Why Can the Yield Curve Predict Output Growth, Inflation, and Interest Rates? An Analysis with an Affine Term Structure Model

Transcription

1 Bank of Japan Woking Pape Seies Why Can he Yield Cuve Pedic Oupu Gowh, Inflaion, and Inees Raes? An Analysis wih an Affine Tem Sucue Model Hibiki Ichiue * hibiki.ichiue@boj.o.jp No.04-E-11 July 2004 Bank of Japan Nihonbashi Hongoku-cho, Chuo-ku, Tokyo * Reseach and Saisics Depamen Papes in he Bank of Japan Woking Pape Seies ae ciculaed in ode o simulae discussion and commens. Views expessed ae hose of auhos and do no necessaily eflec hose of he Bank. If you have any commen o quesion on he woking pape seies, please conac each auho. When making a copy o epoducion of he conen fo commecial puposes, please conac he Public Relaions Depamen (webmase@info.boj.o.jp) a he Bank in advance o eques pemission. When making a copy o epoducion, he souce, Bank of Japan Woking Pape Seies, should explicily be cedied.

2 Why Can he Yield Cuve Pedic Oupu Gowh, Inflaion, and Inees Raes? An Analysis wih an Affine Tem Sucue Model Hibiki Ichiue Depamen of Economics, Univesiy of Califonia, San Diego Reseach and Saisics Depamen, he Bank of Japan Revised: July, 2004 Absac The lieaue povides evidence ha em speads help pedic oupu gowh, inflaion, and inees aes. This pape inegaes and explains hese pedicabiliy esuls by using an affine em sucue model wih obsevable macoeconomic facos fo U.S. daa. The esuls sugges ha consumes ae willing o pay a highe pemium fo a consumpion hedge duing a highe inflaion egime. This causes em speads o eac o inflaion shocks, which poves useful fo pedicion. We also find ha em speads using he sho end of he yield cuve have less pedicive powe han many ohe speads. We aibue his o moneay policy ineia. JEL classificaion: E43; E52 Keywods: Tem sucue, Moneay policy, VAR I am especially gaeful o my disseaion adviso, James D. Hamilon fo his valuable suppo and commens. I would like o hank Majoie Flavin, Buce Lehmann, Alan Timmemann, Keiichi Tanaka, Nobuyuki Oda, Akia Ieda and paicipans in he pesenaions a UCSD and he Bank of Japan. I am also gaeful o Monica Piazzesi fo answeing my quesions on he papes. The views expessed hee ae hose of he auho, and no necessaily of he Bank of Japan. Addess: Nihonbashi-Hongokucho Chuo-ku Tokyo Japan Tel.: (Bank of Japan), Fax: , addess: hibiki.ichiue@boj.o.jp 1

3 1. Inoducion Many sudies in he lieaue povide evidence ha inees ae em speads conain infomaion abou hee diffeen fuue economic vaiables: oupu gowh, inflaion, and inees aes, fo vaious sample peiods and counies. Bu he lieaues examining he pedicabiliy of hese hee vaiables have been quie disincive. Sudies of he pedicabiliy of inees aes have been mainly conduced by financial economiss esing a populaly-held classic heoy, namely he expecaions hypohesis 1. Accoding o his heoy, he long ae is equal o he aveage of expeced fuue sho aes plus a ime-invaian em pemium. Howeve, in spie of is populaiy, his hypohesis has ypically been ejeced. Many economiss ague he expecaions hypohesis fails because of he assumpion of a ime-invaian em pemium 2. The lieaue on he pedicabiliy of inflaion also has a long hisoy following Fama s (1975) classic sudy 3. On he ohe hand, he hisoy of he lieaue sudying he pedicabiliy of oupu gowh is elaively ecen. Afe Sock and Wason (1989) found ha he em spead plays an impoan ole in hei index of economic leading indicaos, many eseaches invesigaed his pedicive elaionship 4. Alhough hee is an exensive lieaue poviding evidence and explanaions fo each of he pedicive elaionships beween em speads on he one hand, and on he ohe, oupu gowh, inflaion, and inees aes, no pape has ye ied o analyze he ineacion beween hese hee elaionships. The main pupose of his pape is o inegae hese pedicabiliy esuls in an aemp o answe o an impoan quesion: why can he em sucue pedic fuue movemens in economic vaiables? This sudy will help us undesand he infomaion conained in he em sucue of inees aes, and he elaionship beween he em sucue and business 1 Fo empiical esuls of ess of he expecaions hypohesis, see, fo example, Campbell and Shille (1991), Hadouvelis (1994), Rudebusch (1995), Campbell, Lo and MacKinlay (1997), Robeds and Whieman (1999), Bekae, Hodick and Mashall (2001), and Cochane (2001). 2 The lieaue povides evidence ha he em pemium is in fac ime-vaying. See, fo example, Mankiw and Mion (1986), Engle, Lilien and Robins (1987), Engle and Ng (1993), Dosey and Ook (1995), and Balduzzi, Beola and Foesi (1997). 3 Fo empiical esuls on he pedicabiliy of inflaion, see, fo example, Mishkin (1988, 1990a, b, 1991), Fama (1990), Joion and Mishkin (1991), Esella and Mishkin (1997), and Kozicki (1997). 4 Fo empiical esuls on he value of em speads fo pedicing oupu gowh o ecessions, see, fo example, Esella and Hadouvelis (1991), Plosse and Rouwenhos (1994), Haubich and Dombosky (1996), Bonse-Neal and Moley(1997), Dueke (1997), Esella and Mishkin (1997), Kozicki (1997), Benad and Gelach (1998), Dosey (1998), and Hamilon and Kim (2002). 2

