Real and Nominal Equilibrium Yield Curves with Endogenous Inflation: A Quantitative Assessment

Transcription

1 Real and Nominal Equilibrium Yield Curves wih Endogenous Inflaion: A Quaniaive Assessmen Alex Hsu, Erica X.N. Li, and Francisco Palomino January 28, 215 Absrac The links beween real and nominal bond risk premia and macroeconomic dynamics are explored analyically and quaniaively in a model wih nominal rigidiies and moneary policy. The ineres-rae policy rule becomes a resricion linking real and nominal risk premia hrough endogenous inflaion. The esimaed model capures macroeconomic and yield curve properies of he U.S. economy, implying significanly posiive real erm and inflaion risk bond premia. Boh premia are induced by wage rigidiies as a compensaion for permanen produciviy shocks. Sronger policy-rule responses o inflaion oupu) increase decrease) boh premia. Policy surprises generae significan yield volailiy bu negligible risk premia. JEL Classificaion: D51, E43, E44, E52, G12. Keywords: Term srucure of ineres raes, bond risk premia, moneary policy, nominal rigidiies. We hank Canlin Li, Eric Swanson, and seminar paricipans a he Universiy of Michigan Finance Brownbag, Federal Reserve Bank of Alana, Banco de la República, Bank of Canada Fixed Income Conference 213, Sociey of Economic Dynamics Meeing 213, 213 China Inernaional Conference in Finance, Conference on Compuing in Economics and Finance 213, Lain American Economeric Sociey Meeing 213, Dynare Conference 213 and CEPR Gerzensee Asse Pricing Meeing 214 for helpful commens and suggesions. Georgia Insiue of Technology, Alex.Hsu@scheller.gaech.edu. Cheung Kong Graduae School of Business, xnli@ckgsb.edu.cn Ross School of Business, Universiy of Michigan, fpal@bus.umich.edu. Francisco Palomino acknowledges suppor from he NTT Research Fellowship.

2 1 Inroducion Undersanding he economic drivers of long-erm real bond yields and risk premia is a fundamenal concern in financial economics. This undersanding is criical, for insance, o explain he dynamics of he real pricing kernel, he diversificaion benefis of real bonds, or he ransmission of moneary policy. Is imporance, however, radically conrass wih our knowledge of he subjec. Insighs from he empirical analysis of inflaion-linked governmen bonds are, a bes, incomplee. These bonds have been raded only recenly in developed economies, are imperfec subsiues of real bonds, and are poenially affeced by illiquidiy and mispricing. 1 In addiion, he available evidence from he Unied Kingdom and Unied Saes inflaion-linked bonds does no presen a clear picure of heir risk properies and link o he economy. Common risk measures such as he average slope of he yield curve, realized excess reurns, or he bond reurn correlaions wih macroeconomic variables and sock reurns provide mixed resuls across counries and sub-periods. For insance, he average inflaion-linked U.K. and U.S. yield curves during are sharply downward and upward sloping, respecively. The same curve in he U.K. is slighly upward sloping for Thus, he limied evidence can be highly benefied from he heoreical analysis of real yields and risk premia. This paper provides such an analysis by developing and esimaing a New Keynesian model ha delivers equilibrium real and nominal yield curves. I focuses on undersanding i) he effec of nominal rigidiies, several fundamenal shocks, and moneary policy on real erm and inflaion risk premia in real and nominal bonds, respecively, and ii) he link imposed by endogenous inflaion on hese wo ypes of premia. The heoreical framework is moivaed by wo reasons. Firs, New Keynesian models have become he workhorse model for undersanding economic dynamics, and are widely used for policy analysis. Second, he framework generaes endogenous inflaion dynamics ha depend on economic fundamenals, linking he properies of real erm and inflaion risk premia. This link 1 See Garcia and van Rixel 27) for recen hisory on inflaion-linked bond markes, D Amico, Kim and Wei 214) for evidence on significan ime-varying liquidiy premia in U.S. and U.K. inflaion-linked bonds, and Fleckensein, Longsaff and Lusig 214) for evidence on mispricing in he TIPS marke. 1

3 acs as an addiional resricion relaive o erm srucure models wih exogenous inflaion. Rudebusch and Swanson 212), and Andreasen 212), among ohers, sudy bond risk premia in his framework wih relaive quaniaive success. I is no well known, however, wha fricions and shocks are essenial or quaniaively imporan in explaining yield dynamics, he decomposiion and link beween real erm and inflaion risk premia, and he economic mechanisms driving he resuls. This paper conribues o fill his gap. The model conains he sandard elemens of he New Keynesian framework, and recursive preferences on consumpion and labor for he represenaive household. These preferences have he abiliy o simulaneously capure macroeconomic and erm srucure dynamics. 2 The analysis is focused on undersanding he conribuion of he following key elemens o real and nominal bond risk premia. Firs, nominal price and wage rigidiies. Boh fricions generae real effecs in moneary policy, bu have differen implicaions for economic dynamics, as highlighed in Chrisiano, Eichenbaum and Evans 25). Second, produciviy, moneary policy, and inflaionarge shocks. Produciviy is modeled o conain permanen difference-saionary) and ransiory rend-saionary) componens. As shown by Campbell 1986) and Labadie 1994), hese wo componens have differen implicaions for bond risk premia. They also have differen effecs on he permanen componen of he marginal uiliy of wealh, and hen in heir abiliy o price oher financial asses such as socks, as demonsraed by Alvarez and Jermann 25). Moneary policy and inflaion-arge shocks are sandard in his framework bu heir implicaions for bond risk premia have been less sudied. Third, a nominal ineres-rae policy rule. The response o economic condiions in his rule has imporan implicaions for he join dynamics of real variables and inflaion, and hen he link beween real erm and inflaion risk premia. 3 The model is esimaed via he Generalized Mehod of Momens GMM) o mach a series 2 See Rudebusch and Swanson 28, 212) for differences in his abiliy beween habi formaion and recursive preferences, respecively. 3 Alernaively, a more srucural approach o moneary policy is o consider he moneary auhoriy as a social planner ha maximizes welfare, as in Palomino 212). This approach may have differen implicaions and is no explored in his paper. 2

