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

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1 Why Can he Yield Cuve Pedic Oupu Gowh, Inflaion, and Inees Raes? An Analysis wih Affine Tem Sucue Model Hibiki Ichiue Depamen of Economics, Univesiy of Califonia, San Diego The Bank of Japan Augus, 2003 Absac The lieaue gives 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. The esuls sugges ha consumes ae willing o pay a highe pemium fo oupu gowh isk hedge duing he highe inflaion egime. This causes em speads o eac o ecen inflaion shocks, which pove 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 suppos and commens. I would like o hank Majoie Flavin, Buce Lehmann, Alan Timmemann, Keiichi Tanaka, and paicipans in he pesenaions a UCSD. I also hank Monica Piazzesi fo answeing my deail quesions on he papes. The views expessed hee ae hose of he auho, and no necessaily of he Bank of Japan. Addess: 2-- Nihonbashi-Hongokucho Chuo-ku Tokyo Japan Tel.: (The Bank of Japan), Fax: , addess: hibiki.ichiue@boj.o.jp

2 . Inoducion Many sudies in he lieaue give evidence ha em speads of inees aes have 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 by esing a vey popula and classic heoy, he expecaions hypohesis. In his heoy, he long ae is equal o an aveage of expeced fuue sho aes plus a ime-invaian em pemium. Howeve, in spie of is populaiy, his hypohesis ypically has been ejeced. Many economiss ague ha his expecaions hypohesis failue is aibuable o he failue of he assumpion of ime-invaian em pemium 2. The lieaue on he pedicabiliy of inflaion also has a long hisoy following Fama s (975) 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 (989) found ha a em spead plays an impoan ole in hei index of economic leading indicaos, many eseaches invesigaed his pedicabiliy 4. Fo empiical esuls of esing he expecaions hypohesis, see, fo example, Campbell and Shille (99), Hadouvelis (994), Rudebusch (995), Campbell, Lo and MacKinlay (997), Robeds and Whieman (999), Bekae, Hodick and Mashall (200), and Cochane (200). 2 The lieaue gives evidence ha em pemium is in fac ime-vaying. See, fo example, Mankiw and Mion (986), Engle, Lilien and Robins (987), Engle and Ng (993), Dosey and Ook (995), and Balduzzi, Beola and Foesi (997). 3 Fo empiical esuls of he pedicabiliy of inflaion, see, fo example, Mishkin (988, 990a, b, 99), Fama (990), Joion and Mishkin (99), Esella and Mishkin (997), and Kozicki (997). 4 Fo empiical esuls of he pedicabiliy of oupu gowh o ecession, see, fo example, Esella and Hadouvelis (99), Plosse and Rouwenhos (994), Haubich and Dombosky (996), Bonse-Neal and Moley(997), Dueke (997), Esella and Mishkin (997), Kozicki (997), Benad and Gelach (998), 2

3 Alhough a huge lieaue gives evidence and explanaions fo each of he pedicabiliies of oupu gowh, inflaion, and inees aes, no pape ies o analyze he elaionship among all of hese hee pedicabiliies. 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 cycle. We use an affine em sucue model (ATSM) wih obsevable economic facos as ou main ool. Afe Ang and Piazzesi (2003) inoduced his ype of model o invesigae he elaionship beween macoeconomic vaiables and he em sucue, he idea has been followed by seveal sudies, 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 as eihe a VAR wih no-abiage esicions o ATSM wih obsevable facos obeying VAR. In his pape, we call his ype of model 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, and exend o he pedicabiliies of no Dosey (998), and Hamilon and Kim (2002). 3

4 only oupu gowh bu also inflaion and sho aes 5. Alhough hei basic idea is vey useful fo analyzing he pedicabiliies, some of hei assumpions and esimaion mehod ae no suiable o ou pupose. Ang, Piazzesi and Wei y o find good foecasing models by compaing pedicive powes, especially olling ou-of-sample foecasing pefomances, of vaious combinaions of egessos. Fo conducing his execise, hei pasimony VAR() model and compuaionally fas, bu less efficien, esimaion mehod may be appopiae. On he ohe hand, we y o eveal 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 moe efficien esimaion mehod, which conibue o he eliabiliy of impulse esponse funcions. We have hee main findings. Fis, he ime-vaying make pice of oupu gowh isk, which is sensiive o he level of inflaion, plays a key ole in he pedicabiliy. When he inflaion ae is highe, consumes ae willing o pay a highe pemium fo oupu gowh isk hedge, which may be explained by a simple model wih a money in he uiliy funcion and a moneay policy ule. This causes em speads o eac o ecen inflaion shocks. Since he inflaion shock has pesisen effecs on no only inflaion bu also oupu gowh and inees aes, he esponse of em spead 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 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 (999), Dai and Singleon (2002), and Duffee (2002) examine whehe ATSM s can fi o he empiical esuls on pedicabiliy of inees aes. Hamilon and Kim (2002) use he Longsaff and Schwaz s (992) em sucue model o explain pedicabiliy of oupu gowh. Bu since hese models use only laen facos, he abiliy o analyze he elaionship among em sucue and macoeconomic vaiables is limied. 4

