Learning, Equilibrium Trend, Cycle, and Spread in Bond Yields

Transcription

1 Learning, Equilibrium Trend, Cycle, and Spread in Bond Yields Guihai Zhao Bank of Canada This version: March 2019 Absrac While he empirical lieraure has shown he imporance of macro rends in modeling he erm srucure of ineres raes, he sandard saionariy assumpion makes i hard for equilibrium models o capure he rend in bond yields. This paper presens an equilibrium model o explain he rend, cycle, and spread in hisorical U.S. Treasury bond yields. The rend in yields is generaed by learning from he sable componens in GDP growh and inflaion, which share similar paerns o he neural rae of ineres (r ) and rend inflaion (π ) esimaes in he lieraure. Cyclical movemens in yields and spread, are mainly driven by learning from he ransiory componens in GDP growh and inflaion. The upward rend in he Treasury yield spread found in he daa and he recen secular sagnaion are ighly coupled due o persisenly negaive shor-run beliefs. The upward-sloping yield curve is mainly driven by he fac ha he amoun of Knighian uncerainy ha invesors face is differen for he long run versus he shor run. Keywords: adapive learning, rend inflaion, equilibrium real ineres rae, bond yields, erm spread, ambiguiy JEL Classificaion: G00, G12, E43 1. Inroducion I is imporan for boh policy makers and academics o undersand he hisorical dynamics of he Treasury yield curve, because of is key role in he ransmission of moneary policy and is igh link wih he sochasic discoun facor. While he empirical Guihai Zhao (gzhao@bankofcanada.ca) is wih he Bank of Canada, 234 Wellingon Sree, Oawa, ON K1A 0G9. I would like o hank Jason Allen, Jonahan Wimer as well as seminar paricipans a he Bank of Canada for heir suggesions. Any remaining errors are my own. The views expressed in his paper are hose of he auhors and do no necessarily reflec hose of he Bank of Canada.

2 lieraure has shown he imporance of accouning for macro rends in models of he erm srucure of ineres raes, 1 bond yields in equilibrium models are generally modeled as saionary, mean-revering processes. Hence, i is hard o explain he low-frequency variaion in ineres raes in such models, and as a resul, cyclical movemens in shorerm yields and in he spread beween long- and shor-erm yields are hard o jusify. 2 In his paper, we provide a join equilibrium for he rend, cycle, and erm spread in hisorical Treasury bond yields. I has long been recognized ha nominal ineres raes conain a slow-moving rend componen (Nelson and Plosser, 1982; Rose, 1988). Recen empirical sudies propose macro rends as he driving forces behind his low-frequency variaion. For example, Kozicki and Tinsley (2001) and Cieslak and Povala (2015) documen he empirical imporance of rend inflaion (π ) for explaining he secular decline in Treasury yields since he early 1980s. Bauer and Rudebusch (2017) show ha i is crucial o include he neural rae of ineres (r ) as well, which has driven he downward rend in long erm yields over he las 20 years since inflaion expecaions have sabilized. Boh π and r (and hence shor raes) are modeled as random walk processes in Bauer and Rudebusch (2017), and heir resuling erm premium componen he difference beween a longerm ineres rae and he model-implied expecaions of average fuure shor-erm raes is relaive small and saionary. In mos equilibrium models, however, shor raes are saionary. Therefore, he large residual erm in yields conaining boh he low-frequency variaion and cyclical movemens, is ypically explained as cerain ypes of risk premium (for example, he inflaion risk premium in Piazzesi and Schneider (2007)). To illusrae he role of learning in generaing he observed low-frequency variaion in yields and in maching he macro rends, we sar wih a simple equilibrium model (model- I) where boh oupu growh and inflaion are exogenous and follow i.i.d. laws of moion. 1 See, for example, Kozicki and Tinsley (2001); Cieslak and Povala (2015); Bauer and Rudebusch (2017). 2 Mos equilibrium erm srucure models are designed o inerpre and quanify he level, volailiy, and average posiive spread in yields. However, given he saionary shor raes, hese models ypically need some exreme assumpions o mach he large volailiy in yields. 2

3 The represenaive agen wih consan relaive risk aversion (CRRA) uiliy does no, however, know he mean inflaion and mean growh rae. The poseriors from sandard Bayesian learning will be random walk processes, which are poenially consisen wih he empirical modeling of macro rends. However, he poserior variance will decline deerminisically o zero and learning will converge (Collin-Dufresne e al., 2016). We assume ha he agen uses a consan-gain learning scheme as proposed by Nagel and Xu (2018) based on a weighed-likelihood approach. This is a modified Bayesian approach where learning is perpeual due o he agen s fading memory. The poseriors for mean inflaion and mean growh rae, as sae variables, capure he rends in inflaion and growh. The shor raes and yields for long-erm bonds implied by he model, as linear funcions of he poseriors, exhibi a low frequency movemen. The model-i implied r, which is a linear funcion of he poserior for mean oupu growh, moves closely wih r esimaes in he lieraure. 3 No only capuring he low-frequency variaion, he model-i implied r also exhibis a moderae business cycle componen, wih dips during recessions, and some degree of recovery aferwards. Figure 3 shows ha he poserior for mean inflaion maches very well he survey-based rend inflaion in Bauer and Rudebusch (2017). As a resul, he hump-shaped rend in he 10-year Treasury yield (from he lae 1960s o he lae 1990s) reflecs an increase in inflaion expecaions before he mid-1980s and a secular decline aferwards. Over he pas wo decades when inflaion expecaions have sabilized, he decline in he poserior for mean oupu growh, and hence he 10-year real yield, has been he main driver of he downrend in nominal yields. 4 However, he poseriors impac yields of all mauriies equally, and yields for shor- and long-erm bonds are almos idenical in model-i. There- 3 We follow Bauer and Rudebusch (2017) and use he r esimaes from Laubach and Williams (2003), Lubik and Mahes (2015), and Kiley (2015). 4 In his model, oupu growh is given exogenously and r is driven by learning abou he mean growh. Alernaive inerpreaions include lower produciviy growh, changing demographics, decline in he price of capial goods, and srong precauionary saving flows from emerging marke economies. See, for example, Summers (2014); Kiley (2015); Rachel and Smih (2015); Carvalho e al. (2016); Hamilon e al. (2016); Laubach and Williams (2016); Johannsen and Merens (2018); Chrisensen and Rudebusch (2019); Holson e al. (2017); Lunsford and Wes (2017); Del Negro e al. (2017). 3