4 of key U.S. macroeconomic momens. The baseline esimaion capures he nominal price and wage rigidiies in he daa, and is affeced by he four shocks described above. I simulaneously capures macroeconomic and nominal bond reurn dynamics beer han all he esed esimaions wih alernaive rigidiy and shock specificaions. I maches he Sharpe raio of he 5-year nominal bond wih a risk aversion in he range of values repored in he erm srucure lieraure. I implies upward sloping real and nominal yield curves, wih average 5-year real and nominal bond spreads of 82 and 112 bps., respecively. This is he resul of posiive real erm and inflaion risk premia. Explaining he economic drivers of posiive real erm and inflaion risk premia in he baseline model is cenral o he analysis. I relies on comparisons wih alernaive model specificaions, and complemened wih approximae closed-form soluions for hese premia. Two properies of he pricing kernel are useful o describe he main findings: Real erm and inflaion risk premia are posiive when he real pricing kernel is negaively auocorrelaed and posiively correlaed wih inflaion, respecively. Consumpion growh is he main driver of he pricing kernel in he baseline model, and hese wo condiions become a negaive auocorrelaion of consumpion growh, and a negaive correlaion of i wih inflaion, respecively. 4 There are hree main findings. Firs, permanen produciviy shocks, in combinaion wih wage rigidiies, are crucial o generae large and posiive real erm and inflaion risk premia. Permanen produciviy shocks conribue wih almos all he variabiliy in he pricing kernel, and hus bond risk premia are mainly a compensaion for his risk. Wage rigidiies, in he presence of hese shocks, generae a consumpion growh process ha is boh negaively auocorrelaed and negaively correlaed wih inflaion. This is explained by he impac of labor dynamics on consumpion wih and wihou rigidiies. In he absence of rigidiies, prices and wages freely adjus afer permanen shocks o preserve produc and labor markups, keeping labor consan. Since consumpion depends on produciviy and labor, consumpion growh inheris he posiive 4 As discussed below, Secion 5 presens a model exension ha breaks he srong link beween consumpion growh and he pricing kernel in he baseline model. I allows us o capure a posiive auocorrelaion of consumpion growh wihou affecing he main erm srucure implicaions of he model. 3

5 auocorrelaion of he permanen shocks. Afer a negaive shock, expeced consumpion growh and inflaion decline, as well as real and nominal yields. Bond reurns are hus high, implying negaive real and nominal erm premia. On he conrary, nominal rigidiies generae procyclical and mean-revering labor ha affecs consumpion growh. Afer a negaive shock, prices and wages are higher han in he fricionless case, depressing labor and reducing consumpion furher. However, expeced fuure labor and consumpion increase since prices and wages gradually adjus downwards. In he baseline model, he effec of wage rigidiies on labor is srong enough o generae a negaive auocorrelaion in consumpion growh, and a negaive correlaion of i wih inflaion. Therefore, real erm and inflaion risk premia become posiive under wage rigidiies. Second, responses o economic condiions in he ineres-rae policy rule, and surprises in moneary policy affec bond risk premia and yield volailiy, respecively. A sronger response o inflaion in he policy rule increases real erm and inflaion risk premia by increasing he sensiiviy of he real rae and pricing kernel) o permanen shocks. A sronger response o he oupu gap, or an increase in he ineres-rae smoohing coefficien, have he opposie effec. In conras, moneary policy and inflaion-arge shocks have a negligible effec on bond risk premia, bu significanly affec real and nominal bond yield volailiy. In he absence of nominal rigidiies, moneary policy does no have any real effecs, and hese shocks do no affec real yields. In he presence of rigidiies, however, moneary policy affecs he real economy and hus real yields for all mauriies. As a resul, moneary policy shocks become he main driver of real yields. Inflaionarge shocks do no have a significan impac on real yields, bu is persisence considerably affec long-erm nominal bond yield volailiy. Third, he ineres-rae policy rule resrics he join behavior of real erm and inflaion risk premia. This resricion is refleced in he baseline model in real erm and inflaion risk premia ha share he same sign. This is explained by he dynamics of he real pricing kernel and inflaion implied by he policy rule. A rule wih a sufficienly srong response o inflaion imposes a posiive relaion beween inflaion and he shor-erm nominal ineres rae. This rae, in urn, 4

6 is negaively relaed o he fuure pricing kernel, as implied by opimaliy condiions and bond marke clearing. Tha is, in equilibrium, inflaion is negaively correlaed wih he real pricing kernel. Inflaion risk premia are hus posiive negaive) when he pricing kernel is negaively posiively) auocorrelaed. This condiion is he same as for posiive negaive) real erm premia. This resricion is no exisen in models wih exogenous inflaion, where parameers are freely se o simulaneously generae negaive real erm and posiive inflaion risk premia. Finally, he paper addresses hree limiaions of he baseline model. Firs, he baseline esimaion does no capure he posiive auocorrelaion of consumpion growh in he daa. A negaive auocorrelaion in consumpion growh induced by permanen produciviy shocks is essenial o generae posiive bond risk premia. Inroducing habi persisence in preferences breaks he igh link beween he pricing kernel and consumpion growh. I generaes negaive auocorrelaion in he former and posiive in he laer, wihou compromising he main macroeconomic and erm srucure implicaions of he model. Second, he baseline model generaes negligible variaion in expeced excess bond reurns, a odds wih he well documened evidence on bond reurn predicabiliy. This shorcoming can be overcome by inroducing counercyclical volailiy in permanen or ransiory produciviy shocks. Third, he baseline model absracs from capial accumulaion. An esimaion of a model wih capial shows ha he main implicaions of he baseline model survive, bu he link beween real erm and inflaion risk premia becomes weaker. This paper joins he lieraure ha analyzes he erm srucure of ineres raes using New Keynesian models see Woodford 23) for he sandard framework). I complemens he curren lieraure by providing an esimaion of he real yield curve, and a deailed analysis of real erm and inflaion risk premia, heir economic link, and he quaniaive conribuion of differen model elemens o he resuls. I is closely relaed o he sudies in Rudebusch and Swanson 212), Andreasen 212), Dew-Becker 214) and Kung 214), which add recursive preferences o he sandard framework wih differen model elemens. Rudebusch and Swanson 212) rely on ransiory produciviy shocks and price rigidiies o capure nominal yield curve properies, and do 5