5 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 displays sylized facs fom simple OLS esuls. In Secion 3, we conside simple ATSM s o undesand basic popeies of ATSM s. 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 infomaion included in 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 pedicabiliies of em speads fo fuue oupu gowh, inflaion, and inees aes wih a common economeic mehod, egessions on he em speads. Howeve, hese egessions do no have exacly he same fom. Fo example, Esella and Mishkin (99) examine oupu gowh pedicabiliy by using egessions of cumulaive oupu gowh fom o +h on a fixed em spead beween en-yea and hee-monh inees aes: g = + +. () (0Y) (3M) + h α β( ) ε+ h On he ohe hand, Mishkin (990a) examines inflaion pedicabiliy by using egessions of 5

6 diffeence beween h-peiod and -yea cumulaive inflaion aes on mauiy maching em speads: π π = α + β + ε. (2) ( h) (Y) + h + Y ( ) + h Campbell and Shille (99) give evidence fo sho ae pedicabiliy by using he mos popula expecaions hypohesis es, egessions of aveage fuue sho ae changes on mauiy maching em speads:. (3) h h () () ( h) () ( + i ) = α + β( ) + ε+ h i= 0 All hee ypes of sudies find ha he slope coefficien β is significanly diffeen fom zeo in many cases, which means ha em speads have pedicive powe fo 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. () uses a fixed egesso, while he egesso of (2) depends on he foecasing hoizon h. So fo analyzing he elaionship among he pedicabiliies, 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 = + + ; (4) ( n) ( m) + h α β( ) ε+ h 6

7 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 =,2,,2; n, m = 2, 4, 8, 2, 6, 20, and n > m), whee g is he eal GDP gowh ae fom - o, π is he inflaion ae of GDP deflao fom - o, and (n) is he n-peiod discoun ae of Teasuy bills o bonds a end of 6. Quaely daa ae used, so we inepe one peiod as one quae. All of g, π, and (n) ae defined as aes pe quae. The sample peiod is 964:Q-200:4Q, following Fama and Bliss (987) who commen ha long ae daa befoe 964 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 ae. Since cumulaive aes ae he aveages of maginal aes, maginal aes ae moe convenien fo specifying which pa of fuue he em speads can pedic well. Second, we use vaious foecasing hoizons h and em speads, so we can specify which componens of he yield cuve pedic a which ( n) ( m) fuue hoizons. Figues and 2 display he -sas and R 2 s of OLS egessions (4)-(6) fo seleced em speads. 20Q-Q spead has significan pedicive powe fo all of oupu gowh, inflaion, and 6 We use discoun ae daa fom CRSP (Cene fo Reseach in Secuiy Pices, Gaduae School of Business, he Univesiy of Chicago: All ighs eseved.) Monhly US Teasuy Daabase wih pemission. We can consuc discoun aes fo, 2, 4, 8, 2, 6, 20 quaes fom he CRSP daa. The 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 2 (hee is no daa on 9/30/987, so we inepolae wih 3 and 2 monh aes). The ohe aes ae obained fom he Fama-Bliss discoun bonds file. 7

8 sho aes a leas fo shoe hoizons. This esul is consisen wih he lieaue, which agues ha em spead beween 5-yea (o 0-yea) and 3-monh aes pedic well. Bu supisingly we found ha mos em speads wihou Q ae ae bee han he 20Q-Q spead in many cases. Fo example, Figue 2 shows ha 2Q-8Q spead is bee excep fo pedicing oupu gowh aes a shoe hoizons. In addiion, 2Q-Q spead, which also uses he Q ae, is almos useless. These facs seem o imply ha em speads using he sho end of he yield cuve educe he pedicabiliies. This is supising because he lieaue does no cae abou he spead wihou he sho end of em sucue so much, and seveal sudies including Ang, Piazzesi and Wei (2003) ague ha he maximal mauiy diffeence is he bes pedico. As anohe noable feaue of he gaphs, we can see ha he R 2 s of oupu gowh egessions ae hump-shaped. Tha is, i is difficul o pedic oupu gowh ae a sho hoizons. Why can em speads pedic he fuue in such ways? Since OLS esuls do no answe his quesion, we need a moe sucued model. A useful mehod o inepe hese OLS esuls is poposed by Ang, Piazzesi and Wei (2003). They use a VAR-ATSM o epesen he model-implied R 2 s fo he egessions of oupu gowh aes and compae pedicive powes of vaious combinaions of egessos. We follow hei basic idea, and exend hei mehods o explain pedicabiliies of all of oupu gowh, inflaion, and sho aes. Alhough hei VAR-ATSM is vey useful fo examining he elaionship among macoeconomic vaiables and he yield cuve, some of hei assumpions and esimaion mehod ae no suiable o ou pupose. So we modify hem in Secion 4 and 5. Then, in Secion 6, we y o eveal 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. 8

9 3. Simple Affine Tem Sucue Models wih Obsevable Facos Befoe inoducing ou VAR-ATSM in he nex secion, le s conside fou simple ATSM s. Since he VAR-ATSM is oo complicaed o give simple inepeaions, we should sa fom hese simple models. In paicula, ime-vaying make pices of isk, which many classic em sucue models assume consan, ae he souce of he complicaion. Bu since hey affec he elaionship beween sho and long aes, i.e. movemens in em speads, hey ae vey impoan fo examining he pedicabiliies of em speads. 3.. An ATSM wih One Faco of Sho Rae Suppose ha quaely daa of sho (3-monh) ae () ae chaaceized by an AR() pocess: = c + φ + σ u, (7) () () +, + whee u ~ (0,), + N i.i.d., and σ > 0. Table epos he OLS esimaes of (7), which show ha he sho ae is pesisen ( φ = ). Suppose ha he sochasic discoun faco M + obeys a condiional log-nomal disibuion: () 2 M exp + = λ, λ, u, + 2, (8) 9