4 cycle. We use an affine em sucue model (ATSM) wih obsevable economic facos as ou main ool, basing ou invesigaion on U.S. daa. Thee have been a numbe of sudies following Ang and Piazzesi s (2003) inoducion of his ype of model o invesigae he elaionship beween macoeconomic vaiables and he em sucue, fo example, Dewache and Lyio (2002), Hodahl, Tisani and Vesin (2002), and Wu (2002). These sudies depend much on macoeconomic heoies o esic hei models so ha he esuls can be inepeed moe easily. Fuhemoe, hese models ypically use laen vaiables ohe han obsevable vaiables, and inepe he laen facos as vaiables such as he moneay policy auhoiy s inflaion age. Convesely, Ang, Piazzesi and Wei (2003) use only obsevable vaiables, and hey do no use macoeconomic heoies ohe han he no-abiage assumpion o esic hei model. This ype of model can be inepeed eihe as a VAR wih no-abiage esicions o as an ATSM wih obsevable facos ha follow a VAR pocess. In his pape, we call his ype of model a VAR-ATSM fo convenience. Ang, Piazzesi and Wei use hei VAR-ATSM o examine he pedicabiliy of oupu gowh using em speads. We follow his basic idea, which we exend o include he pedicive elaionships wih inflaion and sho aes 5. Alhough hei basic idea is vey useful fo invesigaing hese pedicive elaionships, some of hei assumpions and aspecs of hei esimaion mehod ae no suiable o ou pupose hee. Ang, Piazzesi and Wei y o idenify good foecasing models by compaing he pedicive powes, specifically he olling ou-of-sample foecasing pefomances, of vaious combinaions of egessos. Thei pasimonious VAR model and compuaionally fas, hough less efficien, esimaion mehod may be appopiae fo such an execise. Ou aim, howeve, is o shed ligh on he souce of he pedicabiliy by analyzing he elaionship beween impulse esponse funcions and R 2 s. Thus we adop VAR wih moe lags and a moe efficien esimaion mehod, and hese conibue o he eliabiliy of he impulse esponse funcions. We have hee main findings. Fis, he ime-vaying make pice of oupu gowh isk, 5 Befoe Ang, Piazzesi and Wei (2003), seveal papes use em sucue models wih only laen facos fo analyzing pedicabiliy using em speads. Fo example, Robeds and Whieman (1999), Dai and Singleon (2002), and Duffee (2002) examine whehe empiical esuls on he pedicabiliy of inees aes can be fied using ATSMs. Hamilon and Kim (2002) use he Longsaff and Schwaz s (1992) em sucue model o explain he pedicabiliy of oupu gowh. Since hese models use only laen facos, howeve, hey have only 3

5 which is sensiive o he inflaion ae, plays a key ole in he pedicive elaionships. When he inflaion ae is highe, consumes ae willing o pay a highe pemium fo a consumpion hedge, which may be explained by a simple model wih money in he uiliy funcion and a moneay policy ule. This causes em speads o be sensiive o inflaion shocks. Since he inflaion shock has pesisen effecs no only on inflaion bu also on oupu gowh and inees aes, he esponse of em speads o he inflaion shock helps pedic hese vaiables. Second, we also find ha em speads using he sho end of he yield cuve have less pedicive powe han many speads beween longe aes. This fac is aibuable o he ineial chaace of moneay policy. Thid, i is had o pedic oupu gowh wih em speads a sho hoizons, because he moneay policy shock affecs oupu gowh wih a lag while he em sucue esponds o he shock immediaely. The es of his pape is oganized as follows. Secion 2 pesens sylized facs fom simple OLS esuls. In Secion 3, in ode o undesand he basic popeies of ATSMs, we conside some simple epesenaive models. This secion will help o pepae fo he moe complicaed VAR-ATSM inoduced in Secion 4. Esimaion mehods and esuls ae consideed in Secion 5. Hee we discuss he elaionship beween ime-vaying make pices of isk and he infomaion included in he em sucue. In Secion 6, we use impulse esponse funcions and model-implied R 2 s, which can be obained fom he esimaed VAR-ATSM, o explain why em speads pedic well. Secion 7 concludes. 2. Simple OLS Resuls The empiical sudies in he lieaue examine he pedicive powe of em speads fo fuue oupu gowh, inflaion, and inees aes using a common economeic mehod, egessions on he em speads. Howeve, hese egessions do no have exacly he same fom. Fo example, Esella and Hadouvelis (1991) examine oupu gowh pedicabiliy by egessing cumulaive oupu gowh, up o h quaes ahead, on a fixed em spead beween en-yea and limied value fo analyzing he elaionships beween he em sucue and macoeconomic vaiables. 4

6 hee-monh inees aes: g = + + (40) + h α β( ) ε+ h whee g is he oupu gowh ae fom 1 o 2, and 1 2 (n) is he n-peiod nominal discoun ae on Teasuy bills o bonds a he end of. On he ohe hand, Mishkin (1990a) examines inflaion pedicabiliy by egessing he diffeence beween h-quae and 1-yea cumulaive inflaion aes on em speads of maching mauiy: whee 1 2 π π = α + β + ε (2) ( h) (4) + h + 4 ( ) + h π is he inflaion ae fom 1 o 2. Campbell and Shille (1991), meanwhile, povide evidence fo sho ae pedicabiliy by using he mos popula expecaions hypohesis es, egessions of aveage fuue sho ae changes on em speads of maching mauiy: 1. (3) h h 1 ( h) ( + i ) = α + β( ) + ε+ h 1 i= 0 All hee ypes of sudy find ha he slope coefficien β is significanly diffeen fom zeo in many cases, which means ha em speads have pedicive powe fo foecasing movemens in macoeconomic vaiables. Typically hey epo subsanial -sas and R 2 s fo hese egessions. As one can easily see, hese empiical egessions do no have he same fom. Fo example, and (2) do no use he same egesso. Regession uses a fixed egesso, while he egesso in (2) depends on he foecasing hoizon h. In ode o analyze he ineacion beween he pedicive elaionships, heefoe, we need o pu he empiical esuls fo pedicing he diffeen vaiables on a consisen basis. Fo his pupose, we use he egessions below, g = α + β + ε ; (4) ( n) ( m) + h ( ) + h π = α + β + ε ; (5) ( n) ( m) + h ( ) + h = α + β( ) + ε ; (6) ( n) ( m) + h + h fo vaious combinaions of h, n, and m (h = 1,2,,12; n, m = 2, 4, 8, 12, 16, 20, and n > m), whee g is he eal GDP gowh ae fom -1 o, and π is he inflaion ae of GDP deflao 5