7 no sudy real yield curve implicaions. These elemens are presen in our model, bu heir effecs are no as quaniaively imporan as hose of wage rigidiies and permanen shocks. Andreasen 212) incorporaes boh permanen and ransiory componens in produciviy, and Dew-Becker 214) adds wage rigidiies o he analysis. These sudies focus on he ime-variaion in bond risk premia by fiing macroeconomic and yield dynamics. Our model has hese elemens bu wih a differen focus. The GMM approach allows us o arge uncondiional momens, provide quaniaive comparisons across model specificaions, and focus on explaining he economic mechanisms behind he resuls. Kung 214) presens a model wih an endogenous growh channel ha is complemenary o our model srucure. The effec of his channel on our resuls is beyond he scope of he paper. The paper is also relaed o erm srucure models wih exogenous inflaion such as he endowmen economies in Wacher 26), Piazzesi and Schneider 27), and Bansal and Shaliasovich 213), or he real business cycle model in Van Binsbergen e al. 212). These models require a negaive correlaion beween hese variables o mach an average upward sloping nominal yield curve, and have differen implicaions for he real yield curve. 5 Our model generaes he negaive correlaion of consumpion growh and inflaion endogenously, and links real and nominal bond risk premia from firs principles. An advanage of our framework is ha i allows us o predic changes in yield curve dynamics relaed o srucural economic and policy changes. Finally, Buraschi and Jilsov 25) sudies real and nominal bond risk premia in a moneary real business cycle model. This model also generaes endogenous consumpion growh and inflaion. Their moneary policy and fricion specificaions are subsanially differen. The paper is organized as follows. Secion 2 describes he daa and repors descripive saisics for nominal and inflaion-linked yield curves in he Unied Saes and he Unied Kingdom. Secion 3 describes he model. Secion 4 provides deails of he model esimaion and is quaniaive performance, presens is main implicaions, and explores he economic mechanism behind he resuls. Secion 5 presens some model limiaions and exensions, and Secion 6 concludes. 5 The models wih recursive preferences in Piazzesi and Schneider 27) and Bansal and Shaliasovich 213) imply an average downward sloping real yield curve, while he habi model in Wacher 26) implies he opposie. 6

8 2 Empirical Evidence This secion presens descripive saisics of inflaion-linked and nominal governmen bonds in he Unied Kingdom and he Unied Saes. While he empirical properies of nominal bond yields are well known, he sudy of he real yield curve, is risks, and links o he nominal one is limied by daa availabiliy. Inflaion-linked governmen bonds are, a bes, imperfec subsiues of real bonds, and have only been raded in he Unied Kingdom and he Unied Saes since 1981 and 1997, respecively. Their inflaion proecion is limied by a lagged indexaion o price levels and he embedded deflaion opionaliy hey provide. In addiion, pricing in hese markes has been affeced by liquidiy concerns and poenial unexploied arbirage opporuniies. 6 The evidence illusraes several difficulies in exploring properies of real bonds from he available daa, moivaing he join heoreical analysis of real and nominal yields ha follows. 7 We use quarerly daa from January 1985 o Sepember 28 for he U.S. and he U.K., and repor saisics for he periods and The daa sample periods are moivaed by wo reasons. Firs, TIPS daa in he U.S. and inflaion-linked gils daa in he U.K. are only available since 1999 and 1985, respecively. 8 Second, he period Sepember-December 28 coincides wih he collapse of Lehman Brohers ha drove shor-erm ineres raes close o zero, and riggered a swich o unconvenional moneary policies. The period afer Sepember 28 is hen no covered o focus on he effecs on bond yields of a convenional) moneary policy conduced using an ineres-rae rule. 6 See D Amico, Kim and Wei 214) for evidence on significan ime-varying liquidiy premia in U.S. and U.K. inflaion linked bonds. See Fleckensein, Longsaff and Lusig 214) for evidence of mispricing in he TIPS and inflaion swaps markes. 7 There is an exensive empirical lieraure sudying he real erm and inflaion risk premia wih and wihou inflaion-linked bonds wih no-arbirage erm srucure models. This lieraure shows a wide range for he sign and size of inflaion risk premia in he U.K and he U.S. An incomplee lis includes Barr and Campbell 1997) Evans 1998), and Joyce, Lildhold and Sorensen 21) for he U.K., and Ang, Bekaer and Wei 28), D Amico, Kim and Wei 214), Chen, Liu and Cheng 21), Chrisensen, Lopez and Rudebusch 21), Chernov and Mueller 212), Grishchenko and Huang 213), and Abrahams e al. 213) for he U.S. Hördahl and Trisani 212) provide a similar sudy for he Euro zone. 8 Resuls using comparable monhly daa are very similar. We presen resuls for quarerly daa o be consisen wih he model esimaion. The same macroeconomic and erm srucure daa for he Unied Saes are used o esimae he model, for he longer period January 1982 o Sepember 28. 7

9 The consumpion growh and inflaion series for he U.S. are consruced using quarerly daa from he Bureau of Economic Analysis, following he mehodology in Piazzesi and Schneider 27). These series capure only consumpion of non-durables and services and is relaed inflaion, and hen consisen wih he model variables. Wages are real wages per hour of non-farm business from he Federal Reserve Economic Daa FRED) daabase from he Federal Reserve Bank of S. Louis. The daa on U.S. zero-coupon nominal bond and TIPS yields are consruced following he procedure in Gurkaynak, Sack and Wrigh 26, 28), respecively. These daa are obained from he Federal Reserve websie. The shor-erm nominal ineres rae is he 3- monh T-bill rae from he Fama risk-free raes daabase. The hree-monh real rae is esimaed using he mehodology described in Pflueger and Viceira 211). 9 Dividends and sock marke reurns correspond o he marke porfolio obained from he Cener for Research in Securiy Prices CRSP). For he U.K., consumpion growh and inflaion are obained direcly from he FRED daabase. The hisorical yields for U.K. real and nominal bonds are aken from he Bank of England websie. The hree-monh real rae in he U.K. is esimaed using he same mehodology used o esimae he U.S. real rae. Sock reurns are for he UK FTSE All-Shares Index. The bond yields under sudy correspond o mauriies from 2 o 1 years. The long end of he curves has been excluded for comparison purposes across counries. Greenwood and Vayanos 21) documen a significan effec on long-erm inflaion-linked bond yields in he U.K, resuling from he increased demand from pension funds o mee he Minimum Funding Requiremens. Table 1 summarizes he empirical evidence. The properies of bond risk premia are frequenly characerized by he average slope of a yield curve, he average excess bond reurns relaive o a risk-free rae, or he correlaion of excess bond and sock reurns. Panel A repors, a slighly and a significanly upward-sloping average nominal 9 Specifically, he compuaion is based on he regression i π +1 = consan + β i i + β r i 1 π ) + ε, where i is he hree-monh nominal rae and π is he hree-monh inflaion rae. The real rae is hen compued as r = i E [π +1 ] under he assumpion ha he inflaion risk premium in hree-monh nominal bonds is negligible. 8