10 whee λ = γ + δ. (9) (), So in his model, he make pice of isk λ, is ime-vaying, depending on he faco (). Tha is, he sochasic discoun faco M + is affeced by no only he exogenous shock u, + bu also 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 (977) 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 ], (0) ( n) ( n ) + + fo n =, 2,, and all, whee q is he n-peiod bond pice. Noe ha fom (8) and (0), ( n ) q = E[ M q ] () (0) + + = E[ M ] + () = exp( ). () 0

11 This is exacly he definiion of he elaionship beween he -peiod bond pice and coninuously compounded discoun ae. In fac, M + is chosen so ha () holds. By using he fundamenal asse picing equaion (0), we can deive closed foms fo discoun aes as affine funcions of he faco ( n ) () : ( ) ( ) ( ) () ˆ n n n a b = +, n =, 2, (2) whee ( n) ( n) ( n) ( n) a A / n, b B /n = =, (3) ( n+ ) ( n) ( n) 2 ( n)2 A = A + B (c σγ ) + σ B, (4) 2 ( n+ ) ( n) B B ( φ σδ ) =, (5) () A = 0, () B = 7. (6) Fom (2), 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 (3), (5), and (6), we can obain a closed fom fo ( n) b : 7 Since his is one of he simples special cases of VAR-ATSM, i is enough o check he poof fo he geneal model inoduced in Secion 4. Fo he poof, see Ang and Piazzesi (2003).

12 j ). (7) n ( n) b = ( φ σδ n j= 0 Noe ha γ does no appea in (7). Since movemen of sho aes is less volaile han ha of long aes, i is easonable ha he absolue value of ( n) b deceases as n inceases. Fo saisfying his, we need paamee values such ha φ σ δ <. (8) Fom (8), φ + φ < δ <. (9) σ σ Since poin esimaes in Table imply ( + φ) / σ 334, δ can be even hundeds. Fom (7), we can say ha he sensiiviy of long aes o he sho ae is weake, when δ is highe. We can elae his claim wih he expecaions hypohesis. Fom (7), = c + φ + σ u () () + j + j, + j () φ( φ + j 2 σ, + j ) σ, + j = c + c + + u + u M j j i j () i φ φ σ φ, + j i i= 0 i= 0. (20) = c + + u 2

13 So by aking he expecaion, j () i j () [ + j] = φ + φ i= 0 E c. (2) Then, fom (2), (7) and (2), we can obain he em pemium: E a c n n j n ( n) () ( n) i j j () [ + j] = φ + [( φ σδ) φ ] n j= 0 n j= 0 i= 0 n j= 0. (22) So he em pemium is consan, i.e. he expecaion hypohesis holds only when δ = 0. In his case, he movemens of long aes ( n) depend only on hose of aveage expeced sho aes n n () E j 0 = + j [ ]. Since () obeys a pesisen AR() pocess, an incease in aises ( n). () Howeve, when δ > 0, a ise in () also has a negaive effec on hough a decease in ( n ) he em pemium. Theefoe, posiive δ weakens he elaionship beween sho and long aes. Then he sensiiviy of he em spead o he faco () is songe when δ is lage A one faco ATSM wih a consan sho ae Le s conside a model wih consan sho ae () and one faco x obeying AR(): x = c + φ x + σ u, (23) + x x x x, + 3

14 whee u ~ (0,) x, + N i.i.d., and σ x > 0. Suppose ha he sochasic discoun faco obeys M = exp λ λ u 2 () 2 + x, x, x, + (24) whee x, = γx δxx. (25) λ + By using he fundamenal asse picing equaion (0), we can deive expessions fo discoun aes : ( n ) ( ) () ˆ n =, n =, 2, (26) Tha is, when sho ae is consan, yield cuve is always pefecly fla. Moe impoanly, he faco x can no affec he yield cuve, even if he exogenous shock x, u + has a song effec on he sochasic discoun faco M +. This implies ha he sochasic faco x affecs bond pices only hough he movemens in sho aes. So we can no conclude whehe he effec of he faco on he yield cuve is song o no only fom he make pice of isk C-CAPM Le s conside a simple C-CAPM, in which he sochasic discoun faco obeys 4

15 M u'( C ) = δ exp( π ), (27) u'( C ) wih CRRA uiliy funcion ρ C uc ( ) =, (28) ρ whee δ is he subjecive discoun faco, C is consumpion a, and ρ > 0 is he coefficien of elaive isk avesion. Suppose in equilibium, he consumpion C is equal o he oupu Y so ha he consumpion gowh ae gc, + = g 8 +. Then (27) can be ewien as M C = δ exp( π ) = δ exp( ρg π ) + + C + ρ C, + + = δ exp( ρg π ) + + = δ exp( ρ{ E[ g ] + ε } { E[ π ] + ε }) + g, + + π, + = exp(log( δ ) ρe[ g ] E[ π ] ρσ u σ u ). (29) + + ε, g g, + ε, π π, + 8 We assume his jus fo simpliciy, and we can genealize his model o be consisen wih he lieaue, which shows ha dynamics of consumpion gowh ae is smoohe han ha of oupu gowh ae, by assuming ha consumpion gowh ae obeys an affine funcion of oupu gowh ae wih a posiive and less han uniy slope coefficien. Even wih his genealized assumpion, he main popeies of he model do no change. 5

