ +Φ + exp { τ + } )]} exp ( ) Φ X E Ψ C S +Λ U,,,, { [( ψ λ ) } E + +Ψ C w x exp, τ,, τ, +, (30) These errors are all Gaussian and are evaluaed using (7) and (8): { [ ( ) ( P = exp ω + y + γ + ψ θ + ξx +Ψ Θ +Φ x + τ,, τ, τ,, τ, Φ X + Ψ C S +Λ I SC Ψ +Λ )} ( ) ( ),, τ, N,3, τ, Ψ ( δ0 δx, )( ψ, τ λ,, τ C ) } (3) In he case of a one period bond iniial condiions: P = exp { y },,, which gives he γ = 0; ψ = j ; Ψ = J,, (3)

33 Macro-Finance Model of he US Treasury Marke. 33 and he resricion: ω = Λ N Λ + δ + δ λ Subsiuing his and (9) ino (3): { { (, I,3, ( 0 x, ), ) ( ) ( P = exp γ + y + ψ θ + ξx +Ψ Θ +Φ x +Φ τ, τ,, τ,, τ, )} +Ψ Λ + ( + )( +Ψ ) X C S δ δ x ψ C λ,, τ, 0,, τ, τ, ; + Ψ τ Σ +Σ Ψ τ + δ + δ ( x ) ( x ), 0,, 0, (33) ( ψ ), τ +Ψ, τ C } The price parameer sysems are obained by specifying he prices of risk parameers. Following Duffee (00) I define he (N ) deficien vecor Λ = λ,0 and wrie he log SDF as: [ ],, N 4 m = ω + y + λ w +Λ U +,,, +,, + (34) and hen assume: λ = λ + λ x +Λ Χ, 0,, (35) Λ = S C Λ + S C Λ x + S C Λ X, 0,, where: Λ 0 and Λ are (N ) and Λ is (N ). The elemens of he las N n rows of hese marices (and he las N n columns of Λ ) are zero. To obain an affine yield soluion for he EA model i is necessary o assume: λ = 0; Λ = 0 N. To allow he EA model o encompass EA 0 I also assume ha: Λ = 0 N for EA 0, checking ha his was accepable a he 95% significance level. Subsiuing hese formulae ino (33):

34 34 Mulinaional Finance Journal { { [ ( ( ) ( P = exp x ψ ξ + j δ λ + λ x +Ψ Φ Σ Λ τ,,, τ 0,, τ 0 Λ δ λ C Ψ Σ Ψ δ ψ + Ψ ) ( C ) 0, τ, τ, τ, τ + [ +Ψ ( Φ Λ )] Χ + γ + ψ ( θ δ λ ) J, τ, τ, τ 0 0 +Ψ τ ( Θ Σ Λ δ λ C ) Ψ τ Σ Ψ τ δ, , 0, 0 ( ψ τ +Ψ τ C ) }},,. (36) For he EA models M-M3, (0), () and (4) follow by seing λ, = λ 0 and equaing he coefficiens of X in he exponen wih hose in (4). In his case ϒ is inerpreed in erms of he risk parameers as ϒ= ( ΣΛ 0 +Λ +δλ0c ). Seing δ and Σ o zero and equaing coefficiens gives he parameer sysems for he EA 0 model, including (). In his case: ϒ=Λ +δ0λc.() and (4) are shared wih EA, where F =Σ0Λ0 δ0λ0c Α. Forward Raes and Risk Premia The τ period ahead forward ineres rae is defined as f = p p τ, τ, τ+, : f = γ γ + ψ ψ x + Ψ Ψ Χ ; τ =,..., M ( ) [ ] τ, τ+ τ, τ+, τ,, τ+, τ, (37) The risk premia follow by seing he price of risk parameers o zero in (36). This is equivalen o seing M + in (30) o exp{ y, } and gives he discouned expecaion under P: [ y ] E [ P ] exp{ [ γ y ψ ( θ ξx ) exp = Ψ, τ, + τ,, τ,, τ

35 Macro-Finance Model of he US Treasury Marke. ( Θ +Φ x +Φ Χ )] + Ψ ( Σ +Σ x ),,, τ 0, Ψ + + +Ψ ( δ δ x )( ψ C ) }, τ 0,, τ, τ 35 (38) The gross expeced rae of reurn on a τ period bond afer one period is his expecaion E[ P τ, + ] divided by is curren price P τ,. Taking he naural logarihm expresses his as a percenage reurn and subracing he implici one-period yield y, hen gives he expeced excess reurn or risk premium: [ ] log[ ] ρτ, = log E Pτ, + Pτ, y, ( ) ψ ( ξ ξ) = x, Ψ, τ Φ Φ +, τ ( ) +Ψ Φ Φ X, τ, ( ) ψ τ ( θ θ) +Ψ Θ Θ +, τ, (39) (40) (using (36) and (36)). The risk premia implied by model M3 for represenaive mauriies are shown in figure 8. Appendix 3 : The Likelihood Funcion This appendix derives he likelihood funcion and describes he numerical opimizaion procedure. Because he macro and measuremen errors are assumed o be orhogonal, i shows ha he likelihood of he join model is he sum of macro and measuremen componens. Using (4) (9) and (8): w υ A N( 0 n+ k; F) ; where: u = e (4)

36 36 Mulinaional Finance Journal 0 0 year year % p.a. 4 year year FIGURE 8. Risk Premia in model M3 Γ 0 Δ 0 F AD A; A ; D. nk, n, k = = = BΓ I 0 k kn, P so he loglikelihood for period can be wrien as: n+ k υ L = ln( π) ln( F ) [ υ υ ] F u (4) The Gaussian sochasic framework means ha his likelihood funcion normalizes he one-period ahead squared predicion errors using he oneperiod ahead condiional variances in he usual way. However as in E() of Duffee (00) his funcion can be expressed in erms of he macro and measuremen errors using (4): n+ k υ L = ln( π) ln( D ) [ υ e ] D e (43)

37 Macro-Finance Model of he US Treasury Marke. 37 n n+ k = ln( π ) ln( δ ) ( ) i0 + x, δi ν Δ 0 + x, Δ ν i= k ln ( ρτ ) e P e. τ = Summing his over T periods gives he loglikelihood for he esimaion period: T n k T( n+ k) T L= ln π ln δi0 + x, δi ln ρτ ( ) ( ) ( ) = i= τ = Τ ( ) T ν Δ 0 + x, Δ ν ep e. = = Since his is a quadraic in he (inverse) variances of he measuremen errors (ρ,..., ρ 6 ) in (7) his can be concenraed in he usual way by solving for heir opimal values and subsiuing back. (This canno be concenraed wih respec o he parameers of he macro variances since hese also affec he facor loadings in he yield equaions.) This likelihood funcion was maximized using he FindMinimum numerical opimizaion package on Mahemaica. This program checks ha he admissibiliy resricions (0) hold. I also eliminaes observaions for which he parameer esimaes and daa reurn negaive variances, following Chen and Sco (993). The esimaes repored here include all observaions, wih non-negaive definie variance srucures. References Ai-Sahalia, Y Tesing Coninuous Time Models of he Spo Ineres Rae. Review of Financial Sudies, 9, Ang, A., and Piazzesi, M A No-Arbirage Vecor Auoregression of Term Srucure Dynamics wih Macroeconomic and Laen Variables. Journal of Moneary Economics, 43, Campbell, J.; Lo, A.; and MacKinlay, A The Economerics of Financial Markes. Princeon Universiy Press, Princeon, N.J.

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