4 fore, model-i canno explain he cyclical movemens in he shor-erm yield and in he spread beween long- and shor-erm yields. To address hese cyclical movemens as well, we consider an exended version (model- II), where boh GDP growh and inflaion raes are decomposed ino wo componens: one sable componen and one ransiory/volaile componen. Using he same learning scheme, he represenaive agen learns abou he mean oupu growh and inflaion raes from he sable componen, and learns abou he saionary deviaions from mean from he ransiory/volaile componen. As in model-i, he poseriors for he long-run mean inflaion and growh impac shor- and long-erm bonds equally, and hence capure he rend in yields. However, as AR(1) processes, he poseriors for he ransiory deviaions from he long-run mean have larger impacs on he shor-erm yield han on he longerm yield, which implies ha he spread would be posiive (negaive) when he shor-run beliefs are negaive (posiive). Therefore, model-ii can generae cyclical movemens in he shor-erm yield, and hence in he spread, mosly due o variaions in hese shor-run beliefs. The model-ii implied 1-year nominal yield and he spread beween 10- and 1-year nominal yields mach well hisorical movemens in 1-year Treasury yield and he spread beween 10- and 1-year Treasury yields. 5 Ou of recessions, he shor-erm nominal yield sars o rise when he agen begins o revise her beliefs for shor-run growh and inflaion upwards owards heir long-run means (shor-run deviaions are sill negaive and he spread is posiive), and he spread sars o shrink as hese shor-run deviaions are urning ino posiive from negaive. This paern coninues unil he lae expansion sage when he shor-run growh rae and inflaion expecaions are above heir long-run means (shor-run deviaions are posiive now), which implies an invered yield curve. 6 The agen 5 Noe ha he whole nominal yield curve is disored by zero lower bound, quaniaive easing, and oher unconvenional moneary policies in he pos-global financial crisis period. The model-implied yield curve for hese periods is very differen from daa, which can be used as "shadow raes", and hey sared o line up again afer The model is consisen wih our convenional undersanding of business cycle. From rough o peak when inflaion gap and oupu gap (shor-run deviaions in his model) are moving from negaive o posiive, moneary policy urns ino conracionary from accommodaive (shor-erm yields increase) 4

5 hen begins o revise her shor-run beliefs sharply downwards enering a recession (from posiive o negaive), and he spread swiches from negaive o posiive. Furhermore, he model-ii-implied inflaion expecaions (poseriors) closely mach he survey-based shor- and long-run inflaion expecaions. And as in model-i, he model-ii-implied r racks very closely he r esimaes in he lieraure; hence he model- II-implied 10-year nominal yield moves closely wih he daa. The long-erm yield is less sensiive o he shor-run belief movemens, and is business cycle variaions are mainly driven by he beliefs updaing in long-run mean growh and inflaion raes (as in model-i). In addiion, Figure A1 shows ha (1) he poseriors for he shor-run inflaion and growh deviaions move in he opposie direcions before 2000, and hey move in he same direcion aferwards, and (2) he poseriors for boh shor-run inflaion and growh deviaions are persisenly negaive for mos of he pos 2000 periods. These observaions are consisen wih he secular sagnaion saemen in Summers (2014), while he secular decline in r and 10-year Treasury yield in his model are mainly driven by he decline in he poseriors for he long-run means, and he decline in shor- o mid-erm Treasury yields are furhermore due o he persisenly negaive shor-run deviaions. Finally, Figure 10 shows ha boh he nominal spreads in daa and in model-ii have rended upward over he sample period, and we observe less invered curves afer he 1990s. The reason is ha he shor-run inflaion and growh deviaions are srongly posively correlaed (hence move he shor-erm yield and spread in he same direcion) and hey are persisenly negaive (hence posiive spread) for he pas wo decades. However, heir impacs on he nominal spread cancel each oher ou before 2000 when hey move in he opposie direcions. Despie he fac ha he model-ii-implied nominal spread can say posiive or negaive for an exended period of ime a differen phases of he business cycle, he level is almos in parallel lower han daa, due o he saionary assumpion (mean zero) for he shor-run beliefs, as well as he CRRA uiliy. The common equilibrium explanaion for he upward-sloping nominal yield curve is according o Taylor Rule. The difference is ha he poseriors for he long-run mean are unknow and keep moving in his model, while hey are consans in he sandard DSGE models. 5

6 he inflaion risk premium approach by Piazzesi and Schneider (2007), where inflaion is bad news for fuure growh and he agen prefers early resoluion of uncerainy (Epsein and Zin (1989) preference). Zhao (2018) shows ha his approach has been less effecive during he pas wo decades when inflaion has swiched from bad news o good news for fuure growh, providing an alernaive wors-case belief approach hrough ambiguiy. We herefore exend model-ii o model-iii by incorporaing he inuiion in Zhao (2018). The ambiguiy-averse represenaive agen (wih recursive muliple priors, or maxmin, preferences by Epsein and Schneider 2003) has in mind a benchmark or reference measure of he economy s dynamics ha represens he bes esimae of he sochasic process. In model-iii, he reference measure is he full sochasic environmen in model-ii (including he poseriors). Bu he agen is concerned ha he reference measure is misspecified and believes ha he rue measure is acually wihin a se of alernaive measures ha are saisically close o he reference disribuion. Using forecas dispersion o quanify he size of ambiguiy (following Ilu and Schneider 2014), Zhao (2018) finds ha shor-rae expecaions are upward-sloping under invesors wors-case equilibrium beliefs, which generaes upward-sloping nominal and real yield curves even wih a CRRA uiliy. Consisen wih findings in Froo (1989) and Piazzesi e al. (2015), he expecaions hypohesis roughly holds under he subjecive equilibrium belief. Given ha he benchmark measure in model-iii is he same as in model-ii, model-iii can sill mach he rend and cycle in yields as model-ii does. However, by comparing he model-implied 1-year nominal yield wih daa, we observe a recurring paern ha he model-implied shor-erm yields are higher han daa from rough o expansion sage, and hey are lower han daa during he lae expansion periods. Given ha shor-erm yields are conrolled by he Federal Reserve, his suggess ha he Federal Reserve kep he shor-raes low for a longer period han suggesed by he model and here was cerain degree of overshoo during he lae expansion periods (before recessions). In addiion, because he agen s subjecive inflaion and growh expecaions are upward-sloping under her wors-case belief, he model can also generae upward-sloping real and nominal curves 6