10 yield curves in he U.K and he U.S, respecively, suggesing posiive risk premia in nominal bonds. The picure is less clear for inflaion-linked bonds. The average yield curve for hese bonds is slighly upward sloping for he U.K. for he sample , bu becomes drasically downward sloping for he sample During he same period, he comparable average yield curve in he U.S. is significanly upward sloping. Figure 1 shows he average U.K. yield curves for boh samples. These findings sugges negaive and posiive risk premia in inflaion-linked bonds in he U.K. and he U.S. respecively. The average excess reurns in Panel B suppor hese claims. 1 Nominal bonds exhibi posiive average excess reurns for boh counries, and inflaion-linked bonds in he U.K. and he U.S. have negaive and posiive average excess reurns, respecively. However, he correlaions beween excess bond and sock reurns in Panel C sugges a differen sory. While inflaion-linked bond excess reurns in he U.K. have shown posiive correlaions wih sock excess reurns in boh samples, U.K. nominal bonds swich from a posiive correlaion for o a negaive one for The opposie occurs for U.S. nominal bonds, while he correlaion beween TIPS and sock excess reurns is negaive for According o he CAPM, he evidence for he recen sample implies negaive risk premia for U.K. and U.S. nominal and inflaion-linked bonds, a odds wih evidence from panels A and B. 11 The link beween macroeconomic variables and he yield curve also is of ineres o undersand bond risk premia. Panel C repors correlaions of consumpion growh and inflaion wih bond yields. The correlaions of U.K. and U.S. inflaion-linked and nominal bond yields wih consumpion growh are significanly posiive during boh samples. On he oher hand, he correlaions of inflaion wih hese yields change from posiive for o negaive for These changes are accompanied by a reduced auocorrelaion of inflaion in boh he U.K. and he U.S., higher and lower auocorrelaions of consumpion growh in he U.S. and he U.K respecively, and a correlaion beween consumpion growh and inflaion ha is negaive in he U.S. and 1 Excess bond reurns are compued as he difference of realized nominal reurns on inflaion-linked and nominal bonds wih he respecive 3-monh nominal rae for each counry. 11 The ime-varying naure of he correlaion beween nominal bond and sock reurns is highlighed and sudied by Viceira 212) and Campbell, Sunderam and Viceira 213). 9

11 swiching from posiive o negaive in recen years in he U.K. These evidence can be linked o bond risk premia hrough economic heory. According o sandard equilibrium models, a posiive auocorrelaion of consumpion growh implies negaive premia for real bonds, and a negaive correlaion beween consumpion growh and inflaion implies posiive inflaion risk premia. 12 Ineresingly, Panel A also shows ha he sandard deviaions of inflaion-linked and nominal bonds are similar for in boh he U.K. and he U.S. This is inriguing given he addiional exposure of nominal yields o inflaion risk. In summary, he descripive saisics presened here do no provide a clear paern o describe salien properies of real bond risk premia and is link o macroeconomic variables. This is no surprising given he shor-sample for inflaion-linked bonds, he limiaions of hese bonds presened above, and poenial srucural changes in he economy. The heoreical model in Secion 3 allows us o analyze he link beween real and nominal bond risk premia and macroeconomic variables. This analysis can provide esable implicaions o undersand beer he dynamics of real bond yields. 3 Model We model a producion economy wih a represenaive household, a producion secor for differeniaed goods, and moneary policy. The represenaive household derives uiliy from he consumpion of a baske of goods and disuiliy from supplying differeniaed labor o he producion secor. Labor and produc markes are characerized by monopolisic compeiion and nominal wage and price rigidiies, respecively. Moneary policy is modeled as an ineres-rae policy rule ha reacs o economic condiions. All markes are complee. Defaul-free real and nominal bonds are in zero ne supply. The model can be seen as an exension of he sandard New- 12 Campbell 1986) shows he link beween he auocorrelaion of consumpion growh and he real yield curve under consan relaive risk aversion preferences. The same inuiion applies under recursive preferences on consumpion, as shown in Bansal and Shaliasovich 213). The Campbell and Cochrane 1999) habis model can imply he opposie, as shown by Wacher 26). Piazzesi and Schneider 27) highligh he link beween posiive inflaion risk premia and he negaive correlaion beween expeced) consumpion growh and inflaion under recursive preferences. 1