16 ε = σ ε u = g E [ g ], επ, + = σε, πuπ, + = π+ E[ π+ ], and σ ε,g and σ, whee g, +, g g, ε π ae sandad deviaions of ε g, + and ε, + so ha u ~ (0,) g, + N and u, ~ N(0,) π π +. Suppose u g, + and u π, + ae uncoelaed as we ofen obseve empiically. Since ρ > 0, a posiive oupu gowh shock has a negaive effec on M. This is consisen wih a ole of bonds fo consumpion hedge. Tha is, when fuue oupu gowh ae is highe, consumes feel ha fuue cash flows ae less impoan. Noe ha boh of he make pices of isk coesponding o he oupu gowh shock u g, + and inflaion shock, + ae consan + u π (,g ε ρσ and σ, especively). ε π Fom (0), (), and (29), exp( ) = q () () = E[ M + ] = exp(log( δ) ρe[ g ] E[ π ] + { ρ σε + σε π }/ 2). (30) , g, So we can obain 2 2 () ρσg + σ = log( ) + ρe[ g ] E[ π ] π (3) δ 2 Tha is, his C-CAPM implicily assumes (3) holds. This suggess ha he sho ae is highe when expeced oupu gowh and inflaion aes ae highe. An ineesing special case of (3) is 6

17 g = π =, (32) wihou unceainy. In his case, (3) is () = log( ), (33) δ which means ha he sho ae is equal o he subjecive discoun ae C-CAPM wih money-in-he-uiliy funcion Le s conside anohe C-CAPM in which we eplace (28) wih a money-in-he-uiliy (MIU) funcion uc (, m ) whee m is he eal money holding. Even wih he MIU funcion, he sochasic discoun faco obeys he same fom as (27): M u'( C, m ) = δ exp( π ), (34) u'( C, m) whee u'( C, m) u( C, m) C. Suppose ha he fom of he uiliy funcion is ρ C θ uc (, m) = m, (35) ρ 7

18 whee ρ > 0 and 0< θ <. Then if C = Y as befoe, (34) can be ewien as M ρ Y = δ m exp( π ) = δ exp( ρg + θµ π ) Y+ m θ = exp(log( δ) E[ ρg θµ + π ] ρσ u + θε σ u ), (36) ε, g g, + µ, + ε, π π, + whee µ + is he eal money gowh ae fom o +, and ε µ, = µ E[ µ ] Le s conside a case in which ε µ, + can be epesened as a linea combinaion of u g, + and u π, + wih ime-vaying weighs: ε µ, + wg, ug, + wπ, uπ, + = +, (37) whee he weighs w g, and w π, ae affine funcions of g and π : wg, = ωg + ωggg + ωg ππ, (38) w = ω + ω g + ω π. (39) π, π πg ππ The idea behind (37) is simila as Taylo s ule. Bu (37) uses eal money gowh ae insead of 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 obseve u g, + and 8

19 u π, + befoe hei decisions, by which hey can pefecly conol he eal money gowh ae µ + (i.e. ε µ, + ). When oupu gowh ae supisingly inceases ( u g, + > 0 ), he Fed may accommodae he eal money gowh ae o an incease in money demand caused by he oupu gowh shock. Convesely, he Fed may suppess he eal money gowh ae in esponses 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. Equaion (38) implies ha he weigh depends on g and π. We can also discuss he weigh on he inflaion shock w π, in a simila way. Wih (37)-(39), (36) can be ewien as M = exp(log( δ ) E[ ρg θµ + π ] [( ρσ θω ) θω g θω π ] u ε, g g gg gπ g, + [( σ θω ) θω g θω π ] u ). (40) επ, π πg ππ π, + Now in conas wih he simple C-CAPM discussed in pevious subsecion, he make pices of isk coesponding o u g, + and u π, + ae ime-vaying, depending on g and π. Similaly o (3), we can obain () = log( ) + E[ ρg+ θµ + + π+ ] δ 2 2 [( ρσε, g θωg) θωggg θωgππ] + [( σε, π θωπ) θωπgg θωπππ]. (4) 2 9

20 Alhough his ype of MIU funcions is ofen used in he lieaue, he validiy of his heoeical model is unde ciicisms. The uiliy funcion may no depend on money diecly. The 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 all of fou 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 pedicabiliy of oupu gowh ae using em speads. We use he VAR-ATSM fo examining he pedicabiliies of no only oupu gowh, bu also inflaion and sho aes. The VAR-ATSM can be inepeed as eihe a VAR model wih no-abiage esicions o ATSM wih obsevable facos obeying VAR. Le s sa by consideing he VAR of facos. We use fou vaiables: oupu gowh ae g, inflaion ae π, sho ae (), and a benchmak em spead s as facos. As s, we use he em spead beween en-yea Teasuy bond YTM a end of quae and (). These fou macoeconomic vaiables ae assumed o obey VAR(4), x = c+ Φ x + Φ x + Φ x + Φ x + ε, (42) whee π () x = ( g,,, s)' and ε = ε g, επ, ε, εs, (,,, )'. Following he VAR lieaue, le s 20