7 on average. 7 Figure 14 shows ha he model-iii-implied spread maches he daa well. Specifically, compared wih he model-ii-implied nominal spread in Figure 10, he model- III-implied nominal spread is almos in parallel higher. Relaed lieraure The paper is relaed o a large lieraure in equilibrium asse/bond pricing models. 8 While mos equilibrium bond-pricing models focus only on he firs/second momen and he average spread in yields, we go furher in his paper by providing a join equilibrium undersanding of he rend, cycle, and spread in he hisorical Treasury bond yields. The mos closely relaed paper is Piazzesi and Schneider (2007) who show he imporance of he inflaion risk premium for he upward-sloping nominal curve in a saionary sae space model (inflaion and growh). They also show how recursive esimaion (wih more weigh on recen observaions) of he whole sae space model can help improve he model s hisorical performance. This paper differs from hese previous sudies along some imporan dimensions. Firs, he agen in his paper akes ino consideraion he risk of belief updaing as in Collin- Dufresne e al. (2016) and Nagel and Xu (2018). Hence he poseriors are sae variables, which move closely wih macro rend esimaions (r and π ) in he lieraure. Mos imporanly, his is he firs paper ha decomposes boh GDP growh and inflaion 7 The empirical lieraure has found ha professional forecasers, and even cenral banks, make sysemaic forecas errors by comparing he mean forecass and he subsequen realized values (Faus and Wrigh, 2009; Champagne e al., 2018). In his model, he reference measure represens he bes poin esimae from daa and agens would use i for forecasing. However, agens are more cauious when hey make decisions, and hey insead use he wors-case belief for decision making. Therefore, ambiguiy in his model provides a raional explanaion for he expecaion errors. 8 For example, Bansal and Shaliasovich (2013) show ha he long-run risks model of Bansal and Yaron (2004) wih ime-varying volailiy can accoun for bond reurn predicabiliy. Rudebusch and Swanson (2012) show ha a macroeconomic DSGE model wih inflaion non-neuraliy and Epsein and Zin (1989) preference can also generae sizable inflaion risk premium. Wacher (2006) generaes an upward-sloping nominal using an exernal habis model, where innovaions o consumpion and inflaion growh are negaively correlaed. Leau and Wacher (2011) model he erm srucures of equiy and ineres raes by specifying a parsimonious sochasic discoun facor. Albuquerque e al. (2016) show ha he risk o ime preference implies a posiive real erm premium and can generae an upwardsloping real yield curve. Berrada e al. (2018) show he risk o belief updaing implies a posiive real erm premium when agens have beliefs-dependen risk aversion. 7

8 ino wo componens; And shows ha learning abou he long-run mean drives he lowfrequency variaion in yields and learning abou he shor-run deviaion from he mean drives he business cycle movemens in he shor-erm yield, and hence in he spread. Insead of real risk premium (Wacher, 2006; Albuquerque e al., 2016; Berrada e al., 2018) and inflaion risk premium (Piazzesi and Schneider, 2007), he upward-sloping nominal and real curves in he U.S., a leas for he pos 2000 periods, are parially due o he fac ha boh shor-run deviaions for inflaion and growh are negaive for mos of hese periods. However, he shor-run deviaions for inflaion and growh move in he opposie direcions before 2000, which makes i hard for model-ii o generae an average upward-sloping nominal curve. We herefore rely on one addiional wors-case belief approach (Zhao, 2018) o generae upward-sloping nominal and real curves ha are consisen wih he daa. The paper is also relaed o a large empirical lieraure ha links macro informaion and macro rends wih yield curve modeling. 9 The mos closely relaed paper is Bauer and Rudebusch (2017) who model r and π as random walk processes and show he imporance of hese rends in explaining he low-frequency variaion in he long-erm yield. This paper bridges an imporan gap beween he empirical and equilibrium yield curve lieraure by inerpreing macro rends as poseriors for learning abou long-run mean growh and inflaion raes. Furhermore, he paper also provides an equilibrium inerpreaion for he cyclical movemens in he shor-erm yield, and hence in he spread. This paper is relaed o a number of papers ha have sudied he implicaions of ambiguiy and robusness for finance and macroeconomics. 10 Model-III in his paper incorporaes he inuiion in Zhao (2018) o generae he upward-sloping nominal and real curves hrough he upward-sloping shor rae expecaions under he represenaive agen s wors-case belief. Finally, his paper is relaed o some recen developmens ha 9 See Ang and Piazzesi (2003), Diebold e al. (2006), Wrigh (2011), Kozicki and Tinsley (2001), Cieslak and Povala (2015), and Bauer and Rudebusch (2017), among many ohers. 10 See Epsein and Schneider (2003), Epsein and Schneider (2007), Ilu and Schneider (2014), Ulrich (2013), Drechsler (2013), Gagliardini e al. (2009), Zhao (2017), and Zhao (2018), among many ohers. For a deailed survey, see Epsein and Schneider (2010). 8

9 sudy he implicaions of learning in finance. For example, Collin-Dufresne e al. (2016) show how a sandard Bayesian learning can generae subjecive long-run risks when agen prefers early resoluion of uncerainy (Epsein and Zin (1989) preference). Building on he insigh in Nagel and Malmendier (2016) ha he dynamics of he average individuals expecaion can be approximaed closely by a consan-gain learning scheme, Nagel and Xu (2018) show how his consan-gain learning can help separae subjecive and objecive equiy premium and explain he predicabiliy of excess reurns. In his paper, we use he consan-gain learning scheme in a differen seing o explain bond yields dynamics. The paper coninues as follows. Secion 2 oulines and solves model-i in closed form and discusses he model implicaions. Secions 3 and 4 underake he same seps for model-ii and model-iii, respecively. Secion 5 provides concluding commens. 2. Model I - Learning and rend in bond yields In his secion, we consider an endowmen economy wih a represenaive agen who has a CRRA uiliy funcion. She learns wih fading memory abou he mean oupu growh and inflaion raes. Equilibrium prices adjus such ha he agen is happy o consume he oupu as an endowmen Learning wih fading memory Boh oupu growh and inflaion follow i.i.d. laws of moion g +1 = µ c + σ c ε c,+1 π +1 = µ π + σ π ε π,+1, (1) where g +1 is he growh rae of real oupu and π is inflaion. ε c,+1 and ε π,+1 are i.i.d. normal shocks. The represenaive invesor knows ha boh g +1 and π +1 are i.i.d., and also knows σ c and σ π, bu no µ c and µ π. The agen forms expecaions abou µ c and µ π based on he hisory of oupu growh and inflaion realizaions, H g { g 0, g 1,... g } and H π {π 0, π 1,...π }. 9