12 Keynesian framework see Woodford 23), for insance) o capure bond pricing dynamics. I incorporaes recursive preferences for he represenaive household, as in Rudebusch and Swanson 212) and Li and Palomino 214), o disenangle risk aversion from he elasiciy of ineremporal subsiuion of consumpion. This separaion allows us o mach observed macroeconomic dynamics by choosing an appropriae level for he elasiciy of ineremporal subsiuion, while increasing he degree of risk aversion o capure large expeced excess reurns. Nominal prices and/or wages ha are no adjused opimally generae relaive price and wage disorions ha affec producion decisions. In his seing, differen moneary policy rules have differen implicaions on inflaion and real aciviy. As a resul, he dynamics and riskiness of real and nominal bond yields are affeced by boh nominal rigidiies and moneary policy. This secion describes he characerisics of he model economy. 3.1 Household A represenaive agen chooses consumpion C and labor supply N s o maximize he Epsein and Zin 1989) recursive uiliy funcion V = 1 β)uc, N s ) 1 ϕ + βe [V 1 γ 1 ϕ +1 ] 1 ϕ 1 γ, 1) where β > is he subjecive discoun facor, and ϕ and γ deermine he elasiciy of ineremporal subsiuion EIS) and he coefficien of relaive risk aversion, respecively. 13 The recursive uiliy formulaion relaxes he srong assumpion of γ = ϕ implied by consan relaive risk aversion. The inra-emporal uiliy is defined over consumpion and labor supply as UC, N s ) = C 1 ϕ ) 1 ϕ κ N s ) 1+ω 1 1 ϕ, 2) 1 + ω 13 The elasiciy of ineremporal subsiuion of he uiliy bundle of consumpion and labor is ϕ 1. The coefficien of relaive risk aversion is defined in Secion 4. 11

13 where ω 1 > capures he Frisch elasiciy of labor supply, and he process κ is chosen o ensure balanced growh i is specified in he producion secor secion below). The consumpion good is a baske of differeniaed goods produced by a coninuum of firms. Specifically, he consumpion baske is [ 1 C = ] θp C j) θp 1 θp 1 θp dj, 3) where θ p > 1 is he elasiciy of subsiuion across differeniaed goods, and C j) is he consumpion of he differeniaed good j. Labor supply is he aggregae of a coninuum of differen labor ypes supplied o he producion secor, such ha N s = 1 N s k)dk, 4) where N s k) is he supply of labor ype k. The represenaive consumer is subjec o he ineremporal budge consrain [ ] [ ] E M,+sP $ +s C +s E M,+sP $ +s LI +s + D +s ), 5) s= s= where M $,+s is he nominal discoun facor for cashflows a ime + s, P is he nominal price of a uni of he baske of goods, LI is he real labor income from supplying labor o he producion secor, and D is he real dividend from owning he producion secor. Appendix A shows ha he household s opimaliy condiions imply ha he one-period real and nominal discoun facors are M,+1 = β C+1 C ) ϕ [ E V 1/1 ϕ) +1 ] 1/1 γ) V 1 γ)/1 ϕ) +1 ϕ γ ) 1, and M,+1 $ P+1 = M,+1, 6) P 12

14 respecively. The one-period coninuously compounded) real and nominal ineres raes are obained from r = log E [M,+1 ], and i = log E [ M $,+1 ], 7) respecively. The nominal ineres rae i is he insrumen of moneary policy Wage Seing Following Schmi-Grohe and Uribe 27), an imperfecly compeiive labor marke is modeled where he represenaive household monopolisically provides a coninuum of labor ypes indexed by k [, 1]. 14 The supply of labor ype k saisfies he demand equaion ) θw N s W k) k) = N d, 8) W where N d is he aggregae labor demand of he producion secor, W k) is he wage for labor ype k, and W is he aggregae wage index given by [ 1 W = ] 1 1 θw W k) 1 θw dk. 9) The labor demand equaion 8) is obained from he producion secor problem presened in he secion below. The household chooses wages W k) for all labor ypes k under Calvo 1983) saggered wage seing. Specifically, a each ime, he household is only able o adjus wages opimally for a fracion 1 α w of labor ypes. The remaining fracion α w of labor ypes adjus heir previous period wages by he wage indexaion facor Λ w, 1,. The specific funcional form 14 This approach is differen from he sandard heerogeneous households approach o model wage rigidiies in Erceg, Henderson and Levin 2), where each household supplies a differeniaed ype of labor. In he presence of recursive preferences, his approach inroduces heerogeneiy in he marginal rae of subsiuion of consumpion across households since i depends on labor. We avoid his difficuly and obain a unique marginal rae of subsiuion by modeling a represenaive agen who provides all differen ypes of labor. 13

15 of his facor is presened in Secion 4. The opimal wage maximizes 1), subjec o demand funcions 8) for all labor ypes k, and he budge consrain 5). Noice ha real labor income is given by LI = 1 W k) P N s k)dk. 1) Since he demand curve and he cos of labor supply are idenical across differen labor ypes, he household chooses he same wage W for all labor ypes subjec o an opimal wage change a ime. Appendix A shows ha he opimal wage saisfies W P = µ w κ N s ) ω C ϕ G w, H w,, 11) where µ w θw. The recursive equaions describing G θ w 1 w, and H w, are presened in he appendix. Equaion 11) can be inerpreed as follows: In he absence of wage rigidiies α ω = ), he marginal rae of subsiuion beween labor and consumpion is κ N s ) ω C ϕ, and he opimal wage is his rae adjused by he opimal markup µ w. Wage rigidiies generae he ime-varying markup µ w G w, H w,, since he wage of some labor ypes is no adjused opimally. 3.2 Producion Secor The producion of differeniaed goods is characerized by monopolisic compeiion and price rigidiies in a coninuum of firms. Firms se he price of heir differeniaed goods in a Calvo 1983) saggered price seing: A each ime, wih probabiliy α p, a firm ses he price of he good as he previous period price adjused by he price indexaion facor Λ p, 1,. The specific funcional form of his facor is presened in Secion 4. Wih probabiliy 1 α p, he firm ses he produc price o maximize he presen value of profis. The maximizaion problem for firm j can 14

16 be wrien as max {P j)} E { [ αpm s,+s $ Λp,,+s P j)y +s j) W +s j)n+s j) ]} d, 12) s= subjec o he producion funcion Y +s j) = A +s N d +s j), 13) and he demand funcion ) θ P j)λ p,,+s Y +s j) = Y +s. 14) P +s The oupu Y +s j) is he producion of firm j a ime +s given ha he las opimal price change was a ime. The wage W +s j) and he labor demand N+s d j) have a similar inerpreaion. The producion problem akes ino accoun he probabiliy of no being able o adjus he price opimally in he fuure, and he corresponding indexaion Λ p,,+s. The producion funcion depends on labor produciviy A and labor. We assume ha labor produciviy conains difference- and rend-saionary componens. 15 Specifically, A = A p Z, where a log A p and z log Z, are he difference- and rend-saionary componens of produciviy, respecively. These componens follow he processes a +1 = 1 φ a )g a + φ a a + σ a ε a,+1, and z +1 = φ z z + σ z ε z,+1, 15) where is he difference operaor, g a is he average growh rae in he economy, and innovaions ε a, and ε z, IIDN, 1). For simpliciy, hroughou he paper we refer o he difference- and 15 The wo componens are incorporaed given he differen effecs on bond risk premia of hese wo processes for consumpion in endowmen economies. A difference-saionary process for consumpion wih posiive auocorrelaion coefficien generaes negaive erm premia. A rend-saionary process for consumpion wih posiive auocorrelaion coefficien generaes posiive erm premia. 15