21 inepe () as a poxy fo he moneay policy insumen. Ang, Piazzesi and Wei (2003) use a simple model han ous. They use a VAR wih only one lag and hee vaiables which do no include inflaion ae. Bu he VAR lieaue usually uses a leas fou lags fo quaely daa, and indicaes ha he inflaion ae plays an impoan ole. So we follow he VAR lieaue o genealize Ang, Piazzesi and Wei s model. 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 (43) whee exogenous shocks u = ( ug,, uπ,, u,, us, )'~ N( 0, I ) i.i.d., and Σ is lowe-iangula wih posiive diagonal elemens. Since i is no plausible ha g and π espond o conempoaneous inees aes, we ode hem befoe () and s. The ode beween g and π should no have seious effecs on he empiical esuls, since he coelaion beween g, ε and ε π, is small as shown lae. Bu he coelaion beween ε, and ε s, is oo lage o be ignoed. Fo idenifying he las wo exogenous shocks u, and u s,, ypically we need adop one of wo assumpions: he sho ae (he moneay policy auhoiy) does no espond o he em spead (bond make) conempoaily, o vice vesa. Since we ofen obseve ha long aes move immediaely afe changes in moneay policy, he second assumpion seems o be uneasonable. On he ohe hand, hee is no clea evidence ha he moneay policy auhoiy esponds o he bond make conempoaneously. Moeove he lieaue gives evidence fo he 2

22 Fed s ineial behavio, in which he Fed s esponses o new infomaion end o delay. Thus we adop he fis assumpion 9. As will be seen in Secion 6, he impulse esponses o esimaed moneay policy shock u, and spead shock u s, seem o be easonable, and suppo ou ecusive assumpion. Wih his odeing, each componen of u can be inepeed as he exogenous shock o each coesponding vaiable. We call hem, oupu gowh, inflaion, moneay policy, and spead shocks. Now we may inepe he fis hee ows of he sysem (42) as IS cuve, Phillips cuve, and moneay policy ule. The las ow can be inepeed as endogenous esponse funcion of bond make. We can ewie he VAR in (42) ino companion fom, x c Φ Φ2 Φ3 Φ4 x Σ u x 0 I x 2 = (44) x I 0 0 x x I 0 x o X = c% + ΦX % + Σu % %, (45) whee X = ( g, π,, s, K, g, π,, s )' is he 6 sae veco. () () The sochasic discoun faco is defined as 9 Mos sudies in he VAR lieaue using boh sho and long aes choose he fis assumpion. Fo example, Leepe, Sims, and Zha (996) discuss his issue in deail, and conclude ha he fis assumpion is less hamful han he second one. 22

23 M = exp λ ' λ λ ' u 2 () = exp λ ' λ λg, ug, + λπ, uπ, + λ, u, + λs, us, +, (46) 2 () + + whee λ = ( λg,, λπ,, λ,, λs, )' is he make pices of isk. The veco λ is an affine funcion of he cuen economic vaiables x π : () = ( g,,, s)' λ = γ+ δx, (47) fo a 4 veco γ and a 4 4 maix δ. Equaion (46) is a genealizaion of he examples of sochasic discoun facos consideed in Secion 3. Fo example, if we esic he las wo elemens in γ and all elemens in δ o be equal o zeo, we obain he same sochasic discoun faco as he simple C-CAPM inoduced in he pevious secion. (n) 0 : By using he fundamenal asse picing equaion (0), we can obain closed foms fo ˆ ( n) ( n) ( n) = a + b ' X, n =, 2, (48) 0 We deive he closed foms fo discoun aes so ha a esicion = holds. Since we can deive () () ˆ YTM s and s fom he discoun aes, we could also esic he model-implied spead s o be equal o ˆ s. Bu since hee may be measuemen eo of s, we do no use his esicion. 23

24 whee a ( n) ( n) ( n) = A / n, b = B / n, (49) n A B = A + B '( c% Σγ %%) + B ' ΣΣ % % ' B, (50) 2 ( n+ ) ( n) ( n) ( n) ( n) ~ ~~ '( Φ Σδ ) e ( n+ ) ( ) ' = B n 3 ', (5) () () A = 0, B ' e3' =, (52) ~ γ γ = 0 and δ 0 δ % = 0 0, (53) e j is he j h column of he 6 6 ideniy maix. Fom (49), (5) and (52), we can obain ( n) ~ ~~ j b ' = e ( Φ Σδ ). (54) n n 3' j= 0 This is a quie simila fom o (7), and again he em pemium is consan only when δ = Esimaion 5.. Esimaion Mehods The VAR-ATSM has 98 paamees consising of 78 fom he VAR ( c, 24

25 Φ [ ΦΦ2Φ3Φ 4], and Σ ) and 20 in make pices of isk ( γ and δ ). We use GMM o esimae all paamees simulaneously. Momen condiions ae consuced by assuming ha hee ypes of eos ae ohogonal o insumens. The fis ype of he eos ae he eos of he VAR, ε = x ( c+ Φ x + Φ x + Φ x + Φ x ), (55) wih insumens a consan, x, x 2, x 3, and x 4 covaiance maix of he VAR,. The second ype is he eo of he ξ = vech( ΣΣ' εε '). (56) We assume ha he sample mean of ξ is exacly equal o zeo. Noe ha he momen condiions coesponding o (55) and (56) ae exacly same as OLS. The hid ype is he picing eos of discoun aes ν = [ ν (2) ν (4) ν (8) ν (2) ν (6) ν (20) ]' (57) 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 squaed picing eos of discoun aes. This esimaion mehod has an advanage of less compuaional buden ove ou one-sep esimaion. On he ohe hand, since hei esimaion mehod does no use efficien weighs on momen condiions, his is less efficien han ous. In paicula, hei esimaes fo VAR paamees can no have he efficiency gains fom he no-abiage assumpion a all. Since ou lae analyses ae based on impulse esponse funcions calculaed fom he esimaes of VAR paamees, he efficiency gains ae cucial. 25