10 A each ime, a Bayesian agen would updae her prior belief p(µ c ) or p(µ π ) in a way ha assigns each pas observaion g j or π j equal weigh in he poserior probabiliy. The equal-weighing of pas observaions in H g and H π means ha here is no decay of memory as he agen uses all available daa in forming he poserior beliefs. In his paper, we use a consan-gain learning scheme proposed by Nagel and Xu (2018) based on a weighed-likelihood approach used in he heoreical biology lieraure (Mangel, 1990). Compared wih sandard consan-gain learning models, he learning here allows us o derive he full poserior disribuion. Taking oupu growh as an example, wih fading memory, he represenaive agen who has observed an infinie hisory of pas oupu growh g forms her poserior [ p(µ c H g ) p(µ c ) exp ( ( g j µ c ) 2 )] (1 υc) j, (2) j=0 2σ 2 c where 1 υ c is a posiive number close o one, and (1 υ c ) j represens a geomeric weigh on each observaion. The agen weighs recen observaions more han observaions receding ino he pas. We work wih uninformaive priors in he model, µ c N (µ c,0, σ c,0 ) and µ π N (µ π,0, σ π,0 ), wih σ c,0 and σ π,0. 11 The poseriors are given by µ c H g N ( µ ) c,, υ c σc 2 µ π H π N ( µ π,, υ π σπ) 2, (3) where µ c, = µ c, 1 + υ c ( g µ c, 1 ) = υ c (1 υ c ) j g j j=0 µ π, = µ π, 1 + υ π (π µ π, 1 ) = υ π (1 υ π ) j π j. (4) j=0 Unlike sandard Bayesian learning, where he variance of he poserior converges o zero, 11 See Nagel and Xu (2018) for a discussion of he informaive prior case. 10

11 learning is perpeual here. The variance of he poserior is he same as if he agen had observed, and reained fully in memory wih equal weigh, S g 1 υ c (S π 1 υ π ) realized growh rae (inflaion) observaions. Alhough he acual number of observaions is infinie, he loss of memory induced by he geomeric weighed-likelihood implies ha he effecive sample size is equal o a finie number S g (S π ). The poserior µ c, ( µ π, ) resuling from his weighed-likelihood approach is idenical o he poserior ha one obains from a sandard consan-gain updaing scheme wih gain υ c (υ π ). To undersand beer he sochasic naure of he oupu growh and inflaion process from he agen s subjecive viewpoin, we furher ge he predicive disribuion g +j H g N ( µ c,, (1 + υ c ) σ 2 c ) π +j H π N ( µ π,, (1 + υ π ) σ 2 π), (5) where j = 1, 2,... and he variance of he predicive disribuion conains boh he uncerainy due o fuure shocks ε c,+j (ε π,+j ), and he uncerainy abou µ c (µ π ). Denoing expecaions under he predicive disribuion wih Ẽ, we can rewrie he poseriors as µ c,+1 = µ c, + υ c 1 + υc σ c ε c,+1 µ π,+1 = µ π, + υ π 1 + υπ σ π ε π,+1, (6) where ε c,+1 = g +1 µ c, σ c 1+υc and ε π,+1 = π +1 µ π, σ π 1+υπ. ε c,+1 / ε π,+1 is N (0, 1) disribued and hence unpredicable under he ime- predicive disribuion Valuaion wih fading memory The informaion srucure for a sandard Bayesian agen can be represened by a filraion, and poserior beliefs follow a maringale under his filraion. Wih loss of memory, however, he poserior in periods + j will be updaed based on informaion ha is differen, bu no more informaive, han he informaion available o he agen a ime. Thus, he informaion srucure is no a filraion. Nagel and Xu (2018) show ha a ime, he agen knows ha he variaion in µ c,+j / µ π,+j will be saionary 11

12 and perceives fuure incremens ε c,+j / ε π,+j, j = 1, 2,... as negaively serially correlaed. However, he agen canno make use of his serial correlaion by using ε c, / ε π, o forecas ε c,+1 / ε π,+1, because ε c, / ε π, is no observable. To value he zero-coupon bond under his informaion srucure, we use M +j o denoe he one-period sochasic discoun facor (SDF) from +j 1 o +j ha applies given he agen s predicive disribuion a. The ime- price of a zero-coupon bond ha pays one uni of consumpion 2 periods from now is denoed P (2), and i saisfies he recursion P (2) = Ẽ[M +1 P +1] (1) [ ( )] = Ẽ M+1 Ẽ +1 M+2 +1, (7) and he valuaion a is based on he anicipaion ha he value of he asse a dae +1 will be deermined by an agen - or a fuure self of he agen - who perceives ε c,+1 / ε π,+1 as unpredicable Kalman filer alernaive The updaing scheme in (4) is similar in spiri o he opimal filering wih a laen sochasic rend. For oupu growh, if he agen perceives µ c o follow a random walk (µ c, = µ c, 1 + ε µ c,), raher han a consan as in (1), he resuling poserior disribuion from he seady-sae Kalman filer is he same as he adapive learning in (3). Wih an appropriae choice of he volailiy of he ε µ c, shocks, he dynamics of he poserior beliefs from he Kalman filer would be he same as hose in he updaing scheme wih fading memory in (4), however, wih he informaion srucure as a filraion. The one-sepahead predicive disribuion and he valuaion for asses would be he same as well. If he rue law of moion is (1) wih consan µ c, here will be a ime-varying wedge µ c, µ c, beween subjecive and objecive beliefs, which plays an imporan role in generaing excess bond reurn predicions. Given ha here is no srong empirical evidence of GDP growh predicions, especially for long-run growh, we sick o he fading-memory inerpreaion in his model. The same argumen applies o he inflaion process. However, U.S. inflaion is highly 12