17 rend-saionary componens as he permanen and ransiory shocks o produciviy, respecively. Labor demand is a composie of a coninuum of differeniaed labor ypes indexed by k [, 1] via he aggregaor [ 1 N d j) = N d j, k) θw 1 θw ] θw θw 1 dj, 16) where θ w > 1 is he elasiciy of subsiuion across differeniaed labor ypes. All firms ha se prices opimally are idenical and se he same opimal price P. Appendix B shows ha he opimal price saisfies ) P H p, = µ p W G p,, 17) P A P where µ p = θp. The recursive equaions for H θ p 1 p, and G p, are presened in he appendix. Equaion 17) can be inerpreed as follows: In he absence of price rigidiies, he produc price is he markup-adjused marginal cos of producion, wih opimal markup µ p. Price rigidiies generae he ime-varying markup µ p G p, H p,, since some firms do no adjus heir prices opimally. We define κ A p ) 1 ϕ o preserve balanced growh. I can be shown from equaion 11) ha wages and consumpion share he same average rend as long as κ A p ) 1 ϕ, and implies saionary labor. 3.3 Moneary Policy Moneary policy is described by he ineres-rae policy rule i = ρi ρ) [ ī + ı π π π 1) + ı x x x ss ) ] + u. 18) The policy rule has an ineres-rae smoohing componen capured by he sensiiviy ρ o he lagged erm, i 1, and responds o aggregae inflaion π log P P 1, he oupu gap x, and a policy 16

18 shock u. The oupu gap is defined as he log deviaion of oal oupu, Y, from he oupu in an economy under flexible prices and wages, Y f. Tha is, X Y Y f, and x log X. The coefficiens ı π and ı x capure he response of he moneary auhoriy o he deviaions of inflaion and he oupu gap from heir arges, respecively. The consan ī is defined as he nominal rae when he inflaion rae and he oupu gap are a heir arges, i.e., ī log β + ϕg a + g π. The process π denoes he ime-varying inflaion arge. The inflaion arge is ime-varying as in Ireland 27) and Rudebusch and Swanson 212). 16 Is process is π = 1 φ π )g π + φ π π 1 + σ π ε π,, 19) where ε π, IIDN, 1). The oupu gap arge x ss corresponds o he oupu gap in seady sae. The policy shocks u follow he process u +1 = φ u u + σ u ε u,+1, 2) where ε u, IIDN, 1). 3.4 Bond Prices and Yields Real and nominal defaul-free zero-coupon bonds wih mauriy a + n pay a uni of real and nominal consumpion, respecively, a mauriy. Their prices are B c,n) = exp nr n) ) = E [M,+n ], and B $,n) = exp ni n) ) = E [M,+n], $ 21) for real and nominal bonds, respecively, where r n) and i n) are he associaed real and nominal bond yields, and M,+n and M $,+n are he real and nominal discoun facors for payoffs a + n The inflaion arge has also been used in he macro finance lieraure by Bekaer, Cho and Moreno 21), Campbell, Pflueger and Viceira 214) and Dew-Becker 214). 17 Noice ha B c,n) is he real price of he real bond, while B $,n) 17 is he nominal price of he nominal bond.

19 3.5 Equilibrium Equilibrium requires produc, labor, and financial marke clearing. Produc marke clearing is characerized by C j) = Y j) for all j [, 1], and hen C = Y. Labor marke clearing requires ha supply and demand of labor ype k employed by firm j are equal, N s j, k) = N d j, k). As shown in appendix C, i implies he aggregae labor marke clearing condiion N s = N d F w, where N d = Y A F p,. The disorions F w, and F p, measure wage and price dispersion caused by wage and price rigidiies, respecively, and are defined in he appendix. Equilibrium in he financial marke implies ha he nominal ineres rae from household maximizaion in equaion 7) is equal o he ineres rae se by he moneary policy rule in equaion 18). Equilibrium implies he absence of arbirage opporuniies in real and nominal bond markes. Appendix D provides a summary of he equilibrium condiions. 3.6 Expeced Excess Bond Reurns and Risk Premia Risk differences beween shor- and long-erm bonds, and beween real and nominal bonds are analyzed in erms of differences in heir expeced reurns, risk premia, or implied yields. The link beween hese measures is presened in his secion. I allows us o decompose and quanify he compensaions for real and nominal risks in real and nominal bond yields. In paricular, real erm and inflaion risk premia are useful o decompose bond yields ino compensaions for real and nominal risks, respecively. The model deerminans of hese premia are analyzed in Secion 4. One-period gross bond reurns are R l,n),+1 Bl,n 1) +1 B l,n), for l = {c, $}. Real and nominal gross riskfree raes are Rf, c expr ) and R f, $ expi ), respecively. One-period expeced excess reurns [ ] [ ] relaive o he risk-free rae are E XR l,n),+1 = E R l,n),+1 Rf, l, and Sharpe raios are SRl,n) [ ] E ] ) XR l,n),+1 ), for l = {c, $}. In equilibrium, E [XR l,n),+1 = Rf, l cov M,+1, l XR l,n),+1, where σ XR l,n),+1 M c,+1 M,+1. Expeced excess bond reurns capure he compensaion for macroeconomic risk in long-erm bonds. This compensaion depends on he correlaion beween bond reurns and he 18