26 whee ν ( n) = = ( n) ( n) ˆ ( n) ( a ( n) + b ( n) ' X ). (58) We use as insumens a consan, x, and x 2 fo his ype of momen. Now we have 32 momen condiions, which ae sufficien fo idenifying 98 paamees. We use he sample peiod 964:Q-200:4Q, he same as was used fo he OLS egessions in Secion 2. We esic he paamee space by wo ypes of esicions. Fis, he modulus of eigenvalues of Φ ~ ae esiced o less han uniy. Since he sae veco X follows he VAR() of (45) wih he auocoelaion coefficien maix Φ ~, his esicion guaanees he saionaiy of X. In fac, esimaion esuls show ha his esicion does no bind. Second, he modulus of eigenvalues of Φ % Σδ % % ae esiced o be less han o equal o uniy. Fom (54), he faco loading ( n) b can be consideed as he aveage of '( ) j 3 e Φ % Σδ % % ; j = 0,,, n-. So his second esicion guaanees he faco loading no o divege wih he mauiy n. Noe ha his esicion is he genealizaion of (8). In ou esimaion esuls, only one of he esicions binds 2. 2 When a esicion binds, he specal densiy maix a fequency zeo is no guaaneed o be he opimal weighing maix in GMM. Fo solving his poblem, we use he binding esicion o subsiue ou a paamee in advance. Then we use he obained non-esiced GMM o esimae paamees wih coec infeence. The esimae and sandad eo of he subsiued paamee ae obained by subsiuing ou anohe paamee and e-esimaing. 26

27 5.2. Esimaion Resuls The VAR esimaes have gea efficiency gains fom he no-abiage assumpion, alhough poin esimaes ae no so diffeen fom OLS esuls. 42 ou of 68 esimaes fo c and Φ (no epoed) ae significanly diffeen fom zeo a size of 5%, while OLS wihou he no-abiage assumpion gives only 7 significan esimaes. These efficiency gains conibue o he eliabiliy of 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 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 6 esimaes of δ ae significanly diffeen fom zeo a size of 5%. This esul suppos he ime-vaiaion of he make pices of isk, depending on economic vaiables. Among hese significan paamees, he (,) and (,2) elemens of δ, δ and δ 2 ae mos influenial on he movemen of em sucue. The eason fo his is as follows. Given he facos X, he movemen of em sucue depends only on he faco loadings ( n) b, which depend on Φ % Σδ % % fom (54). So he influence of δ on he movemen of em sucue depends on Σ % (i.e. Σ ). As we can see in Table 2, he (,) elemen of Σ, he volailiy of oupu gowh shock, is much lage han he ohes. So he fis ow of δ is mos influenial. Among he esimaes in he fis ow, only δ and δ 2 ae significanly diffeen fom zeo. In fac, as we will discuss in he nex secion, δ 2 plays a key ole in he pedicabiliies, while δ does no. 27

28 The posiive sign of δ 2 implies ha, when inflaion ae π is highe, λ g, is highe and bond holdes ae willing o pay a highe pemium fo oupu gowh isk hedge, which esuls in a lowe em pemium. Why do hey pay he highe pemium duing he highe inflaion egime? A possible explanaion can be obained in he famewok of he C-CAPM wih MIU funcion discussed in subsecion 3.4. Alhough his C-CAPM has only oupu and inflaion shocks, we can genealize his model o be consisen wih he VAR-ATSM by adding moneay policy and spead shocks ino (37) and leing he ime-vaying weighs on shocks depend on all fou VAR vaiables. Fom (40), δ2 = θω gπ. So since θ > 0, δ 2 > 0 implies ω gπ < 0. This means ha, when he inflaion π is high, he weigh on 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 ha he oupu gowh shock duing high inflaion egime ends o cause seious fuue inflaion. Accoding o his consideaion, when inflaion is high, he Fed ends o suppess he eal money gowh ae in esponses o he oupu gowh shock. This less accommodaing esponse of he Fed educes he coelaion beween oupu gowh shock u g, + and he eal money shock ε, +. This educed coelaion causes fuue maginal uiliy, µ uc (, m ) = C m, (59) ρ θ o be moe sensiive o he oupu gowh shock, ha is, bonds ae moe valuable fo consumpion hedge. Theefoe, consumes ae willing o pay moe pemium fo holding bonds duing he highe inflaion egime. We can also discuss he posiive sign of δ in a simila way. 28