7 fom -1 o. We use discoun ae daa fom CRSP 6. U.S. quaely daa ae used, so we inepe one peiod as one quae. g, π, and (n) ae all defined as aes pe quae. The sample peiod is 1964:1Q-2001:4Q, following Fama and Bliss (1987) who commen ha long ae daa befoe 1964 may be uneliable. Thee ae wo ohe popeies of he se of egessions (4)-(6) woh commening on. Fis, egessands ae coninuously compounded maginal aes o one-peiod sho aes. Since cumulaive aes ae he aveages of maginal aes, maginal aes ae moe convenien fo specifying which pa of he fuue he em speads can pedic well. Second, we use vaious foecasing hoizons h and em speads which componens of he yield cuve pedic a which fuue hoizons., so we can specify ( n) ( m) Figues 1 and 2 display he -sas and R 2 s of OLS egessions (4)-(6) fo seleced em speads. The 20Q-1Q spead has significan pedicive powe fo oupu gowh, inflaion, and sho aes, a leas fo shoe hoizons. This esul is consisen wih he lieaue, which agues ha em speads beween 5-yea (o 10-yea) and 3-monh aes pedic well. Bu supisingly we found ha em speads wihou he 1Q ae pefom bee han he 20Q-1Q spead in many cases. Fo example, Figue 2 shows ha he pefomance of he 12Q-8Q spead is supeio, excep fo pedicing oupu gowh aes a shoe hoizons. On he ohe hand, speads beween sho aes, such as he 2Q-1Q spead, ae almos useless. Togehe, hese facs seem o imply ha em speads using he sho end of he yield cuve have less pedicive powe. This is supising because he exising lieaue pays lile aenion o speads ha exclude he sho end of he yield cuve, and seveal sudies including Ang, Piazzesi and Wei (2003) ague ha he bes pedicive pefomance is achieved by maximal mauiy diffeence. Anohe noable feaue of he gaphs is he hump-shape aced ou by he R 2 s of he oupu gowh egessions. This suggess ha i is difficul o pedic he oupu gowh ae a sho hoizons. Why do em speads have his kind of pedicive powe? Since he OLS esuls do no answe his quesion, we need a moe sucued model. A useful mehod fo inepeing hese 6 CRSP (Cene fo Reseach in Secuiy Pices, Gaduae School of Business, he Univesiy of Chicago: All ighs eseved.) Monhly US Teasuy Daabase is used wih pemission. We can consuc discoun aes fo 1, 2, 4, 8, 12, 16, 20 quaes fom he CRSP daa. The 1 quae (3 monh) ae is obained fom aveage aes in he CRSP isk fee aes file. The 2 quae (6 monh) ae is consuced by muliplying aveage-ytm by 12 (hee is no daa on 9/30/1987, so we inepolae wih 3 and 12 monh aes). The ohe aes ae obained fom he Fama-Bliss discoun bonds file. 6

8 OLS esuls is poposed by Ang, Piazzesi and Wei (2003). They inoduce a VAR-ATSM o compae he pedicive powes of vaious combinaions of egessos. We follow hei basic idea, bu exend hei analysis so as o include all hee pedicive elaionships, beween em speads on he one hand, and on he ohe each of oupu gowh, inflaion, and sho aes. Alhough hei VAR-ATSM is vey useful fo examining he elaionships beween macoeconomic vaiables and he yield cuve, some of hei assumpions and aspecs of hei esimaion mehod ae no suiable o ou pupose. We heefoe modify hem in Secions 4 and 5. Then, in Secion 6, we y o shed ligh on he souce of he pedicabiliy by using impulse esponse funcions and R 2 s, which can be calculaed fom he esimaes of he VAR-ATSM. 3. Simple Affine Tem Sucue Models wih Obsevable Facos Befoe inoducing ou VAR-ATSM in he nex secion, le s conside wo simple ATSMs. Since he complexiy of he VAR-ATSM defies easy inepeaions, hese simple models povide a useful saing poin. A paicula complicaion aises as a esul of ime-vaying make pices of isk, which many classic em sucue models assume consan. Since, howeve, hese affec he elaionship beween sho and long aes, i.e. movemens in em speads, hey ae vey impoan fo examining he pedicive powe of em speads An ATSM wih One Sho Rae Faco AR pocess: Suppose ha quaely daa on he sho (3-monh) ae ae chaaceized by an = c + φ + σ u, (7) + 1, + 1 whee u ~ (0,1), + 1 N i.i.d., and σ > 0. Table 1 epos he OLS esimaes fo (7), which demonsae he pesisence of he sho ae ( φ = ). Suppose ha he sochasic discoun faco M + 1 follows a condiional log-nomal disibuion: 7

9 1 2 M 1 exp + = λ, λ, u, + 1 2, (8) whee λ = γ + δ. (9), In his model, heefoe, he make pice of isk λ, is ime-vaying, depending on he faco. In ohe wods, he sochasic discoun faco M 1 is affeced no only by he exogenous shock u, + 1 bu also by he level of he faco + hough he ime-vaying make pice of isk. Thus he effecs of he faco on he yield cuve ae complicaed. Noe ha if δ = 0, i.e. λ, is ime-invaian, his is jus he classic Vasicek (1977) model. Le s assume hee is no abiage oppouniy in he Teasuy make. Since his make is one of he lages and mos highly liquid makes in he wold, he no-abiage assumpion is exemely easonable. Unde he no-abiage assumpion, we can use he fundamenal asse picing equaion fo bond pices, q = E[ M q ], (10) ( n) ( n 1) fo n = 1, 2,, and all, whee (10) lead o q is he n-peiod bond pice wih ( n ) q = 1. Noe ha (8) and (0) q = exp( ). (11) This is exacly he definiion of he elaionship beween he 1-peiod bond pice and he coninuously compounded discoun ae. In fac, M + 1 is chosen so ha (11) holds. By using he fundamenal asse picing equaion (10), we can deive closed foms fo he discoun aes as affine funcions of he faco ( n ) : ( ) ( ) ( ) ˆ n n n a b = +, n = 1, 2, (12) whee ( n) ( n) ( n) ( n) a A / n, b B /n = =, (13) 8

10 ( n+ 1) ( n) ( n) 1 2 ( n)2 A = A + B (c σγ ) + σ B, (14) 2 ( n+ 1) ( n) B B ( φ σδ ) = 1, (15) A = 0, B = 1 7. (16) In (12), he faco loading on he sho ae faco b ( n) can be inepeed as he sensiiviy of longe aes o he sho ae ( n ). Fom (13), (15), and (16), we can obain a closed fom fo ( n) b : 1 j ). (17) n 1 ( n) b = ( φ σδ n j= 0 Noe ha γ does no appea in (17). Since he movemen of sho aes is less volaile han ha of long aes fo U.S. daa, i is easonable ha he absolue value of b ( n) deceases as n inceases. To saisfy his, we need paamee values such ha φ σ δ < 1. (18) Suppose φ σ δ > 0, which guaanees ( n b ) > 0. Fom (17), we can say ha he sensiiviy of long aes o he sho ae is weake when δ is highe. We can elae his claim o he expecaions hypohesis. Fom (7), (12), and (17), we can obain he em pemium: E a c n 1 n 1 j 1 n 1 ( n) ( n) i j j [ + j] = φ + [( φ σδ) φ ] n j= 0 n j= 0 i= 0 n j= 0. (19) The em pemium is heefoe consan, i.e. he expecaion hypohesis holds, only when δ = 0. In his case, movemens in long aes aes n 1 n 1 E j 0 = + j [ ]. Since depend only on movemens in aveage expeced sho ( n ) follows a pesisen AR pocess, an incease in aises. Howeve, when δ > 0, a ise in ( n ) also has a negaive effec on hough a decease ( n ) in he em pemium. Theefoe, posiive δ weakens he elaionship beween sho and long 7 Since his is one of he simples special cases of VAR-ATSM, i is sufficien o check he poof fo he geneal model inoduced in Secion 4. Fo he poof, see Ang and Piazzesi (2003). 9

