Real and Nominal Equilibrium Yield Curves: Wage Rigidities and Permanent Shocks

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Real and Nominal Equilibrium Yield Curves: Wage Rigidities and Permanent Shocks Alex Hsu, Erica X.N. Li and Francisco Palomino, 2016-032 Please cite this paper as: Hsu, Alex, Erica X.N. Li, and Francisco Palomino. (2016). “Real and Nominal Equilibrium Yield Curves: Wage Rigidities and Permanent Shocks,” Finance and Economics Discussion Series 2016-032. Washington: Board of Governors of the Federal Reserve System, http://dx.doi.org/10.17016/FEDS.2016.032. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Real and Nominal Equilibrium Yield Curves: Wage Rigidities and Permanent Shocks ∗ Alex Hsu†, Erica X.N. Li‡, and Francisco Palomino§ April 12, 2016 Abstract The links between real and nominal bond risk premia and macroeconomic dynamics are explored quantitatively in a model with nominal rigidities and monetary policy. The estimated model captures macroeconomic and yield curve properties of the U.S. economy, implying significantly positive real term and inflation risk bond premia. In contrast to previous literature, both premia are positive and generated by wage rigidities as a compensation for permanent productivity shocks. Stronger policy-rule responses to inflation (output) increase (decrease) both premia, while policy surprises generate negligible risk premia. Empirical evidence of the economic mechanism is provided. JEL Classification: D51, E43, E44, E52, G12. Keywords: Termstructureofinterestrates,bondriskpremia,monetarypolicy,nominal rigidities. ∗WethankMaxCroce,CanlinLi,EricSwanson,andseminarparticipantsattheUniversityofMichigan FinanceBrownbag,FederalReserveBankofAtlanta,BancodelaRepu´blica,BankofCanadaFixedIncome Conference 2013, Society of Economic Dynamics Meeting 2013, 2013 China International Conference in Finance, Conference on Computing in Economics and Finance 2013, Latin American Econometric Society Meeting2013,DynareConference2013,CEPRGerzenseeAssetPricingMeeting2014,UBCWinterFinance Conference 2015, and Federal Reserve Board for helpful comments and suggestions. †Georgia Institute of Technology, Alex.Hsu@scheller.gatech.edu. ‡Cheung Kong Graduate School of Business, xnli@ckgsb.edu.cn §Board of Governors of the Federal Reserve System, francisco.palomino@frb.gov. Disclaimer: The material on this manuscript does not represent the views of the Board of Governors of the Federal Reserve System.

1 Introduction What are the economic drivers and sources of risk in real and nominal long-term bonds? Are these bonds risky or hedging instruments with positive or negative risk premia, respectively? Answeringthesequestionswillhelpustobetterunderstandimportantassetpricing dynamics, the portfolio diversification benefits of bonds, and the transmission of monetary policy, among others. Some answers have been recently provided using dynamic stochastic general equilibrium (DSGE) models.1 Most of these models imply risk premia that are (i) small or negative in real bonds (or real term premia), (ii) significantly positive in nominal bonds, and (iii) mainly a compensation for transitory shocks. Thus, positive risk premia in nominal bonds result from substantial positive inflation risk premia offsetting the implied negativerealtermpremia. Inthispaper, weinvestigateanalternativecharacterizationina DSGEframework, andprovidesupportingempiricalevidence: Bothrealtermandinflation risk premia are significantly positive as a compensation for permanent productivity shocks in the presence of nominal wage rigidities. While there is significant empirical support for positive nominal bond risk premia, our knowledge of the sign and size of real term premia is limited. Inflation-linked government bonds are imperfect substitutes of real bonds, their data are affected by illiquidity and mispricing,andevidenceontheirriskpropertiesismixedacrosscountriesandsub-periods.2 For instance, while 1999-2008 data support significantly negative and positive real term premia in the United Kingdom and the United States, respectively, data for 1985-2008 suggest small positive real term premia in the U.K. A theoretical model then can help us shed light into the properties of real term premia. 1See, for instance, Rudebusch and Swanson (2012), Dew-Becker (2014), and Kung (2014). 2See Garcia and van Rixtel (2007) for recent history on inflation-linked bond markets, D’Amico, Kim and Wei (2014) for evidence on significant time-varying liquidity premia in United States and United Kingdom inflation-linked bonds, and Fleckenstein, Longstaff and Lustig (2014) for evidence on mispricing in the TIPS market. 1

To our knowledge, this is the first paper emphasizing the important quantitative role of wage rigidities in combination with permanent productivity shocks to determine bond risk premia. Christiano, Eichenbaum and Evans (2005) show that nominal wage rigidities are both a salient feature of the data and an important element to capture some fundamental dynamics in economic models. Evidence on the relevance of permanent shocks in the economy, however, is mixed. While Campbell and Mankiw (1987) suggest that shocks to GDP are permanent, Christiano and Eichenbaum (1990) find it difficult to conclude whether economic shocks are transitory or permanent given data limitations. To overcome this difficulty, we estimate impulse responses of macroeconomic variables to permanent shocksinthedataandshowthattheirmodelcouterpartsonlyagreewiththeseresponsesin the presence of wage rigidities. This evidence provides support to the underlying economic mechanism generating positive real term and inflation risk premia in the model. Our model contains the standard elements of the New Keynesian framework, adding to the household’s preferences external habit formation in consumption and recursive utility over(habit-adjusted)consumptionandlabor. Andreasen(2012)andRudebuschandSwanson (2012), among others, use these ingredients to study bond risk premia with relative quantitative success.3 Our analysis is focused on understanding the contribution of three key elements to real and nominal bond risk premia. First, nominal price and wage rigidities. Both frictions generate real effects in monetary policy, but have different implications for economic dynamics, as highlighted in Christiano, Eichenbaum and Evans (2005). Second, productivity, monetary policy, and inflation-target shocks. Productivity shocks contain permanent (difference-stationary) and transitory (trend-stationary) components. As shown by Campbell (1986) and Labadie (1994), these two components have different implications for bond 3SeeRudebuschandSwanson(2008,2012)fordifferencesintheabilityofhabitformationandrecursive preferences, respectively, to capture macroeconomic and term structure dynamics simultaneously. 2

risk premia. Third, a nominal interest-rate policy rule. The response to economic conditions in this rule has important implications for the joint dynamics of real variables and inflation, and then for the link between real term and inflation risk premia.4 To assess the contribution of these elements, we use a Generalized Method of Moments (GMM) estimation of the model that matches key U.S. macroeconomic moments and nominal term premia well. There are two main implications from the model. The first implication is that permanent productivity shocks, in combination with wage rigidities, are crucial to generating large and positive real term and inflation risk premia. Permanent productivity shocks contribute to almost all the variability in the pricing kernel, and thus bond risk premia are mainly a compensation for this risk. Positive real term premia are the result of a negative autocorrelation in the pricing kernel induced by wage rigidities. Without rigidities, prices and wages decline after a negative permanent shock, labor supply increases to partially offset the effect of the shock, and habit-adjusted consumption growth drops persistently. The positive autocorrelation in habit-adjusted consumption growth is inherited by the pricing kernel. Inthepresenceofwage rigidities, wagesremainhighafteranegativeshock, leading to a lower increase in labor and lower consumption compared to the no rigidity case. As wages gradually decrease, labor improves and mitigates the negative effect of the shock on next period’s consumption. As a result, the habit-adjusted consumption growth (and the pricing kernel) become negatively autocorrelated, leading to positive real term premia. In addition, due to the higher wages in the presence of wage rigidities, producers set a higher product price to maintain their markups. Inflation then increases after a negative permanent shock and generates positive inflation risk premia. 4Alternatively, a more structural approach to monetary policy is to consider the monetary authority as a social planner that maximizes welfare, as in Palomino (2012). This approach may have different implications and is not explored in this paper. 3

The second implication from the model is that bond risk premia are considerably affected by the response to economic conditions in the interest-rate policy rule, but almost unaltered by surprises in monetary policy. A stronger response to inflation in the policy rule increases real term and inflation risk premia by increasing the sensitivity of the real rate (and pricing kernel) to permanent shocks. A stronger response to the output gap, or an increase in the interest-rate smoothing coefficient, has the opposite effect. In contrast, monetary policy and inflation-target shocks have negligible effects on bond risk premia given their transitory effects on the marginal utility of consumption Finally, we provide empirical support to our model mechanism by comparing data and model impulse responses to permanent productivity shocks for macroeconomic variables of interest. In particular, the response of inflation in the model changes sign in the presence of wage rigidities. We show that the the response of inflation in the data agrees with that of the model under wage rigidities. This paper contributes to the literature in which New Keynesian models (see Woodford 2003 for the standard framework) are used to analyze the term structure of interest rates.5 Rudebusch and Swanson (2012) rely on transitory productivity shocks and price rigidities to capture nominal yield curve properties, and do not study real yield curve implications. These elements, although present in our model, are not as quantitatively important as wage rigidities and permanent shocks. Andreasen (2012) incorporates both permanent and transitory components in productivity, and Dew-Becker (2014) adds wage rigidities. These studies focus on the time-variation in bond risk premia by fitting macroeconomic and yield dynamics. Our focus is different. The GMM approach allows us to target unconditional moments, provide quantitative comparisons across model specifications, and 5Buraschi and Jiltsov (2005) studies real and nominal bond risk premia in a monetary real business cycle model. This model also generates endogenous consumption growth and inflation. Their monetary policyandfrictionspecificationsaresubstantiallydifferentfromthoseofastandardNewKeynesianmodel. 4

focus on explaining the economic mechanisms behind bond risk premia. Kung (2014) uses an endogenous growth channel that is complementary to our model structure. Our paper is also related to term structure models with exogenous inflation.6 As an advantage, ourmodelgeneratesanendogenousnegativecorrelationofconsumptiongrowth andinflation,andlinksrealandnominalbondriskpremiafromfirstprinciples. Thisallows us to predict changes in the yield curve dynamics that are related to structural economic and policy changes. The paper is organized as follows. Section 2 describes the data and reports the descriptive statistics for nominal and inflation-linked bonds in the U.S. and the U.K. Section 3 describesthemodel. Section4providesdetailsofthemodelestimationanditsquantitative performance, presents its main implications, and explores the economic mechanism behind the results. Section 5 validates the impulse responses of the baseline model in the data, and Section 6 concludes. 2 Empirical Evidence This section presents descriptive statistics of inflation-linked and nominal government bonds in the U.K. and U.S. Inflation-linked government bonds are, at best, imperfect substitutes of real bonds, and have only been traded in the U.K. and U.S. since 1981 and 1997,respectively.7 Thereareseveraldifficultieswithexploringthepropertiesofrealbonds from the available data, motivating the joint theoretical analysis of real and nominal yields 6The endowment economy models with recursive preferences in Piazzesi and Schneider (2007) and BansalandShaliastovich(2013)implynegativerealtermpremia,whilethehabitmodelinWachter(2006) implies the opposite. The producton economy model in Van Binsbergen et al. (2012) implies negative real term premia and positive inflation risk premia. 7Theirinflationprotectionislimitedbyalaggedindexationtopricelevelsandtheembeddeddeflation optionality they provide. In addition, pricing in these markets has been affected by liquidity concerns and potentialunexploitedarbitrageopportunities. SeeD’Amico,KimandWei(2014)forevidenceonsignificant time-varying liquidity premia in U.S. and U.K. inflation-linked bonds. See Fleckenstein, Longstaff and Lustig (2014) for evidence of mispricing in the TIPS and inflation swaps markets. 5