13 persisen before he lae 1990s and becomes less predicable hereafer. Alhough he model implied bond prices and yields are he same, i is more reasonable o assume a laen random walk rend wih Kalman filer learning before he lae 1990s and a consan mean wih adapive learning afer he lae 1990s Model soluions Piazzesi and Schneider (2007) show he imporance of Epsein and Zin (1989) preferences in generaing a sizable inflaion risk premium for long-mauriy nominal bonds. To illusrae he key role of rend inflaion and rend oupu growh for long mauriy bond yields, we assume invesors have recursive preferences wih a CRRA uiliy funcion (i.e., hey are indifferen beween early or lae resoluion of uncerainy): V (C ) = Ẽ (U (C ) + βv +1 (C +1 )), (8) 1 1 γ where U (C ) = C1 γ, γ is he coefficien of risk aversion, and β reflecs he invesor s ime preference. Noe ha he agen evaluaes he coninuaion value under her subjecive expecaions Bond pricing Since he represenaive agen forms expecaions under her subjecive beliefs when making porfolio choices, he Euler equaion holds under he subjecive expecaions. Given he CRRA uiliy funcion, he log nominal pricing kernel or he nominal sochasic discoun facor can be wrien as m $ +1 = logβ γ g +1 π c,+1 = logβ v z +1, (9) where v = (γ, 1) and z = ( g, π ) T. The ime- price of a zero-coupon bond ha pays one uni of consumpion n periods from now is denoed P (n), and i saisfies he recursion P (n) = Ẽ[M $ +1 P (n 1) +1 ] (10) 13

14 wih he iniial condiion ha P (0) predicive disribuion. = 1 and Ẽ is he expecaion operaor under he Given he linear Gaussian framework, we assume ha p (n) = log(p (n) ) is a linear funcion of he poseriors µ = ( µ c,, µ π, ) T p (n) = A (n) C (n) µ. (11) When we subsiue p (n) and p (n 1) +1 in he Euler equaion (10), he coefficiens in he pricing equaion can be solved wih C (n) = C (n 1) + v = v n, and A (n) = A (n 1) + A (1) ( ) ( (n 1) (n 1) 0.5 V ar p +1 Cov p +1, m,+1) $ (see he appendix for deails). The log holding period reurn from buying an n period bond a ime and selling i as an n 1 period bond a ime + 1 is defined as r n,+1 = p (n 1) +1 p (n), and he subjecive excess reurn is er n,+1 = Cov ( rn,+1, m $,+1) = v Cov (z +1, µ +1 ) C (n 1). As we can see from he soluion, he yield parameer for poserior µ is consan over horizon n, and all he variance and covariance erms are relaively small in he daa. Hence, given he CRRA uiliy (i.e., no exra erm premium from agen s ime preference, in conras o he Epsein and Zin (1989) case), he erm premium is small in his model, which implies a fla yield curve. To solve he price and yields for real bonds, we can jus replace v wih v = (γ, 0) Empirical findings Using U.S. real GDP growh and he rae of inflaion from he GDP deflaor, we can calculae he poserior beliefs for oupu growh and inflaion. We hen show ha hey closely mach he esimaed r and π in he lieraure. Given he analyical soluions and he poseriors, we can calculae he model-implied 10-year nominal and real bond yields, which mach hisorical movemens in he daa well Daa Real oupu growh and GDP deflaor inflaion are aken from he Bureau of Economic Analysis from 1947.Q2 o 2018.Q2. The end-of-quarer yields for one- o en-year bonds 14

15 γ β υ c υ π σ c σ π corr Table 1: Configuraion of model-i parameers Table 1 repors parameer values for oupu growh and inflaion processes, and for he consan gain in learning. parameers are given in quarerly erms. Mean and sandard deviaion are in percenages. All are from he daily daase consruced by Gürkaynak e al. (2007) from 1961.Q2 o 2018.Q2. The TIPS yields (2003.Q1 o 2018.Q2) and end-of-quarer yields for hreemonh and six-monh Treasury bills are from he U.S. Deparmen of he Treasury via he Fed daabase a he S. Louis Federal Reserve (1969.Q4 o 2018.Q2). The forecass for real oupu growh and inflaion are from he Philadelphia Fed s survey of professional forecasers (SPF) from 1968.Q3 o 2018.Q2. The r and π are from Bauer and Rudebusch (2017) from 1971.Q4 o 2017.Q Parameers The volailiy parameers for oupu growh and inflaion are calibraed o mach heir counerpars in daa. The correlaion beween oupu growh and inflaion in he model is calibraed o mach he correlaion in daa. The consan-gain parameers υ c and υ π are calibraed o mach variaions in r and π. We follow he lieraure and se risk aversion as 3, and ime preference β is calibraed o mach he level of en-year nominal yields in he daa, which are close o he value in Piazzesi and Schneider (2007). 12 The resuling parameer values are repored in Table Poseriors versus r and π The empirical lieraure has shown he imporance of accouning for macro rends in he erm srucure of ineres rae modeling. For example, Kozicki and Tinsley (2001) and Cieslak and Povala (2015) documen he empirical imporance of he inflaion rend for he dynamics of he nominal yields. Bauer and Rudebusch (2017) show ha i is crucial o include r as well, especially for explaining he downrend in yields over he 12 A higher ime preference β helps o lower bond yield levels. Zhao (2018) shows ha ambiguiy abou oupu growh can also lower nominal yields. Therefore, we can also se β o be smaller han 1, bu wih a larger amoun of ambiguiy o mach he bond yield level. 15