11 aes. Then he sensiiviy of he em spead o he faco is songe when δ is lage C-CAPM wih money in he uiliy (MIU) funcion Le s conside a C-CAPM, in which he sochasic discoun faco follows M u ( C, m ) = δ exp( π ), (20) C uc( C, m) whee δ is he subjecive discoun faco, holding a. Suppose ha he fom of he uiliy funcion is C is consumpion and m is he eal money uc (, m) = C m, (21) 1 ρ θ whee 0< ρ < 1 and 0 θ < 1. Then if C = Y 8 in equilibium, (20) can be ewien as M ρ Y = δ m exp( π ) = δ exp( ρg + θµ π ) Y+ 1 m θ = exp(log( δ) E[ ρg θµ + π ] ρσ u + θε σ u ), (22) ε, g g, + 1 µ, + 1 ε, π π, + 1 whee µ + 1 is he eal money gowh ae fom o +1, ε, 1 = µ 1 E[ µ 1], σ ε, gug, + 1 g+ 1 E g+ 1 µ = [ ], σ, u, + 1 = π+ 1 E[ π+ 1], σ, g > 0, σ, > 0, u ~ (0,1) g, + 1 N, and επ π u ~ (0,1) π, + 1 N. Fo simpliciy, le s assume u g, + 1 and u π, + 1 ae uncoelaed, as we ofen obseve empiically. Fis, le s conside a simple case in which θ = 0, i.e. uiliy is independen of money holding. Since ρ > 0, a posiive oupu gowh shock has a negaive effec on M 1. This is consisen wih a ole fo bonds as a consumpion hedge. Tha is, when he fuue oupu gowh ae is highe, consumes feel ha fuue cash flows ae less impoan. Noe ha boh make ε επ + 8 We assume his jus fo simpliciy. We can also genealize his model o be consisen wih he lieaue, which shows ha he dynamics of he consumpion gowh ae ae smoohe han hose of he oupu gowh ae, by assuming ha he consumpion gowh ae follows an affine funcion of he oupu gowh ae wih a posiive slope coefficien of less han uniy. Even in his genealized fom, he main popeies of he model do no change. 10

12 pices of isk, coesponding o he oupu gowh shock u g, + 1 and he inflaion shock, + 1, ae consan ( ρσ ε,g and σ ε, π especively). Nex, le s conside a geneal case in which θ > 0 and ε, + 1 can be epesened as a linea combinaion of u g, + 1 and, + 1 wih ime-vaying weighs: u π ε µ, + 1 wg, ug, + 1 wπ, uπ, + 1 = +, (23) µ u π whee he weighs w g, and w π, ae affine funcions of g and π : wg, = ω g + ωggg + ωg ππ, (24) w = ω + ω g + ω π. (25) π, π πg ππ The idea behind (23) is simila o Taylo s ule. Bu (23) uses he eal money gowh ae insead of he age sho ae, and has ime-vaying weighs. The ime-vaying weighs can be inepeed, fo example, as follows. Suppose ha he moneay policy auhoiy (he Fed) can pefecly conol he eal money gowh ae µ + 1 (i.e. ε, + 1) and can obseve u g, + 1 and, 1 befoe making hei policy decision. In esponse o a supise incease in he oupu gowh ae ( u g, + 1 > 0 ), he Fed may accommodae any incease in money demand caused by he oupu gowh shock by allowing he eal money gowh ae o ise. Convesely, he Fed may suppess he eal money gowh ae in esponse o he shock, if hey conside ha his oupu gowh shock may cause seious inflaion in he fuue. These wo plausible soies imply ha he weigh on he oupu gowh shock w g, can be eihe posiive o negaive. We can also discuss he weigh on he inflaion shock w π, in a simila way. Wih (23)-(25), (22) can be ewien as M+ 1 = exp(log( δ ) E[ ρg+ 1 θµ + 1+ π+ 1] [( ρσ ε, g θω g) θω ggg θω gππ ] ug, + 1 [( σ θω ) θω g θω π ] u ). (26) επ, π πg ππ π, + 1 µ u π + Now, in conas wih he simple C-CAPM wih θ = 0, he make pices of isk coesponding u + and u π, + 1 ae ime-vaying, depending on g and π. Fom (10), (11) and (26), we o g, 1 11

13 can obain 1 = log( ) + E[ ρg+ 1 θµ + 1+ π+ 1] δ [( ρσ θω ) θω g θω π ] + [( σ θω ) θω g θω π ] ε, g g gg gπ ε, π π πg ππ. (27) Alhough his ype of MIU funcion is ofen used in he lieaue, he validiy of his heoeical model has been he subjec of ciicism. Specifically, he uiliy funcion may no depend on money diecly. Also, he ime-sepaable uiliy funcion may be uneasonable due o, fo example, habi fomaion. In Secion 4, we will inoduce a moe geneal and less esiced model, which ness boh models discussed in Secion The VAR-ATSM Now le s inoduce he VAR-ATSM used fo lae analyses. This ype of model is used by Ang, Piazzesi and Wei (2003) o examine he pedicive powe of ems speads fo he oupu gowh ae. We use he VAR-ATSM o examine he pedicabiliy no only of oupu gowh, bu also of inflaion and sho aes. The VAR-ATSM can be inepeed as eihe a VAR model wih no-abiage esicions o an ATSM wih obsevable facos ha follow a VAR pocess. Le s sa by consideing he faco VAR. We use fou vaiables as facos: he oupu gowh ae g, he inflaion ae π, he sho ae, and a benchmak em spead s. Fo s, we use he em spead beween en-yea Teasuy bond YTM a he end of quae and vaiables ae assumed o follow a VAR(4) pocess,. These fou macoeconomic x = c+ Φ x + Φ x + Φ x + Φ x + ε, (28) whee x and ε = ( ε,, ε,, ε,, ε, )'. Following he VAR lieaue, le s = ( g, π,, s)' g π s inepe as a poxy fo he moneay policy insumen. Ang, Piazzesi and Wei (2003) use a simple model han ous. They use a hee vaiable VAR wih only one lag, and do no include he 12