that follows.8 WeusequarterlydatafromJanuary1985toSeptember2008.9 Thisdataperiodismotivatedbytworeasons. First,Britishinflation-linkedGiltsareonlyavailablesince1985. Second,theperiodSeptember-December2008coincideswiththecollapseofLehmanBrothers and a switch to unconventional monetary policy given the zero interest-rate bound. We stop in September 2008, since we want to focus on understanding the effects of a monetary authority setting the level of a short-term rate (conventional monetary policy) on bond yields. We report statistics for the whole sample and for the subsample January 1999 to September 2008, during which TIPS are actively traded in the U.S. The consumption growth and inflation series were constructed using quarterly data from the Bureau of Economic Analysis, following the methodology in Piazzesi and Schneider (2007). These series captureonlyconsumptionofnon-durablesandservicesanditsrelatedinflation. Thesedata are consistent with the variables of the economic model below. The data on zero-coupon nominalbondandTIPSyieldsareconstructedfollowingtheprocedureinGurkaynak, Sack and Wright (2006) and Gurkaynak, Sack and Wright (2008), respectively. The data are obtained from the Federal Reserve website. The short-term nominal interest rate is the 3-month T-bill from the Fama risk-free rates database. Data for British Gilts are obtained from the Bank of England website.10 8Thereisanextensiveempiricalliteratureontherealtermandinflationriskpremiawithandwithout inflation-linked bonds using no-arbitrage term structure models. This literature shows a wide range for the sign and size of inflation risk premia in the U.K. and in the U.S. An incomplete list includes Barr and Campbell (1997) Evans (1998), and Joyce, Lildholdt and Sorensen (2010) for the U.K., and Ang, Bekaert and Wei (2008), D’Amico, Kim and Wei (2014), Chen, Liu and Cheng (2010), Christensen, Lopez andRudebusch(2010),ChernovandMueller(2012),GrishchenkoandHuang(2013),andAbrahamsetal. (2013) for the U.S. Ho¨rdahl and Tristani (2012) provide a similar study for the eurozone. 9Results using comparable monthly data are similar. We present results for quarterly data to be consistent with the model estimation. 10Consumption and inflation data are constructed from Bureau of Economic Analysis (2015). “Personal Consumption Expenditures and ”Price Indexes for Personal Consumption Expenditures.” Accessed January, 2015. http://www.bea.gov/. Nominal bond and TIPS yields are obtained from FEDS Working Papers (2006, 2008). Accessed January, 2015. http://www.federalreserve.gov/pubs/feds/2006/200628/200628abs.html, and 6

Table 1: Descriptive Statistics of U.K. and U.S. Government Indexed and Nominal Bond Yields and Excess Returns, Consumption Growth, and Inflation Yields are annualized rates. Statistics are quarterly, non-annualized. Consumption growth is denoted by ∆c, inflation by πt, and the 3-month nominal rate by it. Excess returns on inflation-linked bonds are computed as logP indexed,t+1 −logP indexed,t +πt+1−it. Excess returns on nominal bonds are computed as logPnom,t+1− logPnom,t−it. Datasources: BEA(2015),FEDS(2015),CRSP(2015),andBankofEngland(2015). UnitedKingdom UnitedStates 1985:Q1-2008:Q3 1999:Q1-2008:Q3 1985:Q1-2008:Q3 1999:Q1-2008:Q3 Indexed Nominal Indexed Nominal Nominal TIPS Nominal Panel A: Bond Yields Average 2.5years 2.85 7.07 2.15 4.77 5.37 3.70 5years 2.88 7.14 2.04 4.81 5.92 2.27 4.24 10years 2.94 7.13 1.91 4.76 6.48 2.64 4.92 Standard Deviations 2.5years 0.94 2.54 0.80 0.73 2.07 1.54 5years 0.86 2.43 0.49 0.60 1.93 1.14 1.10 10years 0.98 2.38 0.40 0.38 1.75 0.88 0.76 Panel B: Bond Excess Returns Average 2.5years -0.22 0.15 0.00 0.06 0.34 0.26 5years -0.17 0.72 -0.04 0.08 0.74 0.79 0.53 10years -0.03 0.36 -0.01 0.02 1.22 1.02 0.77 Sharpe Ratios 2.5years -0.18 0.11 0.00 0.05 0.29 0.24 5years -0.10 0.12 -0.03 0.04 0.25 0.33 0.20 10years -0.01 0.12 -0.01 0.00 0.24 0.30 0.18 Panel C:Correlations with Macroeconomic Variables and Stock Returns Yields and Consumption Growth 2.5years 0.28 0.54 0.06 0.47 0.30 0.51 5years 0.39 0.54 0.16 0.44 0.28 0.42 0.54 10years 0.48 0.56 0.28 0.39 0.24 0.42 0.51 Yields and Inflation 2.5years 0.19 0.46 -0.18 -0.35 0.24 -0.08 5years 0.27 0.46 -0.14 -0.33 0.23 -0.34 -0.18 10years 0.33 0.47 -0.13 -0.28 0.24 -0.37 -0.27 ∆c π ∆c π ∆c π ∆c π Panel D: Macroeconomic Variables Average 1.47 0.77 1.10 0.57 0.43 0.78 0.36 0.78 Std. Deviation 0.90 0.62 0.72 0.49 0.37 0.33 0.37 0.32 Autocorrelation 0.45 0.40 0.11 -0.22 0.40 0.43 0.61 0.18 corr(∆c,π) 0.26 -0.21 -0.24 -0.32 https://www.federalreserve.gov/pubs/feds/2008/200805/200805abs.html. The U.S. 3-month T-bill rate is from The Center for Research in Security Prices (2015). “Fama Risk-Free Rates Database.” Accessed January, 2015. http://www.crsp.com/. British Gilts yields are from Bank of England (2015). “Interest and Exchange Rates.” Accessed January, 2015. http://www.bankofengland.co.uk/statistics/. 7

The properties of bond risk premia are frequently characterized by the average slope of a yield curve, the average excess bond returns relative to a risk-free rate, or the correlation of excess bond and stock returns. Panel A of Table 1 reports a slightly and a significantly upward-sloping average nominal yield curves in the U.K. and the U.S., respectively, indicating positive risk premia in nominal bonds. The picture is less clear for inflation-linked bonds. The average yield curve for these bonds is slightly upward sloping for the U.K. 1985-2008 sample, but becomes drastically downward sloping for the 1999-2008 sample. During the latter period, the comparable average yield curve in the U.S. is significantly upwardsloping. Thesefindingssuggestnegativeandpositiveriskpremiaininflation-linked bondsintheU.K.andtheU.S.respectively. TheaverageexcessreturnsinPanelBsupport these claims.11 Nominal bonds exhibit positive average excess returns for both the U.K. and U.S., and inflation-linked bonds in the U.K. and the U.S. have negative and positive average excess returns, respectively. However, the correlations between excess bond and stock returns in Panel C suggest a different story. While inflation-linked bond excess returns in the U.K. have shown positive correlations with stock excess returns in both samples, U.K. nominal bonds switch from a positive correlation for 1985-2008 to a negative one for 1999-2008. The opposite occurs for U.S. nominal bonds, while the correlation between TIPS and stock excess returns is negativefor1999-2008. AccordingtotheCAPM,theevidencefortherecentsampleimplies negative risk premia for U.K. and U.S. nominal and inflation-linked bonds, at odds with evidence from panels A and B.12 Panel C in Table 1 shows that the correlations of U.K. and U.S. inflation-linked and nominal bond yields with consumption growth are significantly positive during both sam- 11Excessbondreturnsarecomputedasthedifferenceofrealizednominalreturnsoninflation-linkedand nominal bonds with the respective 3-month nominal rate for each country. 12Thetime-varyingnatureofthecorrelationbetweennominalbondandstockreturnsishighlightedand studied by Viceira (2012) and Campbell, Sunderam and Viceira (2013). 8

ple periods. On the other hand, the correlations of these yields with inflation change from positive for 1985-2008 to negative for 1999-2008. These changes are accompanied by a reduced autocorrelation of inflation in both the U.K. and the U.S., higher and lower autocorrelations of consumption growth in the U.S. and the U.K respectively, and a correlation between consumption growth and inflation that is negative in the U.S. and switching from positive to negative in recent years in the U.K. This evidence can be linked to bond risk premia through a standard equilibrium model.13 In summary, the descriptive statistics do not provide a clear pattern to describe the salient properties of real bond risk premia and their links to macroeconomic variables. The theoretical model in Section 3 allows us to analyze the link between real and nominal bond risk premia and macroeconomic variables. This analysis can provide testable implications to better understand the dynamics of real bond yields. 3 Model The model is an extension of the standard New-Keynesian framework with price and wage rigidities (e.g. Woodford (2003)) to capture bond pricing dynamics. It incorporates recursive preferences with habit formation for the representative household. Recursive preferences, as in Rudebusch and Swanson (2012) and Li and Palomino (2014), are used to disentangle risk aversion from the elasticity of intertemporal substitution of consumption. This separation is useful to match observed macroeconomic dynamics, while a high degree 13Under constant relative risk aversion (CRRA) preferences, a positive autocorrelation of consumption growthimpliesnegativepremiaforrealbonds,andanegativecorrelationbetweenconsumptiongrowthand inflationimpliespositiveinflationriskpremia. Campbell(1986)showsthelinkbetweentheautocorrelation of consumption growth and the real yield curve under CRRA. The same intuition applies under recursive preferences on consumption, as shown in Bansal and Shaliastovich (2013). The Campbell and Cochrane (1999) habits model can imply the opposite, as shown by Wachter (2006). Piazzesi and Schneider (2007) highlight the link between positive inflation risk premia and the negative correlation between (expected) consumption growth and inflation under recursive preferences. 9