16 las 20 years as inflaion expecaions have sabilized, and hey model rend inflaion and r as random walk processes. The saionary assumpion in leading equilibrium bond pricing models makes i hard for hem o generae he hisorical observed low-frequency variaion in ineres raes. The poserior mean of oupu growh (inflaion) in Secion 2.1 would be a random walk process under sandard Bayesian learning; however, he poserior variance would decline deerminisically o zero and learning would converge (Collin-Dufresne e al., 2016). In his paper, he represenaive agen updaes her subjecive beliefs wih consan gain, which induces memory loss, and is oherwise sandard Bayesian. However, learning is slow bu perpeual in his model, which generaes low-frequency variaion in poserior beliefs ha closely mach he r and π esimaed in he lieraure and, convenienly, a saionary economy. 13 Figure 1 shows he model-implied 10-year real yield, which is a linear funcion of he poserior mean of oupu growh γ µ c, + Cov, closely racks he esimaed mean r. Given he CRRA uiliy, he erm premium par Cov is very small in he 10-year real yield, and variaions in he model-implied 10-year real yield are mainly driven by variaions in µ c,. The mean r is from Bauer and Rudebusch (2017) and is an average of hree macroeconomic esimaes of r from Laubach and Williams (2003), Lubik and Mahes (2015), and Kiley (2015). Figure 2 shows ha he model-implied 10-year real yield and he hree individual esimaes of r also co-move closely for mos of he sample periods. The hree differen r esimaions diverge from each oher before he 1980s. No only capuring he low-frequency variaion, he model-i implied r also exhibis a moderae business cycle componen, wih dips during recessions, and some degree of recovery aferwards. Figure 3 shows ha he model-implied poserior belief of mean inflaion maches very well he rend inflaion π from Bauer and Rudebusch (2017). The rend inflaion is a survey-based measure, namely, he Federal Reserve s series on he perceived inflaion 13 A each ime, he agen perceives ha shocks oday are negaively serially correlaed wih fuure shocks; see proof in Nagel and Xu (2018). 16

17 4 Figure 1: Average r* and he model implied 10-year real yield year TIPS Mean r* 10-year real yield (model implied) The average r (quarerly daa) are from Bauer and Rudebusch (2017) from 1971:Q4 o 2017:Q2. The model-implied 10 year real yields (quarerly daa) are from 1962:Q1 o 2018:Q2. The end-of-quarer 10-year TIPS yields are from he Fed daabase a he S.Louis Federal Reserve from 2003:Q1 o 2018:Q2. The gray bars represen periods of recession defined by he NBER. arge rae, denoed PTR. I measures long-run expecaions of inflaion in he price index of personal consumpion expendiure (PCE). The resul confirms our assumpion in he model ha he represenaive agen forms her inflaion expecaion based on he hisory of pas inflaion raes year Treasury yield and poseriors While i has long been recognized ha nominal ineres raes conain a slow-moving rend componen (Nelson and Plosser, 1982; Rose, 1988), bond yields in equilibrium models (and no-arbirage erm srucure models in general) are generally modeled as saionary, mean-revering processes. As a resul, low-frequency variaion in ineres raes is hard o explain in such models and is mosly aribued o he erm premium componen, which is a residual erm in empirical models and usually he inflaion risk premium in equilibrium models (Piazzesi and Schneider, 2007). As shown in Figures 1, 2, and 3, he poseriors (γ µ c, and µ π, ) in model-i mach well he macro rends (r and π ). An illusraion of he poenial imporance of hese poseriors in he 10-year nominal yield is provided in Figure 4. The hump-shaped 10-year 17

18 6 Figure 2: Individual r* and he model implied 10-year real yield r* (LW) r* (LM) r* (Kiley) 10-year real yield (model implied) The individual r s (quarerly daa) are from Bauer and Rudebusch (2017) from 1971:Q4 o 2017:Q2. Three macroeconomic esimaes of r s are from Laubach and Williams (2003), Lubik and Mahes (2015), and Kiley (2015), respecively. The model-implied 10-year real yields (quarerly daa) are from 1962:Q1 o 2018:Q2. The gray bars represen periods of recession defined by he NBER. 8 Figure 3: π* and poserior mean inflaion π* Poserior mean inflaion (model implied) The rend inflaion π (quarerly survey-based PTR measure from FRB/US daa) are from Bauer and Rudebusch (2017) from 1971:Q4 o 2017:Q2. The model-implied poserior belief for mean inflaion (quarerly daa) is from 1962:Q1 o 2018:Q2. The gray bars represen periods of recession defined by he NBER. 18

19 15 Figure 4: 10-year nominal yield and macro rends year nominal yield Poserior mean inflaion (model implied) 10-year real yield (model implied) 10-year nominal yield (model implied) The end-of-quarer 10-year nominal yields are from Gürkaynak e al. (2007) from 1962:Q1 o 2018:Q2. The model implied 10-year real yield, 10-year nominal yield, and poserior belief for mean inflaion (quarerly daa) are from 1962:Q1 o 2018:Q2. The gray bars represen periods of recession defined by he NBER. Treasury yield from he lae 1960s o he lae 1990s reflecs an increase in inflaion expecaions before he mid-1980s and a secular decline aferwards. Over he pas wo decades, as inflaion expecaions have sabilized, he pronounced decline in oupu growh expecaions, and hence he 10-year real yield, is he main driver of he downrend in nominal yields. As a resul, he model-implied 10-year nominal yield capures well he rend movemens in he 10-year Treasury yield for he whole sample Model-I limiaions: cycle and spread in yields In conras o he anicipaed uiliy case (Kreps, 1998; Cogley and Sargen, 2008), belief updaing is an imporan risk o he agen, and he poseriors are he only sae variables in model-i (and hence he only source of risk for pricing). We have shown ha, wih fading memory, he modified Bayesian learning is perpeual, and he resuling poserior beliefs rack well he macro rends. Therefore, model-i explains well he rend in he 10-year Treasury yield. However, model-i is silen abou wo oher imporan feaures in yields - he cycle and erm spread. Hisorically, shor-erm bond yields are lower (on average) and much more volaile han long-erm bond yields. The spread beween long-erm and shor-erm bond yields is 19

20 posiive in general, shrinks in he lae expansion sage of he business cycle, and becomes invered before recessions. Given he CRRA uiliy, he erm premium is very small in model-i, and so he model-implied yields for 10-year and 1-year bonds are almos idenical. Hence he model is no able o explain dynamics in he erm spread and cyclical movemens in shor-erm bond yields. We exend model-i in he nex session o overcome hese shorcomings. 3. Model II - Learning, rend, and cycle in bond yields In model-ii, we sill consider an endowmen economy wih a represenaive agen who has a CRRA uiliy funcion. The agen sill learns wih fading memory as in model-i. Bu here, boh GDP growh and inflaion raes are decomposed ino wo componens: one sable componen and one ransiory/volaile componen. The agen learns abou he mean oupu growh and inflaion raes from he sable componen, and learns abou he saionary deviaions from mean from he ransiory/volaile componen. Equilibrium prices adjus such ha he agen is happy o consume he oupu as an endowmen Decomposiion and learning The four componens of GDP - invesmen spending, ne expors, governmen spending, and consumpion - don move in locksep wih each oher. In fac, heir levels of volailiy differ grealy. Consumpion is highly sable and varies less wih he business cycle. In conras, he oher hree componens are exremely volaile, varying grealy during economic conracions and expansions. For his reason, we assume ha here are wo unknowns for he agen o learn in he oupu growh process: he long-run mean and a laen sochasic deviaion from he mean (saionary). The agen learns abou he long-run mean GDP growh only from he sable componen (PCE), and learns abou he saionary deviaion from mean using only he volaile componen (GDP growh excluding PCE). Similarly, for inflaion, he agen learns abou long-run mean inflaion only from core inflaion, and learns he ransiory and saionary deviaion from he mean using only ransiory price changes (GDP deflaor excluding core inflaion). Formally, 20