29 Finally, he J-es suppos ou esimaes wih high p-value of Fo moe evaluaion of he esimaion esuls, le s compae he model-implied discoun aes ( ) ( ) ( ) ˆ n n n a = +b X and he sample aes ( n). Table 4 epos means and sandad deviaions of ( n) and ( ) ˆ n, and coelaions beween hem fo n = 2, 4, 8, 6, 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. 6. Impulse Response Funcions and he Pedicabiliies 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 pedicabiliies of em speads. Fom he VAR-ATSM, we can calculae he opimal foecass condiional on 6 sae vaiables in X. Howeve, ou main inees is no he foecass condiional on hese lage numbes of vaiables, bu on a em spead alone as he egessions (4)-(6) use. Fo ou pupose, in subsecion 6., we fis conside he elaionship beween impulse esponse funcions of vaiables in egessions (4)-(6) and he R 2 s. Since boh egessands and egessos of he egessions 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 consideaions fo he elaionship beween he impulse esponse funcions and he R 2 s will be used fo claifying he 3 The p-value is calculaed fom he J-sa (3.233) and he degee of feedom (23 = ). Noe ha since he one of he esicions on eigenvalues binds, should be subaced fom he degee of feedom. 29

30 souce of pedicabiliies in subsecion Impulse Response Funcions and Model-Implied R 2 s Since x π obeys he VAR in (42), we can calculae hei impulse () = ( g,,, s)' esponse funcions, and epesen he sysem in MA( ) fom wih idenified exogenous shocks. Fo example, g can be epesened as g = g + ψ u + ψ u + ψ u + ψ u, (60) 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 gπ, j, ψ g, j, and ψ gs, j ae funcions of Φ and Σ. So he fuue oupu gowh g + h can be epesened as h h h h (6) 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 g = g + ψ u + ψ u + ψ u + ψ u (62) 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 ) ' 30

31 funcions of X = ( x ', x ', x 2', x 3')', we can also calculae hei impulse esponse funcions, and epesen hem in MA( ) 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 (63) 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 VAR vaiables, he opimal foecass of hem, and em speads. Fom (60), (62) and (63), σ 2 g va( g ) ψgg, j ψgπ, j ψg, j ψgs, j j= 0 j= 0 j= 0 j= 0, (64) = σ va( gˆ ) 2 gh ˆ, + h ψgg, j ψgπ, j ψg, j ψgs, j j= h j= h j= h j= h, (65) = ( 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. (66) =

32 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) ( n) ( m) ( n) ( m) cov( g+ h, ) + h = ( nm, ) σσ g co( g, ) 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. (67) Since he foecasing eo of he opimal foecas g ˆ + h g+ h is unpedicable by any vaiable known a ime such as, ( n) ( m) co( g, ) = co( gˆ, ). (68) ( n) ( m) ( n) ( m) + h + h By squaing he coelaion, we can obain he R 2. Fo example, he R 2 of he egession (4) can be epesened as R = co( gˆ, ). (69) 2( nm, ) ( n) ( m) 2 gh, + h Since he R 2 s ae funcions of paamees in ou VAR-ATSM, we can calculae he R 2 s wih he esimaes of he paamees. We call hem he model-implied R 2 s. Equaion (69) implies ha if ( n) ( m) is a good pedico fo fuue oupu gowh g + h, ( n) ( m) should have simila 32

33 esponses o exogenous shocks as g ˆ+ h has. We invesigae his by looking a he vaiance decomposiion of g ˆ+ h in he nex subsecion. Finally, as we can see fom (67)-(69), he R 2 s depend on he sum of poducs of impulse esponse funcions fo egessands and egessos. Noe ha, in (67), indexes fo ψ s sa fom +h, no, because fuue shocks u,, K u ae + + h unpedicable. This implies ha since ψ s ypically decay wih he hoizon j, is a ( n) ( m) good pedico if his esponds o ecen shocks well, i.e. κ s ae lage fo smalle j Why do em speads help pedic? Figue 3 displays he model-implied R 2 s of he 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 2Q-8Q spead is bee han he 20Q-Q spead excep fo oupu gowh pedicions a shoe hoizons. Second, he 2Q-Q spead is almos useless. Finally, i is difficul o pedic oupu gowh a Q ahead. Theefoe i is easonable o y o explain he sample R 2 s in Figue 2 in ems of he facos ha deemine he model-implied R 2 s in Figue 3. Since he model-implied R 2 s ae funcions of paamees in ou VAR-ATSM, we can analyze how hese paamees affec he R 2 s. Figue 4 shows impulse esponse funcions of VAR vaiables g, π, (), and s 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 33

34 esimaed moneay policy shock shaply educes oupu gowh. This shock also suppesses inflaion aes in he long un. These easonable esuls imply easonable esimaes of he moneay policy shock. 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 he 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 he foecass such as (65) ae nomalized o uniy. As discussed in he pevious subsecion, his indicaes which exogenous shocks should be useful fo pedicion. (5-a) shows ha he oupu gowh shock dominaes oupu gowh pedicabiliy a one quae ahead. Then aound 2-4 quae ahead, he moneay policy shock is he mos impoan. The impoance of he inflaion shock inceases wih he foecasing hoizon, and a las his shock is mos influenial a 2 quaes ahead. These esuls ae consisen wih he impulse esponse funcions in Figue 4. The oupu gowh shock causes a shap jump of oupu gowh only in he sho un. The moneay policy shock has negaive effecs on oupu gowh wih 2-4 quae lags. Bu in he long un, he inflaion shock aises he sho ae pesisenly, which coninues 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, how he em speads espond o he inflaion shock is mos impoan fo specifying he souces of he pedicabiliies especially a longe hoizons. Noe ha, as Figue 4 implies, he effecs of exogenous shocks decay wih he hoizon. So we can also say ha good pedicos should espond o ecen shocks ahe han old shocks. 34