of risk aversion captures large expected excess returns. Nominal price and wage rigidities generate price and wage distortions that affect production decisions. Monetary policy in this setting affects inflation and real activities, thus impacting the riskiness of real and nominal bonds. 3.1 Household A representative household chooses consumption C and labor supply Ns to maximize the t t Epstein and Zin (1989) recursive utility function (cid:20) (cid:21)1−ϕ 1−γ 1−γ V = (1−β)U(C ,Ns)1−ϕ+βE V 1−ϕ , (1) t h,t t t t+1 where β > 0 is the subjective discount factor, ϕ and γ determine the elasticity of intertemporal substitution (EIS) and the coefficient of relative risk aversion, respectively, and C h,t is the habit-adjusted consumption, defined as C ≡ C −b C˜ .14 The external habit is h,t t h t−1 represented by lagged aggregate consumption C˜ , equal to C in equilibrium, but not t−1 t−1 determined directly by the household. This is a simplified Campbell and Cochrane (1999) habit specification. The intra-temporal utility depends on the habit-adjusted consumption and labor supply as (cid:32) C1−ϕ (Ns)1+ω (cid:33) 1− 1 ϕ U(C ,Ns) = h,t −κ t , (2) h,t t 1−ϕ t 1+ω where ω−1 > 0 captures the Frisch elasticity of labor supply and the process κ (specified t below) is chosen to ensure balanced growth. The consumption good is a basket of differentiated goods produced by a continuum of firms. Labor supply is the aggregate of a 14Theelasticityofintertemporalsubstitutionoftheutilitybundleofconsumptionandlaborisϕ−1. The coefficient of relative risk aversion is defined in Section 4. 10

continuum of different labor types supplied to the production sector.15 The representative consumer is subject to the intertemporal budget constraint (cid:34) ∞ (cid:35) (cid:34) ∞ (cid:35) (cid:88) (cid:88) E M$ P C ≤ E M$ P (LI +D ) , (3) t t,t+s t+s t+s t t,t+s t+s t+s t+s s=0 s=0 whereM$ isthenominaldiscountfactorforcashflowsattimet+s,P isthenominalprice t,t+s t ofaunitofthebasketofgoods,andLI andD arethereallaborincomeanddividendsfrom t t the production sector, respectively. The online Appendix shows that optimality implies the one-period real and nominal discount factors  ϕ−γ (cid:18) C (cid:19)−ϕ V 1/(1−ϕ) M = β h,t+1  t+1  , (4) t,t+1 C  (cid:104) (cid:105)1/(1−γ) h,t E V (1−γ)/(1−ϕ) t t+1 (cid:16) (cid:17)−1 and M$ = M Pt+1 , respectively. The one-period (continuously compounded) t,t+1 t,t+1 Pt real and nominal interest rates satisfy (cid:104) (cid:105) r = −logE [M ], and i = −logE M$ , (5) t t t,t+1 t t t,t+1 respectively. The nominal interest rate i is the instrument of monetary policy. t Wage rigidities follow Schmitt-Grohe and Uribe (2007). The representative household monopolistically provides a continuum of labor types indexed by k ∈ [0,1], subject to a demand schedule from the production sector.16 The household chooses wages W (k) for all t 15AdetailedpresentationofthemodelisgivenintheonlineAppendix. Weomitthesedetailsheresince they are standard in the literature. 16Thisapproachisdifferentfromthestandardheterogeneoushouseholdsapproachtomodelwagerigidities in Erceg, Henderson and Levin (2000), where each household supplies a differentiated type of labor. In the presence of recursive preferences, this approach introduces heterogeneity into the marginal rate of substitutionofconsumptionacrosshouseholdssinceitdependsonlabor. Weavoidthisdifficultyandobtain a unique marginal rate of substitution by modeling a representative agent who provides all different types of labor. 11

labor types k under Calvo (1983) staggered wage setting: At each time t, the household is only able to adjust wages optimally for a fraction 1−α of labor types. The remaining w fraction α of labor types adjust their previous period wages by the wage indexation factor w Λ . The online Appendix shows that the household chooses the same optimal wage w,t−1,t W∗ for all labor types subject to an optimal wage change at time t. This wage satisfies t W∗ G t = µ κ (Ns)ωCϕ w,t , (6) P w t t h,tH t w,t where µ ≡ θw , and θ is the elasticity of substitution across labor types. The recursive w θw−1 p equations describing G and H are presented in the Appendix. Equation (6) can be w,t w,t interpreted as follows: In the absence of wage rigidities (α = 0), the marginal rate of ω substitution between labor and consumption is κ (Ns)ωCϕ , and the optimal wage is this t t h,t rateadjustedbytheoptimalmarkupµ . Wagerigiditiesgeneratethetime-varyingmarkup w µ Gw,t, since the wages of some labor types are not adjusted optimally. wHw,t 3.2 Production Sector A continuum of firms indexed by j ∈ [0,1] set the prices of their differentiated goods in a Calvo (1983) staggered price setting. At each time t, with probability α , a firm sets p the price of its good as the previous period price adjusted by the price indexation factor Λ . With probability 1−α , the firm sets the product price to maximize the present p,t−1,t p value of expected profits, subject to a household’s demand function and the production function Y (j) = A Nd (j). (7) t+s|t t+s t+s|t 12

The output Y (j) is the production of firm j at time t+s given that the last optimal t+s|t price change was at time t. The labor demand Nd (j) has a similar interpretation. t+s|t The production function depends on labor productivity A and labor. We assume that t labor productivity contains difference- and trend-stationary components.17 Specifically, A = ApZ , where a ≡ logAp and z ≡ logZ , are the difference- and trend-stationary t t t t t t t components of productivity, respectively. These components follow the processes ∆a = (1−φ )g +φ ∆a +σ ε , and z = φ z +σ ε , (8) t+1 a a a t a a,t+1 t+1 z t z z,t+1 where ∆ is the difference operator, g is the average growth rate in the economy, and a innovations ε and ε ∼ IIDN(0,1). For simplicity, we refer to the difference- and a,t z,t trend-stationary components as the permanent and transitory shocks to productivity, respectively. Labor demand is a composite of a continuum of differentiated labor types. All firms that set prices optimally are identical and set the optimal price P∗. The online t Appendix shows that this price satisfies (cid:18) P∗(cid:19) µ W t H = p t G , (9) p,t p,t P A P t t t where µ = θp , θ is the elasticity of substitution across goods, and W is the aggregate p θp−1 p t wage. The recursive equations for H and G are presented in the Appendix. Equation p,t p,t (9) can be interpreted as follows: In the absence of price rigidities, the product price is the markup-adjusted marginal cost of production, with optimal markup µ . Price rigidip ties generate the time-varying markup µ Gp,t, since some firms do not adjust their prices pHp,t optimally. 17The two components are incorporated given the different effects on bond risk premia of these two processes for consumption in endowment economies. A difference-stationary process for consumption with a positive autocorrelation coefficient generates negative term premia. A trend-stationary process for consumption with positive autocorrelation coefficient generates positive term premia. 13

3.3 Monetary Policy Monetary policy is described by the interest-rate policy rule i = ρi +(1−ρ) (cid:2) ¯ı+ı (π −π(cid:63) )+ı (x −x ) (cid:3) +u . (10) t t−1 π t t−1 x t ss t The policy rule has an interest-rate smoothing component, captured by the sensitivity ρ to i , and responds to aggregate inflation π ≡ log Pt , the output gap x , and the policy t−1 t Pt−1 t shock u . The output gap x ≡ log Yt is the deviation of total output, Y , from the output t t Yf t t in an economy under flexible prices and wages, Yf. The coefficients ı and ı capture the t π x response of the monetary authority to the deviations of inflation and the output gap from their targets, π(cid:63) and x , respectively. The inflation target is time-varying as in Ireland t ss (2007) and Rudebusch and Swanson (2012).18 Its process is π(cid:63) = (1−φ )g +φ π(cid:63) +σ ε , (11) t π(cid:63) π π(cid:63) t−1 π(cid:63) π(cid:63),t where ε ∼ IIDN(0,1). The policy shocks u follow the process π(cid:63),t t u = φ u +σ ε , (12) t+1 u t u u,t+1 where ε ∼ IIDN(0,1). u,t 3.4 Equilibrium Equilibrium requires product, labor, and financial market clearing. Product market clearing implies that consumption is equal to the production of differentiated and final goods. Labor market clearing requires that the supply and demand for all labor types are equal. 18The inflation target has also been used in the macro finance literature by Bekaert, Cho and Moreno (2010), Campbell, Pflueger and Viceira (2014) and Dew-Becker (2014). 14

As shown in the online Appendix, this implies the aggregate labor market clearing condition Ns = NdF where Nd = YtF . The distortions F and F , described in the t t w,t t At p,t w,t p,t Appendix,arecausedbywageandpricerigidities,respectively. Equilibriuminthefinancial market implies that the nominal interest rate from the household maximization problem in equation (5) is equal to the interest rate set by the monetary policy rule in equation (10). Bond market clearing implies the absence of arbitrage opportunities in bond markets. The preference shock in equation (1) is defined as κ ≡ (Ap)1−ϕ to preserve balanced growth. t t The online Appendix provides a summary of the equilibrium conditions. 3.5 Expected Excess Bond Returns and Risk Premia Real and nominal default-free zero-coupon bonds with maturity at t+n pay a unit of real and nominal consumption, respectively, at maturity. Their prices are B c,(n) = E [M ], and B $,(n) = E [M$ ], (13) t t t,t+n t t t,t+n forrealandnominalbonds, respectively, whereM andM$ aretherealandnominal t,t+n t,t+n discount factors for payoffs at t+n.19 The associated real and nominal yields are r (n) = t −1 logB c,(n) and i (n) = −1 logB $,(n) , respectively. n t t n t Real term and inflation risk premia are useful to decompose bond yields and expected excess returns into compensations for real and nominal risks. The one-period real term premium of an n-period (real) bond is defined as (cid:104) (cid:105) rTP (n) ≡ logE R c,(n) −r , (14) t t t,t+1 t c,(n) where R is the one-period gross real bond return. The online Appendix shows that t,t+1 19Notice that Bc,(n) is the real price of the real bond, while B$,(n) is the nominal price of the nominal t t bond. 15

thispremiumcapturesthecorrelationbetweenthemarginalutilityofconsumptionandthe bond’s one-period return. A positive correlation between marginal utility and the bond yield implies low bond real returns during periods of high marginal utility and, therefore, positive expected excess bond returns. The Appendix also shows that the unconditional (n) yield spread r −r can be approximated as an average of one-period real term premia t t during the life of the bond.20 (n) The one-period inflation risk premium πTP is the difference in (log) real return for t investing in an n-period nominal bond over an n-period real bond for one-period. That is, (cid:104) (cid:105) (cid:104) (cid:105) πTP (n) ≡ logE R $,(n) P /P −logE R c,(n) , (15) t t t,t+1 t t+1 t t,t+1 $,(n) whereR istheone-periodgrossnominalbondreturn. TheonlineAppendixshowsthat t,t+1 thispremiumisthenanexpectedreturncompensationinnominalbondsforthecorrelation between the marginal utility of consumption and inflation. If this correlation is positive, the expected real returns of nominal bonds are higher than for real bonds: during periods ofhighmarginalutility, highinflationhasanegativeimpactonnominalbondreturns. The (n) (n) Appendix also shows that the unconditional expectation of the spread i −r between t t nominal and real rates captures average inflation and inflation risk premia. 4 Model Implications and Analysis This section describes the model estimation and its ability to capture macroeconomic and yieldcurvedynamics. Itpresentsthemainfindingsandhighlightstheunderlyingeconomic mechanisms by comparing different model specifications for nominal rigidities, shocks, and 20As shown in the Appendix, this derivation relies on the the assumption of joint normality for the log-pricing kernel and bond yields. This is used only for illustration purposes, since the economic model is solved using a second-order perturbation method, which does not imply log-normality. Equation (14) is used for the computation of real term premia in the quantitative analysis. 16