21 oupu growh and inflaion can be decomposed ino (accoun ideniy) g +1 = g +1 + Gap g +1 π +1 = π +1 + Gap π +1, (12) where g +1 and π +1 are oal real GDP growh and inflaion, respecively. g +1 and π+1 are real consumpion growh (scaled by oal real GDP C +1 C ) and core inflaion (scaled by oal price level P +1 core P core P GDP ), respecively. Gap g +1 and Gap π +1 are he oal GDP growh rae excluding g +1 and he oal inflaion rae excluding π +1, respecively. Boh real consumpion growh and core inflaion follow i.i.d. laws of moion g +1 = µ c + σ c ε c,+1 π +1 = µ π + σ π ε π,+1, (13) where ε c,+1 and ε π,+1 are i.i.d. normal shocks. As wih model-i, he represenaive invesor knows ha boh g+1 and π+1 are i.i.d., and also knows σ c and σ π, bu no he long-run mean µ c and µ π. The agen forms expecaions abou µ c and µ π based on he same learning scheme as in model-i, wih he same poseriors, µ c H g N ( µ c,, υ c σ 2 c ) µ π H π N ( µ π,, υ πσ 2 π), (14) where ( µ c, = µ c, 1 + υc g µ c, 1) ( µ π, = µ π, 1 + υπ π µ π, 1), (15) 21

22 and he same predicive disribuion, g +j H g N ( µ c,, (1 + υ c ) σ 2 c ) π +j H π N ( µ π,, (1 + υ π) σ 2 π), (16) where j = 1, 2,..., H g { g 0, g 1,... g }, and H π {π 0, π 1,...π }. Boh Gap g +1 and Gap π +1 are assumed o conain a laen saionary componen as in Bansal and Yaron (2004) and Piazzesi and Schneider (2007): Gap g +1 = x c,+1 + σc gap ε gap c,+1 Gap π +1 = x π,+1 + σπ gap ε gap π,+1 x c,+1 = ρ c x c, + σ x c ε x c,+1 x π,+1 = ρ π x π, + σ x πε x π,+1, (17) where ε gap c,+1, ε gap π,+1, ε x c,+1, and ε x π,+1 are i.i.d. normal shocks. The represenaive agen knows all he parameers, bu no x c,+1 and x π,+1. She forms expecaions abou x c,+1 and x π,+1 based on he same learning scheme as for he long-run mean, bu wih poenially differen geomeric weighing parameers, υ gap c by and υπ gap. The poseriors are given x c,+1 H gap g, x π,+1 H gap π, N ( ( ρ c x c,, υc gap (σ x c ) 2 + (σ gap N ( ρ π x π,, υ gap π c ) 2)) ( (σ x π ) 2 + (σπ gap ) 2)) x c, = ρ c x c, 1 + υ gap c (Gap g ρ c x c, 1 ) x π, = ρ π x π, 1 + υ gap π (Gap π ρ π x π, 1 ), (18) where H gap g, {Gap g 0, Gap g 1,...Gap g } and H gap π, {Gap π 0, Gap π 1,...Gap π }. As discussed in 2.3, he updaing is same as for an opimal Kalman filering wih an appropriae choice of parameer values. To undersand beer he sochasic naure of he oupu growh and inflaion process 22

23 from he agen s subjecive viewpoin, we furher ge he oal predicive disribuion g +j H g π +j H π N ( µ c, + ρ c x c,, (1 + υc ) σc 2 + (1 + υc gap ) ( (σc x ) 2 + (σc gap ) 2)) N ( µ π, + ρ π x π,, (1 + υπ) σπ 2 + (1 + υπ gap ) ( (σπ) x 2 + (σπ gap ) 2)), (19) where j = 1, 2,... and he variance of he predicive disribuion conains boh uncerainy due o fuure shocks and uncerainy abou µ c/µ π and x c,+1 /x π,+1. H g H g and H gap g,, and H π 3.2. Bond pricing conains boh H π and H gap π,. conains boh The Euler equaion holds under he represenaive agen s subjecive expecaions, and he log nominal pricing kernel is he same as in model-i. The ime- price of a zerocoupon bond saisfies he same recursion in equaion (10). Model-I has wo sae variables (he poserior means for oupu growh and inflaion) ha explain he rend in long-erm yields. However, he differen GDP growh and inflaion componens appear o have very differen dynamics in he daa, hence we allow he agen o learn he long-run mean and he cyclical componen from daa separaely in his model. Model-II has four sae variables: µ c,, µ π,, x c,, and x π,. Given he linear Gaussian framework, we assume ha p (n) variables µ = ( µ c,, µ π,) T and x = ( x c,, x π, ) T : = log(p (n) ) is a linear funcion of hese sae p (n) = A (n) B (n) x C (n) µ. (20) When we subsiue p (n) and p (n 1) +1 in he Euler equaion (10), he coefficiens in he pricing equaion can be solved wih B (n) = B (n 1) ρ+v ρ, C (n) = C (n 1) +v = v n, and A (n) = ( ) ( A (n 1) +A (1) (n 1) (n 1) 0.5 V ar p +1 Cov p +1, m,+1) $ (see he appendix for deails). The subjecive excess reurn is er n,+1 = Cov ( rn,+1, m $,+1) = v Cov ( z+1, µ +1) C (n 1) v Cov (z +1, x +1 ) B (n 1). All he variance and covariance erms are relaively small in he daa. Hence, given CRRA uiliy, he erm premium is small in his model. 23