35 Figue 6 shows impulse esponse funcions of seleced discoun aes. Thee ae hee noable feaues. Fis, he inflaion shock has vey pesisen effecs on levels of discoun aes. Tha is, he discoun aes do no eun o zeo even afe 40 quaes. Since good pedicos should espond o ecen 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 shif he yield cuve almos in paallel. In fac, he uppe gaphs of Figue 7 display ha boh he 20Q-Q and 2Q-8Q speads depend much on ecen shocks. This is a eason why he em speads have pedicive powes. Why do he discoun aes espond in such ways? We find ha he ime-vaying make pice of isk plays impoan oles as follows. As discussed in Secion 5, he paamees coesponding o he effecs of he oupu gowh and inflaion aes on he make pice of oupu gowh isk δ and δ2 ae mos influenial on he movemen of long aes. In fac, only δ 2 plays a suppoive ole in he pedicabiliy. As shown in Figue 5, he inflaion shock is mos impoan fo he pedicabiliy, and he posiive δ 2 causes he make pice of oupu gowh isk o espond posiively o he shock well. On he ohe hand, he posiive δ makes he pedicabiliy even wose. As shown in (4-b), he posiive inflaion shock causes a decease in he oupu gowh ae, which has negaive effecs on he make pice of oupu gowh isk. Since he effec hough δ2 dominaes he effec hough δ, he make pice of oupu gowh isk esponds posiively and so em pemium esponses negaively o he inflaion shock. Fo evaluaing he influence of δ 2, we calculaed he impulse esponse funcions of 35

36 discoun aes when δ 2 = 0 and he 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), he esponses of longe aes ae smalle han he sho ae, and he diffeence beween he long and sho aes almos disappea aound 20 quaes ahead. On he ohe hand, in (8-b), he esponses of longe aes ae songe han he sho ae, and he diffeence does no disappea even aound 40 quae ahead. Why ae hey so diffeen? The expecaions hypohesis says ha he long ae is he aveage of expeced sho aes plus a consan em pemium. Fom (4-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 displayed in (8-b). Bu since in fac δ 2 is posiive, 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 of esponses in (6-b) and (8-b) has lage effecs on he pedicabiliies. Figue 9 shows model-implied R 2 s coesponding o he case of δ 2 = 0. Supisingly, he R 2 s almos disappea. So now we can conclude ha he posiive δ 2, which can be inepeed ha consumes ae willing o pay a highe pemium fo oupu gowh isk hedge duing he highe inflaion egime, is a key explanaion fo he pedicabiliies. The las noable feaue of Figue 6 is he lagged esponses of Q ae (he moneay policy auhoiy) o oupu gowh and inflaion shocks. Panel (6-a) shows ha he immediae esponse of Q ae o oupu gowh shock is smalles among discoun aes, alhough he esponse of Q ae is lages a seveal quaes ahead. Panel (6-b) shows ha he immediae esponse of Q ae o inflaion shock is smalle han 2Q ae, and almos coincides wih 8Q ae. 36

37 These esuls ae consisen wih he moneay policy auhoiy s ineial behavio empiically shown in he lieaue such as Claida, Gali, and Gele (2000). The lowe gaphs in Figue 7 show he impulse esponse funcions of 20Q-Q and 2Q-8Q speads o oupu gowh and inflaion shocks. The nea esponses of 20Q-Q spead ae much weake han 2Q-8Q spead because of he slow esponses of Q ae. Since ecen shocks ae vey impoan fo pedicions, we can conclude ha his is he eason ha 20Q-Q spead is wose han 2Q-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 he oupu gowh and inflaion shocks. 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 fo analyzing why we found he R 2 s shown in Figue 3 o 2. Equaion (67) has fou summed ems, and each of hem can be inepeed as he conibuion of a given exogenous shock o he pedicabiliy. Figue 0 shows he conibuions of exogenous shocks o he absolue values of coelaions fo 20Q-Q and 2Q-8Q speads. The inflaion and oupu gowh shocks conibue o he coelaions wih 2Q-8Q spead ahe han 20Q-Q spead. These diffeences ae he eason fo he usefulness of he 2Q-8Q spead fo pedicion. This esul is consisen wih ou discussion abou he esuls in lowe gaphs in Figue 7. Anohe noable popey in Figue 0 is he hump-shapes of he conibuions of moneay policy shocks o oupu gowh pedicabiliy. So we can conclude ha he hump-shape of R 2 s fo 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 37

38 oupu gowh a sho hoizons. Finally, Figue shows he conibuions in he case of δ 2 = 0. Obviously he shap dops of R 2 s ae aibuable o he diffeen sign of conibuion of he inflaion shock, which ae caused by song long ae esponses o he shock. 7. Conclusion Why do em speads pedic oupu gowh, inflaion, and sho aes? Fo answeing his quesion, we used he VAR-ATSM model wih fou lags and fou vaiables, which is less esiced han hose in he lieaue of affine em sucue models wih obsevable facos. And we succeeded in esimaing his model by using an efficien mehod. We have hee main findings. Fis, he ime-vaying make pice of oupu gowh isk, which is sensiive o he level of inflaion, plays a key ole in explaining why he em spead helps 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 oupu gowh isk hedge, possibly because he maginal uiliy is moe sensiive o he oupu gowh shock due o less accommodaing esponse of he Fed. This causes em speads o eac o ecen inflaion shocks, which also pove 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 shock affecs oupu gowh wih a lag while he em sucue esponses o he shock immediaely. 38

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