monetary policy. 4.1 Estimation Strategy Themodelestimationexaminestheabilityofthemodeltosimultaneouslycaptureobserved macroeconomic and nominal yield curve dynamics, and provides a quantitative framework for the economic analysis of the real yield curve and bond risk premia. Model parameters are chosen to capture key quarterly properties of U.S. data for the period 1982:Q1 to 2008:Q3 using the Generalized Method of Moments (GMM). The sample period is chosen to focus on a monetary policy with a stable response to economic conditions. Clarida, Gal´ı and Gertler (2000) provide empirical evidence of a change in monetary policy after 1979. The monetary experiment period 1979-1981 is excluded since the short-term rate was replaced by monetary aggregates as the policy instrument during this period. Data afterthethirdquarterof2008arenotincludedsincetheabilitytoconductpolicyusingthe Federal Funds rate was limited by the zero bound after December 2008. The data series are described in Section 2. Table 2 reports the parameter values for the baseline model. The model estimation involves three sets of parameters.21 For the first set, parameter values are assigned to match a direct empirical counterpart or to be consistent with the literature. The average productivity growth rate g is chosen to match the average consumption growth during the a period. Non-optimalchangesinpricesandwagesareassumedtobeperfectlyindexedtothe inflationtarget, suchthatlogΛ = π(cid:63), andlogΛ = g +π(cid:63).Thewageindexation p,t,t+1 t w,t,t+1 a t implies no deviation from real wages on average. The price duration of −1/log(α ) ≈ 2.4 p quarters is consistent with the empirical evidence in Bils and Klenow (2004). The wage 21The parametrization has elements of both estimation and calibration. For simplicity, we refer to it as “estimation”throughoutthepaper. ThemethodissimilartothatinAndreasen,Ferna´ndez-Villaverdeand Rubio-Ram´ırez (2014). The model is solved using the Dynare package, available from www.dynare.org. 17

Table 2: Model Parameter Values Parameter values for the baseline estimation of the economic model. Parameter Description Value Panel A: Preferences β Subjective discount factor 0.92107 ϕ Inverse of elasticity of intertemporal substitution 20 γ Risk aversion parameter 600 ω Inverse of Frisch labor elasticity 0.50 b External habit parameter 0.42 h Panel B: Product and Labor Rigidities and Elasticities α Price rigidity parameter 0.66 p θ Elasticity of substitution of differentiated goods 6 p α Wage rigidity parameter 0.78 w θ Elasticity of substitution of labor types 6 w Panel C: Interest Rate Rule ρ Interest-rate smoothing coefficient in policy rule 0.60 ı Response to inflation in the policy rule 1.8 π ı Response to output gap in the policy rule 0.125 x Panel D: Policy and Productivity Shocks φ Autocorrelation of policy shock 0.515 u σ ×102 Conditional vol. of policy shock 0.44 u φ Autocorrelation of permanent productivity shock 0.194 a σ ×102 Conditional vol. of permanent productivity shock 0.42 a φ Autocorrelation of transitory productivity shock 0 z σ ×102 Conditional vol. of transitory productivity shock 1.89 z Panel E: Growth Rates and Inflation Target g ×102 Unconditional mean of productivity growth 0.4695 a g Unconditional mean of inflation target 0.2251 π(cid:63) φ Autocorrelation of inflation target 0.9999 π(cid:63) σ ×102 Conditional volatility of inflation target 0.0010 π(cid:63) duration of −1/log(α ) ≈ 4 quarters is consistent with the evidence in Barattieri, Basu w andGottschalk(2014). Theelasticityparametersθ andθ implypriceandwagemarkups p w of 20%. The value chosen for ω implies a Frisch labor elasticity of 1/ω = 2, which is in the lower range of the values used in the macro literature to capture labor and wage dynamics. The policy responses to inflation ı = 1.8 and the output gap ı = 0.125 are standard in π x the literature. The persistence φ = 0.9999 and volatility σ = 0.001% of the inflation π(cid:63) π∗ target process are chosen to maximize the model’s ability to capture the high volatility of 18

long-term yields, and are in line with the ones used by Rudebusch and Swanson (2012), and the unit root process in Campbell, Pflueger and Viceira (2014). Forasecondsetofparameters,valuesareestimatedusingGMM.Thisprocedurefocuses on maximizing the model’s ability to capture macroeconomic dynamics. Nine parameter values are chosen to minimize the percentage deviations of nine model moments from their datacounterparts.22 Themomentsarethevolatilitiesandautocorrelationsofconsumption growth, inflation, wage growth, and the short-term nominal interest rate, and the correlation of consumption growth and inflation. The estimated parameters are ϕ, b , ρ, and h the persistence and volatility parameters of productivity and monetary policy shocks.23 The elasticity of intertemporal substitution 1/ϕ = 0.05, is in the lower range of values in the macroeconomic literature. The habit parameter value b = 0.42 is lower than those h reported in habit models, but not directly comparable given the recursive utility specification. The interest-rate smoothing coefficient ρ = 0.6 in the policy rule is slightly lower than the one estimated by Clarida, Gal´ı and Gertler (2000) for the period, but in line with the literature. The persistence of policy shocks φ ≈ 0.5 is in the upper range of values u estimated in the literature. The persistence parameters for both permanent and transitory productivity components are lower than those in Andreasen (2012). Finally, values for the subjective discount factor β, the average inflation target g , and π the risk aversion parameter γ are chosen to match the average (annualized) inflation rate of 3.26%, the short-term nominal (annualized) interest rate of 5.20%, and the Sharpe ratio and average spread implied by excess returns of the 5-year bond simultaneously.24 The 22The estimation is restricted within a range of parameter values that are economically sensible. 23Allowingω,φ π(cid:63),andσ π(cid:63) tobeestimatedimpliesaverysimilarperformancematchingthesemoments. 24The model is solved using a second-order approximation around the non-stochastic steady state. The high value for γ generates large precautionary savings terms that affect the means of inflation and the short-term interest rate. The precautionary savings terms are offset by a large values for g , reducing π its interpretation as a long-term inflation target. The approach does not generate distortions in expected excess bond returns. 19

policy rule constant¯ı ≡ −logβ+ψg +g is the nominal rate when both inflation and the a π output gap are at their respective targets. The average coefficient of risk aversion in the presence of leisure preferences, as shown by Swanson (2012), is given by25 ϕ γ −ϕ + ≈ 44. 1+ ϕµw 1− 1−ϕµw ω µp 1+ω µp Thisvalueiscomparabletothoseusedintermstructuremodelswithrecursivepreferences. For instance, Piazzesi and Schneider (2007) estimate a value of 59 in an endowment economy, andRudebuschandSwanson(2012)andAndreasen(2012)usevaluesbetween75and 110 in models with price rigidities.26 4.2 Quantitative Performance of the Model This section describes the model’s ability to simultaneously match macroeconomic and yield curve properties of the economy. The estimation centers almost entirely on matching macroeconomic moments, and uses only yield curve information to match the Sharpe ratio andaverageyieldspreadofthe5-yearnominalbond. Itisthenimportanttoverifythatthe estimationcancaptureotherpropertiesofthenominalyieldcurveandprovideareasonable baseline for the analysis of the implied real yield curve. Table 3 reports moments for the baseline model and their empirical counterparts in columns (2) and (1), respectively. Panel A reports the macroeconomic moments. The 25In the presence of recursive preferences on consumption and labor, the coefficient of relative risk aversion is not solely determined by γ, since the ability to smooth consumption using labor changes the representativeagent’sattitudestowardsrisk. Thecoefficientiscomputedrelativetointertemporalgambles on consumption-related wealth, since the coefficient related to total wealth (including the value of leisure) is not well defined. 26This value could be reduced by incorporating persistent sources of long-run risk as in Bansal and Shaliastovich (2013), or Kung (2014). Bansal and Shaliastovich (2013) achieve this in an endowment economywithexogenousinflation. Kung(2014)introducesendogenousgrowthtoaNewKeynesianmodel andgeneratesanendogenouspersistentsourceoflong-runrisk. Wedonotfollowthisapproachtohighlight thedifferenteffectsofpriceandwagerigiditiesanddifferentshocksinastandardNewKeynesianframework. 20

Table 3: Data and Baseline Model Implied Statistics - The Effect of Rigidities and Shocks The data statistics are for the 1982:Q1 to 2008:Q3 period. The parameter values of the baseline model are reported in Table 2. The operators E[·], σ(·), and AC(·) denote the unconditional mean, volatility, and first-order autocorrelation, respectively. rTP(20) and πTP(20) are the 5-year bond real term and inflation risk premia, respectively. “Baseline” indicates an economy with both price and wage rigidities and all four exogenous shocks. “WR” indicates no price rigidities (α = 0). “PR” indicates no wage p rigidities (α =0). “NR” indicates no price and wage rigidities (α = α = 0). “Only Ap” indicates only w p w permanent productivity shocks (σ z = σ u = σ π(cid:63) = 0). “Only Z” indicates only transitory productivity shocks (σ a = σ u = σ π(cid:63) = 0). “Only u” indicates only policy shocks (σ a = σ z = σ π(cid:63) = 0). “Only π(cid:63)” indicatesonlyshockstotheinflationtarget(σ =σ =σ =0). Thebaselinemodelstatisticcorrespondsto a z u theclosed-formaverageofthesecond-orderapproximationofthesolution. Volatilities,yields,and(excess) returns are in percentage terms. The inflation rate, yields, excess returns, and risk premia are annualized. Thedatastatisticsrelatedtotherealrater areobtainedfromtheestimatedrealrate. Valuesnotreported are not available. The values of β and g are adjusted across columns to match the average inflation and π short-term nominal rate. Data sources: BEA (2015), FEDS (2015), and CRSP (2015). Model Rigidities Shocks (1) (2) (3) (4) (5) (6) (7) (8) (9) Statistic Data Baseline WR PR NR Only Ap Only Z Only u Only π(cid:63) Panel A: Macroeconomic Variables σ(∆c) 0.38 0.34 0.35 0.32 0.32 0.34 0.01 0.05 0.00 σ(π) 1.36 1.83 10.80 3.13 12.38 0.16 1.02 0.89 0.65 σ(∆w) 0.66 0.52 2.71 2.59 2.71 0.36 0.21 0.14 0.00 σ(x) 0.00 0.13 0.10 0.09 0.00 0.07 0.08 0.10 0.02 AC(∆c) 0.42 0.41 0.35 0.44 0.48 0.42 0.01 0.25 0.00 corr(∆c,π) -0.15 -0.08 -0.29 0.35 0.01 -0.99 -0.96 0.41 0.00 Panel B: Real and Nominal Yield Curves E[i] 5.20 5.20 5.20 5.20 5.20 5.20 5.20 5.20 5.20 E[i(20)−i] 1.38 1.28 2.46 -4.78 -13.20 1.08 0.00 0.05 0.00 E[r] 1.98 1.37 -0.64 5.57 17.71 1.51 1.93 1.94 1.93 E[r(20)−r] 0.99 3.42 -5.41 -20.63 0.71 0.00 0.05 0.00 σ(i) 2.59 2.34 6.16 1.96 7.49 0.18 0.79 2.31 0.65 σ(r) 2.09 3.17 12.98 2.87 12.97 0.12 1.03 3.00 0.01 σ(r)/σ(i) 0.81 1.36 2.11 1.47 1.73 0.69 1.31 1.30 0.01 corr(i,r) 0.99 0.96 0.99 0.88 0.98 0.89 1.00 1.00 0.00 σ(i(20))/σ(i) 1.02 0.30 0.11 0.34 0.10 0.17 0.07 0.15 1.00 σ(r(20))/σ(r) 0.13 0.04 0.11 0.05 0.20 0.08 0.15 0.10 Panel C: Expected Excess Returns E[XR$,(20)] 4.28 2.27 4.05 -7.69 -20.03 1.90 0.00 0.09 0.00 E[XRc,(20)] 1.47 4.59 -7.06 -25.49 1.05 0.01 0.11 0.00 SR$,(20) 0.32 0.34 0.67 -1.69 -1.73 3.04 0.00 0.02 0.00 SRc,(20) 0.19 0.36 -1.16 -1.60 3.04 0.00 0.02 0.00 E[rTP(20)] 1.23 3.84 -5.96 -21.74 0.88 0.01 0.11 0.00 E[πTP(20)] 0.86 1.58 -2.91 -7.64 0.82 0.00 -0.04 0.00 21