24 As we can see from he soluion, he yield parameer ( C(n) ) for he poserior mean n µ is consan over horizon n; herefore, he impacs of µ on long-erm and shor-erm yields are he same, which explains he low frequency movemens (rend) in yields. However, he yield parameer for x, B(n), is decreasing over horizon n. Hence, he impac of x n on he shor-erm yield is bigger han on he long-erm yield, which capures he cyclical movemens in shor-erm yield. The spread beween long-erm and shor-erm yields is mainly driven by he cyclical componen x, which could be posiive or negaive for many periods (depending on he persisence parameer ρ). Sill, in conras wih daa, he spread is mean zero because of he saionariy assumpion for x. To solve he price and yields for real bonds, we can jus replace v wih v = (γ, 0) Empirical findings We use he same daa ses as in model-i, bu real GDP growh and he rae of inflaion from he GDP deflaor are decomposed ino one sable componen and one ransiory componen. We can hen calculae he poserior beliefs for he long-run mean and he saionary deviaion from he mean. Again, we show ha he model-implied 10-year real yield closely maches he esimaed r sar in he lieraure and he oal poserior belief for inflaion closely maches he survey-based rend inflaion. As a resul, he modelimplied 10-year nominal bond yields mach he hisorical rend movemens in he 10-year Treasury yields well. In addiion, he model can also capure he cyclical movemens in 1-year Treasury yields Parameers The volailiy parameers for consumpion growh and core inflaion are calibraed o mach heir counerpars in he daa. As shown in he appendix, even hough we have wo differen parameers for he growh gap (inflaion gap) volailiy, σ x c and σ gap c and σπ gap ), hese parameers can be considered as one parameer for he model soluion; hence σ cx = (σc x ) 2 + (σc gap ) 2 (σ πx = (σπ) x 2 + (σπ gap ) 2 ) is calibraed o mach volailiy in ransiory GDP growh (ransiory inflaion). In he model, he correlaion in sable (σ x π 24

25 υ gap π γ β υc υπ υc gap corr ρ c ρ π σ c σ π σ cx σ πx corr gap Table 2: Configuraion of model-ii parameers Table 2 repors parameer values for oupu growh and inflaion processes, and for he consan gain in learning. parameers are given in quarerly erms. Mean and sandard deviaion are in percenages. All and ransiory componens beween oupu growh and inflaion is calibraed o mach he correlaion in he daa The parameers for learning in inflaion υ π, ρ π, and υ gap π are calibraed o mach variaions in π and he one-quarer-ahead mean survey inflaion expecaion (persisence and volailiy). The parameers for learning in growh υ c, ρ c, and υ gap c are calibraed o mach variaions in r and one-year nominal yields (persisence and volailiy). We follow he lieraure and se risk aversion as 3, and ime preference β is calibraed o mach he level of en-year nominal yields in he daa. The resuling parameer values are repored in Table Poseriors versus r and π I is well undersood by invesors ha some componens of inflaion and GDP growh are more volaile han ohers; in fac, heir levels of volailiy differ grealy. Hence, i is naural for he agen o learn he long-run mean and a saionary deviaion from he mean separaely. The agen uses a longer effecive sample size o learn he long-run mean and a shorer effecive sample size o learn he shor-run deviaion (υ c < υ gap c υπ < υπ gap ). Wih he same learning wih memory loss, he represenaive agen updaes her poserior beliefs slowly bu perpeually. Figure 5 shows ha he model-implied oal poserior belief for 10-year-ahead mean inflaion maches very well he survey-based PTR rend inflaion π. Similarly, Figure 6 shows ha he model-implied oal poserior belief for 1-quarer-ahead mean inflaion also racks very closely he survey mean for 1-quarer-ahead inflaion from SPF. The resul confirms our assumpion ha he agen forms her inflaion expecaions differenly and 25

26 8 Figure 5: π* and 10-year poserior mean inflaion π* 10-year poserior mean inflaion (model implied) The rend inflaion π (quarerly survey-based PTR measure from FRB/US daa) are from Bauer and Rudebusch (2017) from 1971:Q4 o 2017:Q2. The model-implied oal poserior belief for 10-year-ahead mean inflaion (quarerly) is from 1968:Q3 o 2018:Q2. The gray bars represen periods of recession defined by he NBER. for long-run versus shor-run. Figure 7 shows he model-implied r, which is a linear funcion of only he poserior for he long-run mean growh γ µ c, + Cov, and closely racks he hree macroeconomic esimaes of r year and 10-year Treasury yields, and heir spread In model-i, we have shown he imporance of macro rends in explaining he movemens in he long-erm nominal yield. For he same reason, he poseriors for long-run mean growh and inflaion move slowly and mach well he macro rends in model-ii, and Figure 8 shows he poenial imporance of hese poseriors in he 10-year nominal yield. The secular decline in he 10-year Treasury yield afer he 1980s is mainly driven by a combinaion of wo phenomena: firs a downrend in inflaion expecaions and hen a seady decline in r. Due o he lack of cyclical movemens in he poseriors, model-i is limied o explain only he rend in long-erm yields. In model-ii, however, learning abou he deviaion 14 To be consisen wih he concep of r, he shor-run effec from x c, on he real yield is no included for calculaion of he model-implied r, because i will evenually vanish. Therefore he model-implied r impacs he real yields of any mauriy equally. 26

27 12 Figure 6: One-quarer-ahead inflaion expecaions - poserior vs. survey Survey mean inflaion expecaions Poserior mean inflaion expecaions (model implied) The mean survey one-quarer-ahead inflaion (quarerly daa) are from he Philadelphia Fed s SPF from 1968:Q3 o 2018:Q2. The model-implied oal poserior belief for 1-quarer-ahead mean inflaion (quarerly) is from 1968:Q3 o 2018:Q2. The gray bars represen periods of recession defined by he NBER. 6 Figure 7: Individual r* and model implied r* r* (LW) r* (LM) r* (Kiley) r* (model implied) The individual r s (quarerly daa) are from Bauer and Rudebusch (2017) from 1971:Q4 o 2017:Q2. Three macroeconomic esimaes of r s are from Laubach and Williams (2003), Lubik and Mahes (2015), and Kiley (2015). The model-implied r (quarerly) is from 1968:Q3 o 2018:Q2. The gray bars represen periods of recession defined by he NBER. 27