14 inflaion ae. The VAR lieaue, howeve, usually uses a leas fou lags fo quaely daa, and indicaes ha he inflaion ae plays an impoan ole. Ou genealizaion of Ang, Piazzesi and Wei s model is in line wih his lieaue. To give a sucual inepeaion o he VAR, we need idenifying assumpions. We use a ecusive sucue wih he vaiables odeed as ( g, π,, s). Tha is, ε = Σu (29) whee exogenous shocks u = ( u,, u,, u,, u, )'~ N( 0, I ) i.i.d., and Σ is lowe-iangula g π s wih posiive diagonal elemens. Since significan esponses of g and π o conempoaneous inees aes ae implausible, ou odeing places hem befoe and s. The ode of g and π should no seiously affec he empiical esuls, since he coelaion beween g, ε and ε is small as shown lae. The coelaion, howeve, beween ε, and ε, is oo lage o be π, ignoed. Fo idenifying he las wo exogenous shocks u, and u s, s, ypically we need o adop one of wo assumpions: he sho ae (he moneay policy auhoiy) does no espond o he em spead (bond make) conempoaneously, o vice vesa. Since we ofen obseve long aes moving immediaely afe changes in moneay policy, he second assumpion seems uneasonable. In addiion, hee is no clea evidence suppoing a conempoaneous moneay policy esponse o he bond make. In fac, he lieaue povides evidence ha he Fed s behavio is ineial: he Fed s esponses o new infomaion end o be delayed. Thus we adop he fis assumpion 9. As will be seen in Secion 6, he impulse esponses seem o be easonable, and suppo ou ecusive assumpion. Wih his odeing, each componen of u can be inepeed as he exogenous shock o he coesponding vaiable. We call hem oupu gowh, inflaion, moneay policy, and spead shocks, especively. Now we may inepe he fis hee ows of sysem (28) as an IS cuve, a Phillips cuve, and a moneay policy ule, especively. The las ow can be inepeed as an endogenous esponse funcion fo he bond make. We can ewie he VAR in (28) ino companion fom, 9 Mos sudies in he VAR lieaue using boh sho and long aes choose he fis assumpion. Fo example, Leepe, Sims, and Zha (1996) discuss his issue in deail, and conclude ha he fis assumpion is less hamful han he second. 13

15 x c Φ1 Φ2 Φ3 Φ4 x 1 Σ u x 1 0 I x 2 = (30) x I 0 0 x x I 0 x o X = c + ΦX + Σu, (31) 1 whee X = ( g, π,, s,, g, π,, s )' is he 16 1 sae veco The sochasic discoun faco is defined as M 1 = exp λ ' λ λ ' u 2 1 = exp λ ' λ λg, ug, + 1 λπ, uπ, + 1 λ, u, + 1 λs, us, + 1, (32) whee λ = ( λg,, λπ,, λ,, λs, )' ae he make pices of isk. The veco λ is an affine funcion of he cuen economic vaiables x g s : = (, π,, )' λ = γ+ δx, (33) fo a 4 1 veco γ and a 4 4 maix δ. (n) 10 : By using he fundamenal asse picing equaion (10), we can obain closed foms fo ˆ ( n) ( n) ( n) = a + b ' X, n = 1, 2, (34) whee a ( n) ( n) ( n) = A / n, b = B / n, (35) n 10 We deive he closed foms fo discoun aes so ha he esicion = ˆ holds. Since we can calculae YTMs fom he discoun aes, we could also esic he model-implied spead s o be equal o s ˆ. Bu since hee may be a lage measuemen eo fo s, we do no use his esicion. 14

16 A B 1 = A + B '( c Σγ ) + B ' ΣΣ ' B, (36) 2 ( n+ 1) ( n) ( n) ( n) ( n) ~ ~~ '( Φ Σδ ) e ( n+ 1) ( ) ' = B n 3 ', (37) 0 A =, B ' e3' =, (38) ~ γ γ = 0 and δ 0 δ = 0 0, (39) e j is he j h column of he ideniy maix. Fom (35), (37) and (38), we can obain ( n) 1 ~ ~~ j b ' = e ( Φ Σδ ). (40) n n 1 3' j= 0 This is a quie simila fom o (17), and again he em pemium is consan only when δ = Esimaion 5.1. Esimaion mehods The VAR-ATSM has 98 paamees consising of 78 fom he VAR ( c, Φ [ Φ1Φ2Φ3Φ 4], and Σ ) and 20 in make pices of isk ( γ and δ ). We use GMM o esimae all paamees simulaneously 11. Momen condiions ae consuced by assuming ha he hee ypes of eo ae ohogonal o hei insumens. The fis of hese ae he VAR eos, ε = x ( c+ Φ x + Φ x + Φ x + Φ x ), (41) Ang, Piazzesi and Wei (2003) use wo-sep esimaion, in which he VAR paamees ae esimaed by OLS, and hen, given hese poin esimaes, γ and δ ae esimaed by minimizing he sum of he squaed picing eos of he discoun aes. This esimaion mehod has he advanage of having a smalle compuaional buden han ou one-sep esimaion. On he ohe hand, since hei esimaion mehod does no use efficien weighs on he momen condiions, i is less efficien han ous. In paicula, hei esimaes fo VAR paamees ae unable o aain any of he efficiency gains fom he no-abiage assumpion. Since ou lae analyses ae based on impulse esponse funcions calculaed fom he esimaes of VAR paamees, hese 15

17 whee he insumens ae a consan, x 1, x 2, x 3, and x 4 he covaiance maix of he VAR,. The second ype is he eo of ξ = vech( ΣΣ' εε '). (42) We assume ha he sample mean of ξ is exacly equal o zeo. Noe ha he momen condiions coesponding o (41) and (42) ae exacly he same as in OLS. The hid ype consiss of he discoun ae picing eos ν = [ ν (2) ν (4) ν (8) ν (12) ν (16) ν (20) ]' (43) whee ν ( n) = = ( n) ( n) ˆ ( n) ( a ( n) + b ( n) ' X ). (44) We use as insumens a consan, x 1, and x 2 fo his ype of momen. Now we have 132 momen condiions, which ae sufficien fo idenifying 98 paamees. We use he sample peiod 1964:1Q-2001:4Q, he same as was used fo he OLS egessions in Secion 2. We esic he paamee space wih wo ypes of esicion. Fis, he moduli of he eigenvalues of Φ ~ ae esiced o be less han uniy. Since he sae veco X follows he VAR pocess descibed in (31) wih an auocoelaion coefficien maix Φ ~, his esicion guaanees he saionaiy of X. In fac, esimaion esuls show ha his esicion does no bind. Second, he moduli of he eigenvalues of Φ Σδ ae esiced o be less han o equal o uniy. Fom (40), he faco loading e Φ Σδ ; j = ( n) b can be consideed as he aveage of '( ) j 3 0, 1,, n-1. So his second esicion guaanees ha, wih mauiy n, he faco loading does no divege. Noe ha his esicion is he genealizaion of (18). In ou esimaion esuls, only one of he esicions binds 12. efficiency gains ae cucial. 12 When a esicion binds, he specal densiy maix a fequency zeo is no guaaneed o be he opimal weighing maix in GMM. To solve his poblem, we use he binding esicion o subsiue ou a paamee in advance. Infeence will hen be coec when we use he obained non-esiced GMM o esimae paamees. The esimae and sandad eo of he subsiued paamee ae obained by subsiuing ou anohe paamee and e-esimaing. 16