model captures well the volatilities and autocorrelations of consumption and wage growth andinflation,aswellasthenegativecorrelationbetweenconsumptiongrowthandinflation. This correlation is important in explaining a positive inflation risk premium. Panels B and C of Table 3 report yield curve and bond excess return statistics, respectively. The baseline model implies an average 5-year nominal bond spread of 128 bps. vs. 138 bps. in the data, and a positive 5-year real bond spread of 99 bps. The model does a reasonable job at capturing the volatility of the short-term nominal interest rate but fails to reproduce the high volatility of long-term nominal yields. This is a well-known shortcoming of most equilibrium models. The Sharpe ratio of the 5-year nominal bond is higher than the implied Sharpe ratio for the comparable real bond. The one-quarter expected bond excess return in the model is small relative to the average realized excess return in thedata. Thisreflectsthemodel’slimitationtocapturethehighvolatilityofbondreturns. The positive 5-year real bond spread implies a real term premium of 1.23%. The higher expected excess return for the comparable nominal bond reflects a positive inflation risk premium of 86 bps. In summary, the baseline model provides a reasonable description of U.S. macroeconomic and yield dynamics, and thus it provides a good framework for the quantitative analysis of the real term structure. 4.3 Bond Risk Premia Thissectiondescribesthecontributionofnominalrigiditiesandshockstorealandnominal bond risk premia in the baseline model. The findings are illustrated using the results of alternative model specifications in Table 3. This table presents statistics of models that share the same parameter values with the baseline estimation, except for rigidity or shock parameters. It highlights the contribution of each rigidity and shock to the quantitative results. Columns (2)-(5) in this table are related to parameterizations where price or wage 22

rigidities or both are shut down, but all shocks affect the economy. Columns (6)-(9) are related to parameterizations where the economy is exposed only to one source of risk, but both rigidities are active.27 The main finding is that positive real term and inflation risk premia are mostly a compensation for permanent productivity shocks as a result of wage rigidities. Permanent productivity shocks generate most of the variability in consumption growth and the real and nominal pricing kernels.28 Consistent with this, column (6) in Table 3 shows that the 5-year real term and inflation risk premia for permanent productivity shocks are 88 and 82 bps., respectively, while columns (7)-(9) show that these premia are minor or negligible for the other shocks. A similar pattern is seen in bond spreads and expected excess returns. Columns (2)-(4) show that only wage rigidities generate significantly positive real and nominal risk premia in the baseline model. 4.4 The Mechanism Behind Bond Risk Premia Permanent productivity shocks, in combination with wage rigidities, lead to positive real termandinflationriskpremia. Themechanismbehindtheseresultscanbeillustratedusing model’simpulseresponses. Figure1presentstheimpulseresponsesofdifferentendogenous variables in the baseline model after a negative one-standard deviation permanent shock under four specifications: no nominal rigidities, only price rigidities, only wage rigidities, and both price and wage rigidities. Positive inflation risk premia can be understood from the responses of habit-adjusted consumption growth and inflation to permanent productivity shocks. After a negative shock, the supply of labor increases to mitigate the negative effect of productivity on 27Forcomparisonpurposes,βandg areadjustedacrossparametrizationstomatchtheaverageinflation π and short-term nominal rates. This adjustment has a minor effect on second moments. 28A variance decomposition of the pricing kernels shows that more than 99% of their variability is the result of permanent productivity shocks. 23

Figure 1: Impulse responses to a one-standard deviation negative permanent productivity shock. The parameter values of the baseline model are reported in Table 2. 24

consumption. Intheabsenceofwagerigidities,wagesdropsharplyandfirmsadjustproduct prices downwards to get closer to their optimal markups, resulting in lower inflation. With wage rigidities, wages do not decline as much, and increase relative to labor productivity. Firms increase product prices in response to higher marginal costs and inflation goes up. Therefore, inflationincreasesafteranegativepermanentshockonlyinthepresenceofwage rigidities, as shown in Figure 1. Inflation is high then in states of high marginal utility, resulting in positive inflation risk premia. Positive real term premia can be understood from the effect of wage rigidity on the autocorrelation of the habit-adjusted consumption growth. The magnitude of the labor response to permanent shocks is critical to understand their effect on habit-adjusted consumption. The labor increase after a negative permanent shock is more significant in the absence of rigidities than under price or wage rigidities. With price rigidities, some firms cannot reduce prices which reduces the demand of output and labor. With wage rigidities, the wages of some labor types cannot be adjusted down, which leads to lower labor demand. The impulse response of labor, n in Figure 1, indicates that the dampening effect of wage rigidities on labor is quantitatively stronger than that of price rigidities. Therefore, consumptionandhabit-adjustedconsumptiondeclinebymoreinthepresenceofwage rigidities after the negative productivity shock. To understand the effect of wage rigidity on the autocorrelation of habit-adjusted consumption growth, consider a shock at time t. The permanent nature of the negative shock implies that consumption continues to drop after the shock, leading to a positively autocorrelated consumption growth rate. However, the autocorrelation of the habit-adjusted consumption growth rate, ∆c , could be positive or negative, depending on the magh,t+1 nitude of the reduction in C . A lower C leads to lower ∆c but higher ∆c , which t t h,t h,t+1 decreases the autocorrelation of the habit-adjusted consumption growth rate. If the reduc- 25

tion in C is large enough, its positive impact on ∆c outweighs the negative effect of t h,t+1 the permanent shock. Consequently, after the negative permanent shock, current habitadjusted consumption growth declines, but the next period’s habit-adjusted consumption growth is expected to rise, consistent with the impulse responses of ∆c and E[∆c ] in h h Figure 1, respectively. Therefore, the autocorrelation of ∆c becomes negative. h,t Consistent with the response of labor, the impulse responses of ∆c in Figure 1 show h thatthehabit-adjustedconsumptioninthepresenceofwagerigiditiesreturnsmorequickly to its steady state after the initial drop. In our calibration, habit-adjusted consumption growth becomes negatively autocorrelated. The autocorrelation of the real pricing kernel largelydependsontheautocorrelationof∆c . Itthenbecomesnegativelyautocorrelated, h,t real interest rates rise, as shown in Figure 1, and real term premia are positive. It is important to notice the role of the habit in the results. In the absence of habit formation, the autocorrelation of the pricing kernel largely depends on the autocorrelation of consumptiongrowth. Wagerigiditiesandpermanentshocksleadtoprocyclicallaborinthis setting, and consumption growth becomes counterfactually negatively autocorrelated. The model, however, is still able to generate positive real term premia under wage rigidities.29 Habit formation is hence critical to simultaneously match the positive autocorrelation of consumption growth in U.S. data and obtain positive real term premia. 4.5 Bond Risk Premia and Monetary Policy The response to economic conditions in the policy rule affect both real term and inflation risk premia. Comparative statics for policy rule parameters are computed to quantify the sensitivity of these premia. The parameters are modified individually, keeping the remaining parameters at their baseline model levels. Selected moments are computed 29The results of the model without habit are presented in Section C of the online Appendix. 26

Table 4: Data and Baseline Model Implied Statistics - The Effects of Monetary Policy The data statistics are for the 1982:Q1 to 2008:Q3 period. The parameter values of the baseline model are reported in Table 2. The model columns report statistics for the baseline model estimation and for parametrizations where individual parameters in the policy rule i =ρi +(1−ρ)[¯ı+ı (π −π(cid:63) )+ı (x −x )]+u t t−1 π t t−1 x t ss t are modified to the values reported in each column. The operators E[·], σ(·), and AC(·) denote the unconditional mean, volatility, and first-order autocorrelation, respectively. rTP(20) and πTP(20) are the 5-year bond real term and inflation risk premia, respectively. The model statistic corresponds to the closed-form average of the second-order approximation of the solution. Volatilities, yields, and (excess) returns are in percentage terms. The inflation rate, yields, excess returns, and risk premia are annualized. The data statistics related to the real rate r are obtained from the estimated real rate. Values not reported are not available. Thevaluesofβandg areadjustedacrosscolumnstomatchtheaverageinflationandshort-term π nominal rate. Data sources: BEA (2015), FEDS (2015), and CRSP (2015). Model (1) (2) (3) (4) (5) Statistic Data Baseline ı =1.9 ı =0.25 ρ=0.7 π x Panel A: Macroeconomic Variables σ(∆c) 0.38 0.34 0.34 0.34 0.34 σ(π) 1.36 1.83 1.79 1.82 2.19 σ(∆w) 0.66 0.52 0.52 0.52 0.56 σ(x) 0.13 0.13 0.13 0.15 AC(∆c) 0.42 0.41 0.41 0.41 0.41 corr(∆c,π) -0.15 -0.08 -0.09 -0.09 -0.04 Panel B: Real and Nominal Yield Curves E[i] 5.20 5.20 5.20 5.20 5.20 E[i(20)−i] 1.38 1.28 1.37 1.20 1.11 E[r] 1.98 1.37 1.36 1.35 1.39 E[r(20)−r] 0.99 1.07 0.87 0.91 σ(i) 2.59 2.34 2.34 2.33 2.37 σ(r) 2.09 3.17 3.15 3.16 3.63 σ(r)/σ(i) 0.81 1.36 1.35 1.35 1.53 corr(i,r) 0.99 0.96 0.96 0.96 0.96 σ(i(20))/σ(i) 1.02 0.30 0.29 0.30 0.31 σ(r(20))/σ(r) 0.13 0.13 0.13 0.15 Panel C: Expected Excess Returns E[XR$,(20)] 4.28 2.27 2.37 2.13 2.02 E[XRc,(20)] 1.47 1.58 1.31 1.42 SR$,(20) 0.32 0.34 0.36 0.32 0.27 SRc,(20) 0.19 0.21 0.17 0.15 E[rTP(20)] 1.23 1.33 1.10 1.19 E[πTP(20)] 0.86 0.87 0.91 0.70 27