20 marginal uiliy of consumpion. The one-period real erm premium of an n-period real) bond is defined as rt P n) log E [ R c,n),+1 ] log Rf,. c 22) Appendix E shows ha his premium and he average spread r n) r can be approximaed as 18 rt P n) = cov m,+1, n 1)r n 1) +1 ) [ ], and E r n) r = J.I. r n) + 1 n 2 n s= [ E rt P n s) +s ], 23) respecively, where m,+1 log M,+1, and J.I. denoes Jensen s inequaliy erms no imporan for he analysis. The real erm premium capures he correlaion beween he marginal uiliy of consumpion and he bond one-period reurn. This reurn depends on he bond yield a he end of he period. A posiive correlaion beween marginal uiliy and he bond yield implies low bond real reurns during periods of high marginal uiliy and, herefore, posiive expeced excess bond reurns. The uncondiional yield spread can be seen as an average of one-period real erm premia during he life of he bond. The one-period inflaion risk premium πt P n) is he difference in log) real reurn for invesing in an n-period nominal bond over an n-period real bond for one-period. Tha is, πt P n) log E [ R $,n),+1p /P +1 ] log E [ R c,n),+1 ], 24) Appendix E shows ha his premium and he average spread i n) πt P n) = cov m,+1, ) n π +s s=1 [, and E i n) r n) ] r n) = E[π ] + J.I. π n) + 1 n can be approximaed as n [ ] E πt P n s) +s,25) 18 As shown in he appendix, his derivaion relies on he he assumpion of join normaliy for he log-pricing kernel and bond yields. This is used only for illusraion purposes, since he economic model is solved using a secondorder perurbaion mehod, which does no imply log-normaliy. Similar approximaions are used hroughou he paper for illusraion purposes only. Equaion 22) is used for he compuaion of real erm premia in he quaniaive analysis. 19 s=

21 The inflaion risk premium is hen an expeced reurn compensaion in nominal bonds for he correlaion beween he marginal uiliy of consumpion and inflaion. If his correlaion is posiive, he expeced real reurns of nominal bonds are higher han for real bonds: during periods of high marginal uiliy, high inflaion has a negaive impac on nominal bond reurns. The uncondiional spread beween nominal and real raes capures average inflaion and inflaion risk premia. 4 Model Implicaions and Analysis This secion repors and analyzes he main model implicaions for bond yields and risk premia. I describes firs he model esimaion and is quaniaive performance capuring macroeconomic and yield curve dynamics simulaneously. The main findings are highlighed by comparing he baseline model s performance wih alernaive model specificaions for nominal price and wage rigidiies, model shocks, and moneary policy. The economic mechanisms behind he quaniaive findings are explained based on approximae analyical soluions for real erm and inflaion risk premia, and heir link. 4.1 Esimaion Sraegy The purpose of he model esimaion is i) o examine he model s quaniaive abiliy o simulaneously capure observed macroeconomic and nominal yield curve dynamics, and ii) o provide a quaniaive framework for he economic analysis of he real yield curve and bond risk premia. Model parameers are chosen o capure key quarerly properies of U.S. daa for he period 1982:Q1 o 28:Q3 using he Generalized Mehod of Momens GMM). The sample period is chosen o focus on a moneary policy wih a sable response o economic condiions, which can be described by an ineres-rae policy rule. Clarida, Galí and Gerler 2) provide empirical evidence of a change in moneary policy afer The moneary experimen period is excluded since he shor-erm rae was replaced by moneary aggregaes as he policy insru- 2

22 men during his period. Daa afer he hird quarer of 28 are no included since he abiliy o conduc policy using he Federal Funds rae was limied by he zero bound afer December 28. Table 2 repors he parameer values for he baseline model. The model esimaion involves hree ses of parameers. 19 For he firs se, parameers values are assigned o mach a direc empirical counerpar or o be consisen wih he lieraure. The average produciviy growh rae g a is chosen o mach he average consumpion growh during he period. Non-opimal changes in prices and wages are assumed o be perfecly indexed o he inflaion arge, such ha log Λ p,,+1 = π, and log Λ w,,+1 = g a + π. The wage indexaion implies no deviaions from real wages on average. The price duraion of 1/ logα p ) 2.4 quarers is consisen wih he empirical evidence in Bils and Klenow 24). The wage duraion of 1/ logα w ) 4 quarers is consisen wih he evidence in Baraieri, Basu and Goschalk 214). The elasiciy parameers θ p and θ w imply price and wage markups of 2%. The value chosen for ω implies a Frisch labor elasiciy of 1/ω = 2, in he lower range of he values used in he macro lieraure o capure labor and wage dynamics. The policy responses o inflaion ı π = 1.5 and he oupu gap ı x =.125 are sandard in he lieraure. The persisence φ π =.9999 and volailiy σ π =.1% of he inflaion arge process are chosen o maximize he model s abiliy o capure he high volailiy of long-erm yields, and are in line wih he ones used by Rudebusch and Swanson 212), and he uni roo process in Campbell, Pflueger and Viceira 214). For a second se of parameers, values are esimaed using he Generalized Mehod of Momens. This procedure focuses on maximizing he model s abiliy o capure macroeconomic dynamics. Eigh parameer values are chosen o minimize percenage deviaions of nine model momens from heir daa counerpars. 2 The momens are he volailiy and auocorrelaion of consumpion growh, inflaion, wage growh, and he shor-erm nominal ineres rae, and he correlaion of 19 The paramerizaion has elemens of boh esimaion and calibraion. For simpliciy, we refer o i as esimaion hroughou he paper. The mehod is similar o ha in Andreasen, Fernández-Villaverde and Rubio-Ramírez 214). The model is solved using he Dynare package, available from 2 The esimaion is resriced wihin a range of parameer values ha are economically sensible. 21