18 5.2. Esimaion esuls The VAR esimaes achieve significan efficiency gains fom he no-abiage assumpion, alhough poin esimaes ae no so diffeen fom he esuls wihou he assumpion. 42 ou of 68 esimaes fo c and Φ (no epoed) ae significanly diffeen fom zeo a a size of 5%, while OLS wihou he no-abiage assumpion gives only 17 significan esimaes. These efficiency gains conibue o he eliabiliy of he impulse esponse funcions used lae. The esimae of Σ is epoed in Table 2. The diagonal elemens of Σ ae much highe han he ohes in geneal, which implies ha coelaions among he educed VAR eos ae small, bu he conempoaneous effec of he sho ae shock u, on he em spead s is oo lage o be ignoed. The oupu gowh shock has he lages volailiy, and his is abou hee imes as lage as he second lages volailiy, ha fo he inflaion shock. Table 3 epos he esimaes fo γ and δ. Seven ou of 16 esimaes of δ ae significanly diffeen fom zeo a size of 5%. This esul suppos he idea ha he make pices of isk ae indeed ime-vaying, depending on economic vaiables. Among hese significan paamees, he (1,1) and (1,2) elemens of δ, δ 11 and δ 12 have he mos influence on he em sucue. The eason fo his is as follows. Given he facos on he faco loadings ( n) b, which depend on Φ X, he em sucue depends only Σδ fom (40). So he influence of δ on he em sucue depends on Σ (i.e. Σ ). As we can see in Table 2, he (1,1) elemen of Σ, he volailiy of oupu gowh shock, is much lage han he ohes. So he fis ow of δ is he mos influenial. Among he esimaes in he fis ow, only δ 11 and δ 12 ae significanly diffeen fom zeo. In fac, as we will discuss in he nex secion, δ 12 plays a key ole in he pedicive elaionships, while δ11 does no. The posiive sign of δ 12 implies ha, when he inflaion ae π is highe, λ g, is highe and bond holdes ae willing o pay a highe pemium fo an oupu gowh isk hedge, which esuls in a lowe em pemium. Why do hey pay a highe pemium duing a highe inflaion egime? A possible explanaion can be obained fom hec-capm famewok wih 17

19 MIU funcion discussed in subsecion 3.2. Alhough his C-CAPM conains only oupu and inflaion shocks, we can genealize he model o be consisen wih he VAR-ATSM by adding moneay policy and spead shocks in (23) and leing he ime-vaying weighs on shocks depend on all fou VAR vaiables. Fom (26), δ12 = θω. So since θ > 0, δ 12 > 0 implies ω < 0. gπ gπ This means ha, when he inflaion ae π is high, he weigh on he oupu gowh shock g, w is small and he Fed is less accommodaing owad he oupu gowh shock. This esul makes sense if he Fed consides an oupu gowh shock duing a high inflaion egime likely o cause seious fuue inflaion. In such a siuaion, when inflaion is high, he Fed ends o suppess he eal money gowh ae in esponse o an oupu gowh shock. This less accommodaing Fed esponse educes he coelaion beween he oupu gowh shock u g, + 1 and he eal money shock ε, + 1. This educed coelaion causes fuue maginal uiliy, µ uc (, m ) = (1 ρ) C m, (45) ρ θ o be moe sensiive o he oupu gowh shock, which is a desied popey fo a consumpion hedge. Theefoe, consumes ae willing o pay a lage pemium o hold bonds duing a highe inflaion egime. We can also discuss he posiive sign of δ 11 in a simila way. Finally, he J-es suppos ou esimaes wih a high p-value of To ge a fuhe sense of he obusness of he esimaion esuls, le s compae he model-implied discoun aes ( ) ( ) ( ) ˆ n n n a ( n) = +b X and he sample aes and ( ) ˆ n ( n). Table 4 epos means and sandad deviaions of, and coelaions beween hem fo n = 2, 4, 8, 16, 20. Since hey have vey simila values fo means and sandad deviaions and he coelaions ae close o uniy, we can conclude ha ( ) ˆ n appoximaes ( n) vey well. 13 The p-value is calculaed fom he J-sa (3.2313) and he degees of feedom (23 = ). Noe ha since he one of he esicions on he eigenvalues binds, 1 should be subaced fom he degees of feedom. 18

20 6. Impulse Response Funcions and he Pedicive Powe of Tem Speads In he pevious secion, we obained esimaes fo ou VAR-ATSM wih gea efficiency gains fom he no-abiage assumpion. Le s use his model o examine he pedicive powe of em speads. vaiables in Fom he VAR-ATSM, we can calculae he opimal foecass condiional on he 16 sae X. Howeve, ou main inees lies no in foecass condiional on his lage numbe of vaiables, bu in foecass condiional on he em spead alone, in line wih he egessions in (4)-(6). Fo ou pupose, in subsecion 6.1, we fis conside he elaionship beween he impulse esponse funcions of vaiables in egessions (4)-(6) and he R 2 s. Since boh egessands and egessos can be epesened as affine funcions of X, we can calculae he impulse esponse funcions and he R 2 s fom paamees in he VAR-ATSM. The elaionship beween he impulse esponse funcions and he R 2 s will be used o shed ligh on he souce of he pedicive powe of em speads in subsecion Impulse esponse funcions and model-implied R 2 s Since x g π s follows he VAR pocess specified in (28), we can calculae = (,,, )' he coesponding impulse esponse funcions, and epesen he sysem in VMA( ) fom wih idenified exogenous shocks. Fo example, g can be epesened as g = g + ψ u + ψ u + ψ u + ψ u, (46) gg, j g, j gπ, j π, j g, j, j gs, j s, j j= 0 j= 0 j= 0 j= 0 whee g is he uncondiional mean of g, and impulse esponse funcions gg, j ψ, ψ g, j, ψ g, j, π and ψ gs, j ae funcions of Φ and Σ. So fuue oupu gowh g + h can be epesened as h 1 h 1 h 1 h 1 (47) g = gˆ + ψ u + ψ u + ψ u + ψ u + h + h gg, j g, + h j gπ, j π, + h j g, j, + h j gs, j s, + h j j= 0 j= 0 j= 0 j= 0 whee 19