and compared with the baseline estimation in Table 4. Column (3) reports results when the response to inflation in the policy rule ı is increased from 1.8 to 1.9 in the baseline π estimation. Similarly, column (4) reports results when the response to the output gap ı is x increased from 0.125 to 0.25. A comparison of both columns with the baseline estimation in column (2) shows opposite effects of the two policy changes. While a stronger response to inflation increases real and nominal bond spreads, expected excess returns and Sharpe ratios, a stronger response to the output gap has the opposite effect. For instance, the 0.1 increase in ı is reflected in an increase in expected excess returns on real and nominal π 5-yearbondsof8and9bps., respectively. Anincreaseinı of0.125reducestheseexpected x returns by 14 and 16 bps., respectively. Changes in expected excess returns are explained by the effects of the policy rule on labor dynamics and then on interest rates. An increase in the response to inflation increases the response of labor to productivity shocks and real termpremia. Anincreaseintheresponsetotheoutputgaphastheoppositeeffect. Column (5)presentsresultsforapolicythatincreasestheinterest-ratesmoothingcoefficientρfrom the baseline value of 0.6 to 0.7. Similar to a reduction in the response to inflation, this policydecreasesrealandnominalbondspreads,expectedexcessreturns,andSharperatios. Finally, reducing the autocorrelation of the inflation target from 0.9999 to 0.9 (not shown in the table) does not have any significant effects on bond risk premia, given the small price of risk of inflation target shocks. 4.6 Model Extensions This section extends the model to incorporate (i) capital accumulation in the economy, and (ii) time variation in bond risk premia. 28

4.6.1 Capital Accumulation and Bond Risk Premia The baseline model economy has an only-labor production function. It is of interest to learn whether the bond risk premia mechanism and the results above hold under capital accumulation. Table 5 reports results for an estimation of a model with capital K and the t production function Y = (A Nd)1−αKα. Capital follows the process t t t t (cid:18) J (cid:19) (cid:18) J (cid:19) b (cid:18) J (cid:19)1−1/ζ t t 2 t K = (1−δ)K +Φ K , where Φ = b + t+1 t t 1 K K 1−1/ζ K t t t captures capital adjustment costs through ζ ≥ 0, J is investment, δ is the depreciation t rate, and b and b are parameters chosen to preserve balanced growth.30 The model 1 2 has a reasonable macroeconomic performance using an adjustment cost ζ = 4, similar to values reported in the literature. It matches the volatility of output and investment growth,thepositiveautocorrelationofconsumptiongrowth,andthenegativecorrelationof consumption growth and inflation. The output gap is highly volatile, and the correlations between real and nominal yields are lower. As in the baseline estimation, the real and nominalaverageyieldcurvesareupwardsloping, butwithsubstantiallysmallerandlarger, respectively, real term and inflation risk premia. 4.6.2 Stochastic Volatility and Time-Varying Bond Risk Premia The low volatility in bond risk premia in the baseline model is at odds with the welldocumented empirical evidence on deviations from the expectations hypothesis and bond return predictability. Adding time-varying volatility to productivity shocks captures substantialvariationinbondriskpremia. Considerthemodifiedspecificationsforproductivity 30Thecompletemodelspecification,equilibriumconditionsandestimatedparametersareavailableunder request. 29

Table 5: Data and Model Implied Statistics - Capital Accumulation Thedatastatisticsareforthe1982:Q1to2008:Q3period. Theparametervaluesofthebaselinemodelarereportedin Table2. TheoperatorsE[·],σ(·),andAC(·)denotetheunconditionalmean,volatility,andfirst-orderautocorrelation, respectively. rTP(20),andπTP(20) arethe5-yearbondrealtermandinflationriskpremia,respectively. “Baseline” indicates an economy with both price and wage rigidities and all four exogenous shocks. “Capital” indicates an economywithcapitalaccumulation. Themodelstatisticcorrespondstotheclosed-formaverageofthesecond-order approximationofthesolution. Volatilities, yields, and(excess)returnsareinpercentageterms. Theinflationrate, yields,excessreturns,andriskpremiaareannualized. Thedatastatisticsrelatedtotherealraterareobtainedfrom the estimated real rate. Values not reported are not available. All estimations use γ =400. The objective value is thesumofsquaredpercentagedifferencesbetweenthemodel-anddata-impliedmomentstargetedintheestimation. Datasources: BEA(2015),FEDS(2015),andCRSP(2015). Model Statistic Data Baseline Capital Panel A: Parameter values b 0.42 0.90 h ζ - 4.00 Panel A: Macroeconomic Variables Objectivevalue 0.29 0.67 σ(∆c) 0.38 0.35 0.34 σ(π) 1.36 1.73 1.90 σ(∆w) 0.66 0.51 0.29 σ(∆y) 0.65 0.35 0.82 σ(∆j) 2.45 - 2.46 σ(x) 0.14 1.55 AC(∆c) 0.42 0.41 0.40 AC(∆c ) 0.00 0.00 -0.23 h corr(∆c,π) -0.15 -0.08 -0.12 Panel B: Real and Nominal Yield Curve E[i] 5.20 5.20 5.20 E[i(20)−i] 1.38 0.65 0.58 E[r] 1.98 1.61 1.65 E[r(20)−r] 0.11 0.05 σ(i) 2.59 2.36 1.49 σ(r) 2.09 2.91 1.23 σ(r)/σ(i) 0.81 1.24 0.83 corr(i,r) 0.99 0.93 0.35 σ(i(20))/σ(i) 1.02 0.39 0.88 σ(r(20))/σ(r) 0.13 0.08 Panel C: Expected Excess Returns E[XR$,(20)] 4.28 1.14 0.94 E[XR(20)] 0.64 0.21 SR$,(20) 0.32 0.18 0.31 SR(20) 0.10 0.11 E[rTP(20)] 0.56 0.20 E[πTP(20)] 0.54 0.98 shocks in equation (30) given by ∆a = (1−φ )g +φ ∆a +σ eνa∆aε , and z = φ z +σ eνzztε , t+1 a a a t a a,t+1 t+1 z t z z,t+1 30

Table6: Data and Model Implied Statistics - The Effects of Stochastic Volatility in Shocks The operators E[·], and σ(·) denote the unconditional mean and volatility, respectively. rTP(20), and πTP(20) are the 5-year bond real term and inflation risk premia, respectively. The parameter values of the baseline model are reported in Table 2. “Baseline” indicates an economy with both price and wage rigidities and all four exogenous shocks. “Capital” indicates an economy with capital accumulation. Columns labeled as “No SV” corresponds to the case νa = νz = 0. Columns labeled νa = −100 and νz = 100 correspond to the specifications with stochastic volatilityinthepermanentandtransitorycomponentsinproductivity,respectively. Themodelstatisticcorresponds to the simulated average statistics for a sample of 1,000 periods of the third-order approximation of the solution. Volatilities, yields, (excess) returns, and risk premia are in percentage terms. The excess returns, and risk premia areannualized. Allestimationsuseγ=400. Datasources: BEA(2015),FEDS(2015),andCRSP(2015). Baseline Capital NoSV νa=−100 νz =−100 NoSV νa=−100 νz =−100 Panel A: Means E[XR$,(20)] 1.15 1.15 1.15 0.94 0.94 0.94 E[XRc,(20)] 0.64 0.64 0.64 0.20 0.20 0.21 E[rTPc,(20)] 0.56 0.56 0.56 0.20 0.20 0.21 E[πTPc,(20)] 0.54 0.54 0.54 0.98 0.98 0.98 Panel B: Standard Deviations σ(XR$,(20)) 0.03 0.40 0.17 0.12 0.13 0.12 σ(XRc,(20)) 0.02 0.18 0.21 0.01 0.10 0.08 σ(rTP(20)) 0.02 0.16 0.18 0.01 0.10 0.08 σ(πTP(20)) 0.02 0.22 0.08 0.13 0.15 0.13 where ν (cid:54)= 0 and ν (cid:54)= 0 capture time-variation in the conditional volatility of the shocks. a z Volatility depends on the level of the productivity component, avoiding the need for new statevariables. Table6reportsresultsfortwospecificationswithonlytime-varyingvolatility in only one productivity component at a time: ν = −100, or ν = −100, respectively. a z The negative signs capture the fact that volatility tends to increase during periods of high marginal utility. The magnitude implies that the level of volatility is 40% higher if a positive shock of size σ or σ , respectively, hits the economy. The table shows that bond risk a z premia become time-varying in specifications with stochastic volatility.31 In particular, volatility in permanent shocks in the model with habit persistence generates the largest variability in bond risk premia. Real term premia are more or less volatile than inflation riskpremiadependingonwhetherthestochasticvolatilityisinthepermanentortransitory 31A third-order perturbation of the model solution is required to capture the effects of time-varying volatility. Themodelmomentsarecomputedbasedonsimulationsthatusethepruningmethoddescribed in Andreasen, Ferna´ndez-Villaverde and Rubio-Ram´ırez (2014). 31

productivity components, respectively. 5 Permanent Shocks in the Data We construct empirical impulse responses to permanent shocks of several macroeconomic variables, and compare them with their model counterparts. The purpose of this exercise is to provide empirical support to the model mechanism generating bond risk premia. We expect to see an increase in inflation after a negative permanent shock to productivity. This only occurs in the model as a result of wage rigidities. Empirically, the first challenge is to determine a methodology that allows us to separately identify the permanent component and the transitory component of the productivity shock. Our identification strategy for the permanent productivity shock follows Altig, Christiano, Eichenbaum and Linde (2011) (ACEL henceforth), among others, where a ten-variable VAR is used to separate innovations to neutral and capital embodied technology growth. We use a modified seven-variable VAR to identify neutral shocks to productivity growth by making the first difference of log GDP per hour the first element in the VAR. The remaining six variables, in order, are: inflation, labor hours, wage growth, consumption growth, the output gap, and the Fed funds rate. The sample period spans from 1982:Q1 to 2008:Q3 to match the 32

estimation sample. Specifically, the VAR is:     ∆ln(GDP/Hours) ∆ln(GDP/Hours)          ∆ln(GDP deflator)   ∆ln(GDP deflator)           ln(Hours)   ln(Hours)            ∆ln(Wage)   = B¯ +B(L)  ∆ln(Wage)   +ε VAR,t ,          ∆ln(Consumption)   ∆ln(Consumption)           OutputGap   OutputGap          FFR FFR t t−1 where ∆ln(GDP deflator) is inflation and L is the lag operator. We choose to include four lags in the VAR, consistent with ACEL. The permanent component of productivity shock is the identified shock to the first element of the VAR, which means it impacts all the remaining variables without delay. For all details on the estimation strategy, see ACEL. Figure 2 shows the impulse responses of the VAR estimation. The left column shows that the empirical impulse responses agree with the theoretical IRs after a negative permanent productivity shock is realized: per hour output growth declines, inflation and hours increase, whilewagegrowth, consumptiongrowth, andtheoutputgapalldecrease, andthe Fed funds rate nudges up slightly. Inflation increases after the negative shock, consistent with the model response under wage rigidities. However, the 95% confidence band is too wide afterthe initialshock to definitivelyreport thatthis is thecase. This is notsurprising given the fact that there is some tension from price and wage rigidities on inflation. The theoretical impulse response of inflation in the figure also shows that inflation increases only slightly after a negative permanent shock when both types of rigidities are activated in the model. Finally, the right column of Figure 2 provides further evidence that our proposed esti- 33