23 consumpion growh and inflaion. The daa series are described in Secion The esimaed parameers are ϕ, ρ, and he persisence and volailiy parameers of produciviy and moneary policy shocks. The esimaed elasiciy of ineremporal subsiuion 1/ϕ =.5, is in he lower range of values in he macroeconomic lieraure, and conrass dramaically wih he values used in he asse-pricing long-run risk lieraure. The ineres-rae smoohing coefficien ρ.62 in he policy rule is slighly lower han he one esimaed by Clarida, Galí and Gerler 2) for he period, bu in line wih values used in he lieraure. The persisence of policy shocks φ u.4 is in he upper range of values esimaed in he lieraure. The persisence parameers for boh permanen and ransiory produciviy componens are lower han hose in Andreasen 212). Finally, values for he subjecive discoun facor β, he average inflaion arge g π, and he risk aversion parameer γ are chosen o mach he average annualized) inflaion rae of 3.26%, he shor-erm nominal annualized) ineres rae of 5.2%, and he Sharpe raio of.32 implied by excess reurns of he 5-year bond simulaneously. 22 The policy rule consan ī log β +ψg a +g π is he nominal rae when boh inflaion and he oupu gap are a heir respecive arges. The average coefficien of risk aversion in he presence of leisure preferences, as shown by Swanson 212), is given by 23 ϕ 1 + ϕ µ w + γ ϕ ω µ p 1 1 ϕ µ w ω µ p This value is comparable o hose used in models of he erm srucure wih recursive preferences. For insance, Piazzesi and Schneider 27) esimae a value of 59 in an endowmen economy, and Rudebusch and Swanson 212) and Andreasen 212) use values beween 75 and 11 in models 21 Allowing ω, φ π, and σ π o be esimaed implies a very similar performance maching hese momens. 22 The model is solved using a second-order approximaion around he non-sochasic seady sae. The high value for γ generaes large precauionary savings erms ha affec he means of inflaion and he shor-erm ineres rae. The precauionary savings erms are offse by a large values for g π, reducing is inerpreaion as a long-erm inflaion arge. The approach does no generae disorions in expeced excess bond reurns. 23 In he presence of recursive preferences on consumpion and labor, he coefficien of relaive risk aversion is no solely deermined by γ, since he abiliy o smooh consumpion using labor changes he represenaive agen s aiudes owards risk. The coefficien is compued relaive o ineremporal gambles on consumpion-relaed wealh, since he coefficien relaed o oal wealh including he value of leisure) is no well defined. 22

24 wih price rigidiies Quaniaive Performance of he Baseline Model This secion describes he model s abiliy o simulaneously mach macroeconomic and yield curve properies of he economy. The esimaion ceners almos enirely on maching macroeconomic momens, and uses only yield curve informaion o mach he Sharpe raio of he 5-year nominal bond. I is hen imporan o verify ha oher properies of he nominal yield curve are capured by he esimaion and provide a reasonable baseline for he analysis of he implied real yield curve. Table 3 repors momens for he baseline model and heir empirical counerpars. Simulaed 9% confidence inervals are added from samples wih he size of he daa sample 17 quarers). Panel A repors he macroeconomic momens. The model capures well he volailiies of consumpion growh and inflaion, he auocorrelaions of inflaion and wage growh, and he negaive correlaion beween consumpion growh and inflaion. This correlaion is imporan for explaining a posiive inflaion risk premium. The model, however, generaes lower volailiy in wage growh han in he daa, and fails o capure he repored empirical posiive auocorrelaion of consumpion growh. 25 As explained below, a negaive auocorrelaion of consumpion growh is crucial in he baseline model o generae negaive auocorrelaion in he pricing kernel and hen a posiive real erm premium. Secion 5 shows ha an exernal habi in consumpion preferences can simulaneously generae negaive and posiive auocorrelaions in he pricing kernel and consumpion growh, respecively, while preserving he main implicaions of he baseline model. Panels B and C of Table 3 repor yield curve and bond excess reurn saisics, respecively. 24 This value could be reduced by incorporaing persisen sources of long-run risk as in Bansal and Shaliasovich 213), or Kung 214). Bansal and Shaliasovich 213) achieve his in an endowmen economy wih exogenous inflaion. Kung 214) inroduces endogenous growh o a New Keynesian model and generaes an endogenous persisen source of long-run risk. We do no follow his approach o highligh he differen effecs of price and wage rigidiies and differen shocks in a sandard New Keynesian framework. 25 I is well known however, ha he auocorrelaion of consumpion growh is poorly measured and differen empirical sudies show mixed evidence. While Campbell and Mankiw 1989) and Cochrane 1994), for insance, find ha U.S. consumpion growh is almos unforecasable, Kandel and Sambaugh 1991) and Mehra and Presco 1985) imply a small persisen predicabiliy componen in his variable. 23

25 The baseline model implies an average 5-year nominal bond spread of 112 bps. vs. 138 bps. in he daa, and a posiive 5-year real bond spread of 82 bps. The model does a reasonable job capuring he volailiy of he shor-erm nominal ineres rae bu fails o reproduce he high volailiy of long-erm nominal yields. This is a well-known shorcoming of mos equilibrium models, given he lack of enough persisence in he explanaory macroeconomic variables. The model implicaions for he volailiy of real and nominal yields are explored in he analysis below. By consrucion, he model reproduces he Sharpe raio of he 5-year nominal bond, higher han he implied Sharpe raio for he comparable real bond. However, he one-quarer expeced bond excess reurn in he model is small relaive o he average realized excess reurn in he daa. I reflecs he model limiaion o capure he high volailiy of bond reurns. The posiive 5-year real bond spread implies a real erm premium and expeced excess reurn of around 1%. The higher expeced excess reurn for he comparable nominal bond reflecs a posiive inflaion risk premium of 86 bps. I is worh menioning addiional model implicaions no argeed in he esimaion. Panel E of Table 3 shows a volailiy of dividend growh in he model, σ d ), similar o he one in he daa. I implies an equiy premium E[XR d ] slighly below he daa counerpar. The R 2 s of regressions of yields on consumpion growh and inflaion also are presened in he able o highligh ha he model can reasonably capure he join dynamics of yields and macroeconomic variables. In summary, he baseline model provides a reasonable descripion of U.S. macroeconomic and yield dynamics, and hen a good framework for he quaniaive analysis of he real erm srucure. The model, however, has some limiaions ha will be addressed in Secion Alernaive model specificaions and comparison o similar models This secion presens resuls of alernaive esimaions for models wih only one rigidiy prices or wages) or none, and exposed o one or wo componens of produciviy shocks permanen and ransiory). A baseline model wih boh rigidiies and boh produciviy componens is chosen 24