21 g = g + ψ u + ψ u + ψ u + ψ u (48) ˆ + h gg, j g, + h j g π, j π, + h j g, j, + h j gs, j s, + h j j= h j= h j= h j= h is he opimal foecas of g condiional on + h X. Since discoun aes = a +b X and em speads ( n) ( m) ae affine ( n) ( n) ( n ) ' funcions of X = ( x ', x 1', x 2', x 3')', we can also calculae coesponding impulse esponse funcions, and epesen hem in VMA( ) fom. Fo example, can be epesened as ( n) ( m) ( n) ( m) ( n) ( m) ( n, m) ( n, m) ( n, m) ( n, m) = + κg, j g, j + κπ, j π, j + κ, j, j + κs, j s, j j= 0 j= 0 j= 0 j= 0, u u u u (49) whee is he uncondiional mean of ( n) ( m) ( n) ( m), and impulse esponse funcions κ, ( nm, ) g, j κ, ( nm, ) π, j κ and ( nm κ, ) ae funcions of Φ, Σ, and δ. ( nm, ), j s, j Since u ~ N( 0, I ) i.i.d., we can calculae he uncondiional vaiances of he VAR vaiables, opimal foecass fo hem, and em speads. Fom (46), (48) and (49), σ 2 g va( g ) ψgg, j ψgπ, j ψg, j ψgs, j j= 0 j= 0 j= 0 j= 0, (50) = σ va( gˆ ) 2 gh ˆ, + h ψgg, j ψgπ, j ψg, j ψgs, j j= h j= h j= h j= h, (51) = ( nm, ) 2 ( n) ( m) ( σ ) va( ) ( nm, )2 ( nm, )2 ( nm, )2 ( nm, )2 κg, j κπ, j κ, j κs, j j= 0 j= 0 j= 0 j= 0. (52) = Similaly we can calculae he coelaions among hese vaiables. The coelaion beween fuue oupu gowh g + h and he cuen em spead can be epesened as ( n) ( m) 20

22 ( n) ( m) co( g+ h, ) ψ κ ψ κ ψ κ ψ κ = ( nm, ) ( nm, ) ( nm, ) ( nm, ) gg, j + h g, j gπ, j + h π, j g, j + h, j gs, j + h s, j ( nm, ) ( nm, ) ( nm, ) ( nm, ) j= 0 σσ g j= 0 σσ g j= 0 σσ g j= 0 σσ g. (53) Since he foecasing eo of he opimal foecas g ˆ + h g+ h is unable o be pediced by any vaiable known a ime, such as, ( n) ( m) co( g, ) = co( gˆ, ). (54) ( n) ( m) ( n) ( m) + h + h By squaing he coelaion, we can obain he R 2. Fo example, he R 2 fo egession (4) can be epesened as R = co( gˆ, ). (55) 2( nm, ) ( n) ( m) 2 gh, + h Since he R 2 s ae funcions of paamees in ou VAR-ATSM, we can calculae hem fom he esimaes of he paamees. We call hese he model-implied R 2 s. Equaion (54) implies ha if ( n) ( m) is a good pedico fo fuue oupu gowh g + h, ( n) ( m) should espond o exogenous shocks in a simila way o g ˆ+ h. We invesigae his by looking a he vaiance decomposiion of g ˆ+ h in he nex subsecion. Finally, as we can see fom (53)-(55), he R 2 s depend on he sum of poducs of he impulse esponse funcions fo egessands and egessos. Noe ha, in (53), indexes fo ψ s sa fom +h, no, because fuue shocks u,, 1 u ae + + h unpedicable. This implies ha since he ψ s ypically decay wih he hoizon j, good pedico if i is esponsive o ecen shocks, i.e. κ s ae lage fo smalle j. is a ( n) ( m) 6.2. Why do em speads have pedicive powe? Figue 3 displays he model-implied R 2 s fom egessions (4)-(6) fo hee seleced em speads, and is he model-calculaed analog of Figue 2. The esuls show ha he model-implied R 2 s eplicae hee popeies of he sample R 2 s in Figue 2 vey well. Fis, he 12Q-8Q spead pefoms bee han he 20Q-1Q spead, excep fo oupu gowh pedicions a shoe hoizons. Second, he 2Q-1Q spead is almos useless. Finally, i is difficul o pedic oupu gowh a 1Q ahead. I is heefoe easonable o y o explain he sample R 2 s in Figue 2 in ems of he 21

23 facos ha deemine he model-implied R 2 s in Figue 3. Since he model-implied R 2 s ae funcions of he paamees in ou VAR-ATSM, we can analyze how hese paamees affec he R 2 s. s Figue 4 shows he impulse esponse funcions of he VAR vaiables 22 g, π,, and o one uni exogenous shocks. These ae based on he esimaes fom he esiced GMM esimaion of he VAR-ATSM. In geneal, hese esuls ae consisen wih hose in he VAR lieaue. Fo example, (4-a) and (4-b) show ha he sho ae, he insumen of he moneay policy auhoiy, esponds posiively o oupu gowh and inflaion shocks. Panel (4-c) demonsaes ha he esimaed moneay policy shock shaply educes oupu gowh. This shock also suppesses inflaion aes in he long un. These easonable esuls imply ha esimaes of he moneay policy shock ae easonable. Fuhe suppo is povided by Panel (4-d). As we discussed in Secion 3, he mos quesionable pa of ou idenificaion saegy may come fom he conaminaion beween he moneay policy shock and he spead shock. Panel (4-d) indicaes ha he esimaed spead shock aises oupu gowh and suppesses inflaion. Since oupu gowh and inflaion should espond o a moneay policy shock in he same diecion, he esuls in (4-d) sugges ha he spead shock is no measuing a change in moneay policy. Figue 5 shows vaiance decomposiions of he opimal foecass, whee he vaiances of foecass such as (51) ae nomalized o uniy. As discussed in he pevious subsecion, his indicaes which exogenous shocks should be useful fo pedicion. Panel (5-a) shows ha he oupu gowh shock dominaes pedicions of oupu gowh a one quae ahead. Then aound 2-4 quaes ahead, he moneay policy shock is he mos impoan. The impoance of he inflaion shock inceases wih he foecasing hoizon, and his shock finally becomes mos influenial a 12 quaes ahead. These esuls ae consisen wih he impulse esponse funcions in Figue 4. The oupu gowh shock causes a shap jump in oupu gowh, bu only in he sho un. The moneay policy shock has a negaive effec on oupu gowh, bu wih 2-4 quae lags. In he long un, he ise in he sho ae induced by he inflaion shock is pesisen, and his acs o suppess oupu gowh. Panels (5-b) and (5-c) show ha he inflaion shock is mos impoan fo pedicing inflaion and sho aes a mos hoizons. Accodingly, he esponse of he em spead o he inflaion shock is cucial fo specifying he souce of is pedicive powe, especially a longe hoizons. Noe ha, as Figue 4 implies, he effecs of exogenous shocks decay wih he