IR from Real Data IR from Simulated Data h 0 tw o 0 rG -0.2 tu-0.2 p tu -0.4 O-0.4 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0.05 0.04 n o ita 0 0.02 lfn I -0.05 0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0.15 s ru 0.4 0.1 o H0.2 0.05 0 0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 h tw o 0 -0.1 rG e -0.2 -0.2 g a W -0.4 -0.3 h 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 tw 0 o rG 0.05 n 0 -0.1 o-0.05 itp m -0.1 -0.2 u-0.15 s n -0.3 o C 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 p a 0.2 G -0.02 tu p 0 tu -0.04 O -0.2 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0.03 0.2 0.02 R 0 F F 0.01 -0.2 0 -0.4 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 Figure 2: Impulse responses to a one-standard deviation negative permanent productivity shock constructed from a seven-variable VAR employing data from Altig, Christiano, Eichenbaum and Linde (2011) and a similar identification strategy. The left column uses real macroeconomic data, and the right column uses corresponding simulated data from the baseline model. The parameter values of the baseline model are reported in Table 2. 34

mationstrategyiscapturingthepermanentshocktoproductivityasallsimulatedvariables respond in the same fashion as we would expect from the theoretical impulse responses. Moreover, a comparison between the two sets of impulse responses indicates that the estimated reactions following the negative shock are much more precise on the simulated data as evidenced by the narrow confidence bands. Overall, the empirical exercise provides quantitative support to the model mechanism of permanent productivity shocks in combination with wage rigidities generating positive real term and inflation risk premia. 6 Conclusion This paper provides a quantitative analysis of bond risk premia in the presence of nominal rigidities, permanent shocks, and monetary policy. The model estimation implies positive real term and inflation risk premia for permanent productivity shocks as a result of wage rigidities. These properties are (i) consistent with those observed in the U.S. for inflationlinked and nominal bonds in recent years, (ii) different from previous literature where real term premia are negative, and (iii) supported by empirical evidence on macroeconomic variable responses to permanent shocks. The results have implications for the riskiness of nominal bonds and the effects of monetary policy on bond risk premia. Nominal bonds are risky not only because they involve a substantial positive compensation for inflation risk, but also because of positive real term premia. Regarding the interest-rate policy rule, a stronger response to inflation or a weaker response to output increase real term and inflation risk premia. The analysis can be extended in several dimensions. For instance, an empirical study of the model’s testable implications across countries. The model predicts lower real yield curve slopes in economies with more flexible wages. This is consistent with the average inverted real yield curve in the U.K., and the findings in Smith (2000) and Dickens et al. 35

(2007)oflessrigidwagesintheU.K.thaninU.S.Also, theframeworkcanbeusedtolearn about the effects of optimal monetary policy on real rates and their economic content. 36

A U.S. and U.K. Inflation-Linked Bonds and Macroeconomic Data WeusequarterlydatafromJanuary1985toSeptember2008fortheU.S.andtheU.K.,andreportstatistics for the periods 1985-2008 and 1999-2008. The data sample periods are motivated by two reasons. First, TIPS data in the U.S. and inflation-linked gilts data in the U.K. are only available since 1999 and 1985, respectively.32 Second,theperiodSeptember-December2008coincideswiththecollapseofLehmanBrothers that drove short-term interest rates close to zero, and triggered a switch to unconventional monetary policies. The period after September 2008 is then not covered to focus on the effects on bond yields of a (conventional) monetary policy conducted using an interest-rate rule. TheconsumptiongrowthandinflationseriesfortheU.S.areconstructedusingquarterlydatafromthe Bureau of Economic Analysis, following the methodology in Piazzesi and Schneider (2007). These series captureonlyconsumptionofnon-durablesandservicesanditsrelatedinflation,andthenconsistentwiththe model variables. Wages are real wages per hour of non-farm business from the Federal Reserve Economic Data (FRED) database from the Federal Reserve Bank of St. Louis. The data on U.S. zero-coupon nominal bond and TIPS yields are constructed following the procedure in Gurkaynak, Sack and Wright (2006, 2008), respectively. These data are obtained from the Federal Reserve website. The short-term nominal interest rate is the 3-month T-bill rate from the Fama risk-free rates database. The three-month realrateisestimatedusingthemethodologydescribedinPfluegerandViceira(2011).33 Dividendsandstock marketreturnscorrespondtothemarketportfolioobtainedfromtheCenterforResearchinSecurityPrices (CRSP). For the U.K., consumption growth and inflation are obtained directly from the FRED database. The historical yields for U.K. real and nominal bonds are taken from the Bank of England website. The three-month real rate in the U.K. is estimated using the same methodology used to estimate the U.S. real rate. Stock returns are for the UK FTSE All-Shares Index. The bond yields under study correspond to maturitiesfrom2to10years. Thelongendofthecurveshasbeenexcludedforcomparisonpurposesacross countries. GreenwoodandVayanos(2010)documentasignificanteffectonlong-terminflation-linkedbond yieldsintheU.K,resultingfromtheincreaseddemandfrompensionfundstomeettheMinimumFunding Requirements. Table 1 summarizes the empirical evidence. B Model We model a production economy with a representative household, a production sector for differentiated goods,andmonetarypolicy. Therepresentativehouseholdderivesutilityfromtheconsumptionofabasket of goods and disutility from supplying differentiated labor to the production sector. Labor and product marketsarecharacterizedbymonopolisticcompetitionandnominalwageandpricerigidities,respectively. Monetary policy is modeled as an interest-rate policy rule that reacts to economic conditions. All markets are complete. Default-free real and nominal bonds are in zero net supply. The model can be seen as an extension of the standard New-Keynesian framework (see Woodford (2003), for instance) to capture bond pricing dynamics. It incorporates recursive preferences with habit formation for the representative 32Results using comparable monthly data are very similar. We present results for quarterly data to be consistent with the model estimation. The same macroeconomic and term structure data for the United States are used to estimate the model, for the longer period January 1982 to September 2008. 33Specifically, the computation is based on the regression i −π =constant+β i +β (i −π )+ε , t t+1 i t r t−1 t t where i is the three-month nominal rate and π is the three-month inflation rate. The real rate is then t t computedasr =i −E [π ]undertheassumptionthattheinflationriskpremiuminthree-monthnominal t t t t+1 bonds is negligible. 37

household. Recursive preferences, as in Rudebusch and Swanson (2012) and Li and Palomino (2014), are used to disentangle risk aversion from the elasticity of intertemporal substitution of consumption. This separation allows us to match observed macroeconomic dynamics by choosing an appropriate level for the elasticity of intertemporal substitution, while increasing the degree of risk aversion to capture large expected excess returns. Nominal prices and/or wages that are not adjusted optimally generate relative price and wage distortions that affect production decisions. In this setting, different monetary policy rules havedifferentimplicationsoninflationandrealactivity. Asaresult,thedynamicsandriskinessofrealand nominal bond yields are affected by both nominal rigidities and monetary policy. This section describes the characteristics of the model economy. B.1 Household A representative agent chooses consumption C and labor supply Ns to maximize the Epstein and Zin t t (1989) recursive utility function V =(1−β)U(C ,Ns)1−ϕ+βE (cid:20) V 1 1 − − ϕ γ(cid:21)1 1 − − ϕ γ , (16) t h,t t t t+1 whereβ >0isthesubjectivediscountfactor,ϕandγ determinetheelasticityofintertemporalsubstitution (EIS)andthecoefficientofrelativeriskaversion,respectively,andC isthehabit-adjustedconsumption, h,t defined as C ≡ C −b C˜ .34 The external habit is represented by lagged aggregate consumption h,t t h t−1 C˜ , equal to C in equilibrium, but not determined directly by the household. This is a simplified t−1 t−1 Campbell and Cochrane (1999) habit specification. The recursive utility formulation relaxes the strong assumption of γ =ϕ implied by constant relative risk aversion. The intra-temporal utility is defined over the habit-adjusted consumption and labor supply as (cid:32) C1−ϕ (Ns)1+ω (cid:33) 1− 1 ϕ U(C ,Ns)= h,t −κ t , (17) h,t t 1−ϕ t 1+ω whereω−1 >0capturestheFrischelasticityoflaborsupply,andtheprocessκ ischosentoensurebalanced t growth (it is specified in the production sector section below). Theconsumptiongoodisabasketofdifferentiatedgoodsproducedbyacontinuumoffirms. Specifically, the consumption basket is (cid:20)(cid:90) 1 θp−1 (cid:21) θp θ − p 1 C t = C t (j) θp dj , (18) 0 where θ > 1 is the elasticity of substitution across differentiated goods, and C (j) is the consumption of p t thedifferentiatedgoodj. Laborsupplyistheaggregateofacontinuumofdifferentlabortypessuppliedto the production sector, such that (cid:90) 1 Ns = Ns(k)dk, (19) t t 0 where Ns(k) is the supply of labor type k. t 34Theelasticityofintertemporalsubstitutionoftheutilitybundleofconsumptionandlaborisϕ−1. The coefficient of relative risk aversion is defined in Section 4 of the paper. 38