24 hoizon. So we can also say ha good pedicos should espond o ecen shocks ahe han old shocks. Figue 6 shows impulse esponse funcions of seleced discoun aes. Thee ae hee noable feaues. Fis, he effec of he inflaion shock on he levels of he discoun aes is highly pesisen. In fac, he discoun aes do no eun o zeo even afe 40 quaes. Since good pedicos should espond o ecen shocks ahe han old shocks, his is an impoan eason why levels of yield cuves do no have gea pedicive powe. Second, discoun aes wih diffeen mauiies display diffeen esponses o ecen shocks, while hey espond o old shocks in simila ways. This implies ha mos movemens in em speads ae due o ecen shocks, because old shocks esul in almos paallel shifs of he yield cuve. In fac, he uppe gaphs of Figue 7 illusae he consideable dependence of boh he 20Q-1Q and 12Q-8Q speads on ecen shocks. This is one eason why em speads have pedicive powe. Why do discoun aes espond like his? We find ha he ime-vaying make pice of isk plays he following impoan ole. As discussed in Secion 5, i is he paamees coesponding o he effecs of he oupu gowh and inflaion aes on he make pice of oupu gowh isk, δ11 and δ 12, ha have he mos influence on movemens in long aes. Of hese, only δ 12 has a suppoive ole o play in he pedicive elaionship. As shown in Figue 5, he inflaion shock is he cucial elemen in he pedicive elaionship, and a posiive δ 12 causes he make pice of oupu gowh isk o espond posiively o he shock. In conas o his, a posiive δ 11 educes pedicive powe. As shown in (4-b), a posiive inflaion shock causes a decease in he oupu gowh ae, which has a negaive effec on he make pice of oupu gowh isk. Since he effec fom δ12 dominaes he effec fom δ 11, he make pice of oupu gowh isk esponds posiively and so he em pemium esponses negaively o he inflaion shock. In evaluaing he effec fom δ 12, we calculaed he impulse esponse funcions of discoun aes when δ 12 = 0 and ohe paamees ae unchanged in Figue 8. The main change in he impulse esponse funcions appeas in (8-b), which is oally diffeen fom (6-b). In (6-b), 23

25 he esponses of longe aes ae smalle han hose of he sho ae, and he diffeence beween hei esponses almos disappeas aound 20 quaes ahead. On he ohe hand, in (8-b), he esponses of longe aes ae songe han hose of he sho ae, and his dispaiy does no disappea even aound 40 quae ahead. Why ae hei esponses so diffeen? The expecaions hypohesis saes ha he long ae is he aveage of expeced sho aes plus a consan em pemium. Whehe we look a (6-b) o (8-b), he inflaion shock coninues o aise he sho ae up o aound 20 quaes ahead. So, accoding o he hypohesis, he iniial esponses of long aes wih mauiies up o 20 quaes should be songe han he esponse of he sho ae, as illusaed in (8-b). Since δ 12 is posiive, howeve, he inflaion shock aises he make pice of oupu gowh isk, and so educes he em pemium. This is why long aes espond less songly han he sho ae in (6-b). The diffeence beween he esponses in (6-b) and (8-b) has a significan effec on he pedicive powe of em speads. Figue 9 gives model-implied R 2 s fo he case when δ 12 = 0. Supisingly, he R 2 s almos disappea. This enables us o conclude ha a posiive δ 12, which can be inepeed in ems of consumes willingness o pay a highe pemium fo an oupu gowh isk hedge duing a highe inflaion egime, is a key explanaion fo he pedicive powe of he em spead. The las noable feaue of Figue 6 is he lagged esponse of he 1Q ae (he moneay policy auhoiy) o oupu gowh and inflaion shocks. Panel (6-a) shows ha he immediae esponse of he 1Q ae o an oupu gowh shock is he smalles among he discoun aes, alhough he esponse of he 1Q ae is lages seveal quaes ahead. Panel (6-b) shows ha he immediae esponse of he 1Q ae o an inflaion shock is smalle han ha of he 2Q ae, and almos coincides wih he esponse of he 8Q ae. These esuls ae consisen wih ineial behavio by he moneay policy auhoiy, as empiically shown by, among ohes, Claida, Gali, and Gele (2000). The lowe gaphs in Figue 7 show he impulse esponse funcions of 20Q-1Q and 12Q-8Q speads o oupu gowh and inflaion shocks. The immediae esponse of he 20Q-1Q spead is much weake han ha of he 12Q-8Q spead because of he slow esponse of he 1Q ae. Since ecen shocks ae vey impoan fo pedicive puposes, we can conclude ha his is he eason behind he infeio pefomance of he 20Q-1Q spead compaed o he 12Q-8Q spead. Tha is, he moneay auhoiy s ineial behavio disubs he esponses of em speads using he sho end of he yield cuve o oupu gowh and inflaion shocks. 24

26 Fuhe suppo fo his view is povided by he coelaions beween fuue pediced vaiables and cuen em speads. Since model-implied R 2 s ae squaes of hese model-implied coelaions, we can use he coelaions o analyze why we found he R 2 s shown in Figues 2 and 3. Equaion (53) has fou summed ems, each of which can be inepeed as he conibuion of he coesponding exogenous shock o he pedicive elaionship. Figue 10 shows he conibuions of exogenous shocks o he absolue values of he coelaions wih he 20Q-1Q and 12Q-8Q speads. The oupu gowh and inflaion shocks conibue o he coelaions wih he 12Q-8Q spead ahe han wih he 20Q-1Q spead. These diffeences explain why he 12Q-8Q spead is useful fo pedicion. This esul is consisen wih ou discussion of he lowe gaphs in Figue 7. Anohe noable feaue of Figue 10 is he hump-shaped conibuion of moneay policy shocks o oupu gowh pedicions. So we can conclude ha he hump-shape of he R 2 s fo he oupu gowh pedicions is aibuable o he moneay policy shock. Tha is, he moneay policy shock affecs oupu gowh wih a lag, while he em sucue esponds o he shock immediaely. This diffeence in iming makes i hade fo em speads o help foecas oupu gowh a sho hoizons. Finally, Figue 11 shows he conibuions in he case whee δ 12 = 0. Obviously he shap dops of R 2 s ae aibuable o he diffeen sign of he conibuion of he inflaion shock, which is caused by he song long ae esponse o he shock. 7. Conclusion Why do em speads pedic oupu gowh, inflaion, and sho aes? In answeing his quesion, we used a VAR-ATSM model wih fou lags and fou vaiables, which is less esiced han simila affine em sucue models wih obsevable facos in he exising lieaue. We succeeded in esimaing his model using an efficien mehod. We have hee main findings. Fis, he ime-vaying make pice of oupu gowh isk, which is sensiive o he inflaion ae, plays a key ole in explaining why he em spead helps 25

27 foecas oupu gowh, inflaion, and inees aes. This finding can be inepeed as follows. When he inflaion ae is highe, consumes ae willing o pay a highe pemium fo an oupu gowh isk hedge, possibly because, wihin his highe inflaion envionmen, he Fed s esponse o an oupu gowh shock is less accommodaing and so maginal uiliy is moe sensiive o he shock. This causes em speads o eac o ecen inflaion shocks, which also poves useful fo foming longe-un foecass. Second, we also found ha em speads using he sho end of yield cuve have less pedicive powe han many speads beween longe aes. This fac is aibuable o he ineial chaace of moneay policy. Finally, i is had o pedic oupu gowh wih em speads a sho hoizons, because moneay policy shocks affec oupu gowh wih a lag while he em sucue esponds o he shock immediaely. 26

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