The representative consumer is subject to the intertemporal budget constraint (cid:34) ∞ (cid:35) (cid:34) ∞ (cid:35) E (cid:88) M$ P C ≤E (cid:88) M$ P (LI +D ) , (20) t t,t+s t+s t+s t t,t+s t+s t+s t+s s=0 s=0 where M$ is the nominal discount factor for cash flows at time t+s, P is the nominal price of a unit t,t+s t of the basket of goods, LI is the real labor income from supplying labor to the production sector, and D t t is the real dividend from owning the production sector. Itcanbeshownthatthehousehold’soptimalityconditionsimplythattheone-periodrealandnominal discount factors are  ϕ−γ (cid:18) C (cid:19)−ϕ V1/(1−ϕ) (cid:18) P (cid:19)−1 M =β h,t+1  t+1  , and M$ =M t+1 , (21) t,t+1 C  (cid:104) (cid:105)1/(1−γ) t,t+1 t,t+1 P h,t E V(1−γ)/(1−ϕ) t t t+1 respectively. Theone-period(continuouslycompounded)realandnominalinterestratesareobtainedfrom (cid:104) (cid:105) r =−logE [M ], and i =−logE M$ , (22) t t t,t+1 t t t,t+1 respectively. The nominal interest rate i is the instrument of monetary policy. t B.1.1 Wage Setting FollowingSchmitt-GroheandUribe(2007),animperfectlycompetitivelabormarketismodeledwherethe representativehouseholdmonopolisticallyprovidesacontinuumoflabortypesindexedbyk∈[0,1].35 The supply of labor type k satisfies the demand equation (cid:18) W (k) (cid:19)−θw Ns(k)= t Nd, (23) t W t t where Nd is the aggregate labor demand of the production sector, W (k) is the wage for labor type k, and t t W is the aggregate wage index given by t W = (cid:20)(cid:90) 1 W1−θw(k)dk (cid:21) 1− 1 θw . (24) t t 0 The labor demand equation (23) is obtained from the production sector problem presented in the section below. ThehouseholdchooseswagesW (k)foralllabortypeskunderCalvo(1983)staggeredwagesetting. t Specifically, at each time t, the household is only able to adjust wages optimally for a fraction 1−α of w labor types. The remaining fraction α of labor types adjust their previous period wages by the wage w indexationfactorΛ . ThespecificfunctionalformofthisfactorispresentedinSection4ofthepaper. w,t−1,t The optimal wage maximizes (16), subject to demand functions (23) for all labor types k, and the budget 35ThisapproachisdifferentfromthestandardheterogeneoushouseholdsapproachtomodelwagerigiditiesinErceg,HendersonandLevin(2000),whereeachhouseholdsuppliesadifferentiatedtypeoflabor. In thepresenceofrecursivepreferences,thisapproachintroducesheterogeneityinthemarginalrateofsubstitution of consumption across households since it depends on labor. We avoid this difficulty and obtain a uniquemarginalrateofsubstitutionbymodelingarepresentativeagentwhoprovidesalldifferenttypesof labor. 39

constraint (20). Notice that real labor income is given by (cid:90) 1 W (k) LI = t Ns(k)dk. (25) t P t 0 t Sincethedemandcurveandthecostoflaborsupplyareidenticalacrossdifferentlabortypes,thehousehold choosesthesamewageW∗ foralllabortypessubjecttoanoptimalwagechangeattimet. Itcanbeshown t that the optimal wage satisfies W∗ G t = µ κ (Ns)ωCϕ w,t , (26) P w t t h,tH t w,t where µ ≡ θw . The recursive equations describing G and H are presented in the appendix. w θw−1 w,t w,t Equation (26) can be interpreted as follows: In the absence of wage rigidities (α =0), the marginal rate ω ofsubstitutionbetweenlaborandconsumptionisκ (Ns)ωCϕ ,andtheoptimalwageisthisrateadjusted t t h,t by the optimal markup µ . Wage rigidities generate the time-varying markup µ Gw,t, since the wage of w wHw,t some labor types is not adjusted optimally. B.2 Production Sector The production of differentiated goods is characterized by monopolistic competition and price rigidities in a continuum of firms. Firms set the price of their differentiated goods in a Calvo (1983) staggered price setting: At each time t, with probability α , a firm sets the price of the good as the previous period price p adjusted by the price indexation factor Λ . The specific functional form of this factor is presented in p,t−1,t Section 4 of the paper. With probability 1−α , the firm sets the product price to maximize the present p value of profits. The maximization problem for firm j can be written as max E (cid:40) (cid:88) ∞ αsM$ (cid:104) Λ P (j)Y (j)−W (j)Nd (j) (cid:105) (cid:41) , (27) t p t,t+s p,t,t+s t t+s|t t+s|t t+s|t {Pt(j)} s=0 subject to the production function Y (j)=A Nd (j), (28) t+s|t t+s t+s|t and the demand function (cid:18) P (j)Λ (cid:19)−θ Y (j)= t p,t,t+s Y . (29) t+s|t P t+s t+s TheoutputY (j)istheproductionoffirmj attimet+sgiventhatthelastoptimalpricechangewasat t+s|t timet. ThewageW (j)andthelabordemandNd (j)haveasimilarinterpretation. Theproduction t+s|t t+s|t problem takes into account the probability of not being able to adjust the price optimally in the future, and the corresponding indexation Λ . p,t,t+s The production function depends on labor productivity A and labor. We assume that labor product tivity contains difference- and trend-stationary components.36 Specifically, A =ApZ , where a ≡logAp t t t t t and z ≡ logZ , are the difference- and trend-stationary components of productivity, respectively. These t t 36The two components are incorporated given the different effects on bond risk premia of these two processes for consumption in endowment economies. A difference-stationary process for consumption with positive autocorrelation coefficient generates negative term premia. A trend-stationary process for consumption with positive autocorrelation coefficient generates positive term premia. 40

components follow the processes ∆a =(1−φ )g +φ ∆a +σ ε , and z =φ z +σ ε , (30) t+1 a a a t a a,t+1 t+1 z t z z,t+1 where ∆ is the difference operator, g is the average growth rate in the economy, and innovations ε and a a,t ε ∼ IIDN(0,1). For simplicity, throughout the paper we refer to the difference- and trend-stationary z,t components as the permanent and transitory shocks to productivity, respectively. Labordemandisacompositeofacontinuumofdifferentiatedlabortypesindexedbyk∈[0,1]viathe aggregator N t d(j)= (cid:20)(cid:90) 1 N t d(j,k) θw θw −1 dj (cid:21) θw θw −1 , (31) 0 where θ >1 is the elasticity of substitution across differentiated labor types. w AllfirmsthatsetpricesoptimallyareidenticalandsetthesameoptimalpriceP∗. Appendix??shows t that the optimal price satisfies (cid:18) P∗(cid:19) µ W t H = p tG , (32) P p,t A P p,t t t t where µ = θp . The recursive equations for H and G are presented in the appendix. Equation (32) p θp−1 p,t p,t can be interpreted as follows: In the absence of price rigidities, the product price is the markup-adjusted marginal cost of production, with optimal markup µ . Price rigidities generate the time-varying markup p µ Gp,t, since some firms do not adjust their prices optimally. pHp,t We define κ ≡(Ap)1−ϕ to preserve balanced growth. It can be shown from equation (26) that wages t t and consumption share the same average trend as long as κ ∝(Ap)1−ϕ, and implies stationary labor. t t B.3 Monetary Policy Monetary policy is described by the interest-rate policy rule i =ρi +(1−ρ)[¯ı+ı (π −π(cid:63) )+ı (x −x )]+u . (33) t t−1 π t t−1 x t ss t Thepolicyrulehasaninterest-ratesmoothingcomponentcapturedbythesensitivityρtothelaggedterm, i , and responds to aggregate inflation π ≡ log Pt , the output gap x , and a policy shock u . The t−1 t Pt−1 t t outputgapisdefinedasthelogdeviationoftotaloutput,Y ,fromtheoutputinaneconomyunderflexible t prices and wages, Yf. That is, X ≡ Yt , and x ≡logX . The coefficients ı and ı capture the response t t Yt f t t π x ofthemonetaryauthoritytothedeviationsofinflationandtheoutputgapfromtheirtargets,respectively. Theconstant¯ıisdefinedasthenominalratewhentheinflationrateandtheoutputgapareattheirtargets, i.e.,¯ı≡−logβ+ϕg +g . The process π(cid:63) denotes the time-varying inflation target. The inflation target a π t is time-varying as in Ireland (2007) and Rudebusch and Swanson (2012).37 Its process is π t (cid:63) =(1−φ π(cid:63))g π +φ π(cid:63)π t (cid:63) −1 +σ π(cid:63)ε π(cid:63),t , (34) where ε π(cid:63),t ∼IIDN(0,1). The output gap target x ss corresponds to the output gap in steady state. The policy shocks u follow the process t u =φ u +σ ε , (35) t+1 u t u u,t+1 37The inflation target has also been used in the macro finance literature by Bekaert, Cho and Moreno (2010), Campbell, Pflueger and Viceira (2014) and Dew-Becker (2014). 41

where ε ∼IIDN(0,1). u,t B.4 Bond Prices and Yields Real and nominal default-free zero-coupon bonds with maturity at t+n pay a unit of real and nominal consumption, respectively, at maturity. Their prices are (cid:16) (cid:17) (cid:16) (cid:17) Bc,(n) =exp −nr(n) =E [M ], and B$,(n) =exp −ni(n) =E [M$ ], (36) t t t t,t+n t t t t,t+n for real and nominal bonds, respectively, where r(n) and i(n) are the associated real and nominal bond t t yields, and M and M$ are the real and nominal discount factors for payoffs at t+n.38 t,t+n t,t+n B.5 Equilibrium Equilibriumrequiresproduct,labor,andfinancialmarketclearing. Productmarketclearingischaracterized byC (j)=Y (j)forallj ∈[0,1],andthenC =Y . Labormarketclearingrequiresthatsupplyanddemand t t t t of labor type k employed by firm j are equal, Ns(j,k) = Nd(j,k). It implies the aggregate labor market t t clearingconditionNs =NdF whereNd = YtF . ThedistortionsF andF measurewageandprice t t w,t t At p,t w,t p,t dispersion caused by wage and price rigidities, respectively, and are defined in the appendix. Equilibrium in the financial market implies that the nominal interest rate from household maximization in equation (22) is equal to the interest rate set by the monetary policy rule in equation (33). Equilibrium implies the absence of arbitrage opportunities in real and nominal bond markets. Here we provide a summary of the equilibrium equations for the model. These conditions need to be expressed in terms of de-trended variables. In order to obtain balanced growth, κ ≡ κ (Ap)1−ϕ. This t 0 t condition ensures that Y , W , W∗, C , and C share the same average trend. Therefore, the equations t t t t h,t can be written in stationary form in terms of Yˆ = Yt, Wˆ = Wt, Wˆ∗ = Wt ∗ , Cˆ = Ct, and Cˆ = Ch,t t Ap t t Ap t t Ap t t Ap t h,t Ap t Wage setting W∗ G t = µ κ (Ns)ωCϕ w,t . P w t t h,tH t w,t (cid:34) (cid:18) Nd (cid:19)(cid:18) W (cid:19)−θw (cid:35) H = 1+α E M$ Λ−θw t+1 t H , w,t w t t,t+1 w,t,t+1 Nd W w,t+1 t t+1 (cid:34) (cid:18) P (cid:19)(cid:18) C (cid:19)ϕ(cid:18) Nd (cid:19)(cid:18) κ (cid:19)(cid:18) Ns (cid:19)ω(cid:18) W (cid:19)−θw (cid:35) G = 1+α E M$ Λ−θw t+1 h,t+1 t+1 t+1 t+1 t G . w,t w t t,t+1 w,t,t+1 P C Nd κ Ns W w,t+1 t h,t t t t t+1 Price dispersion (cid:90) 1(cid:18) P (j) (cid:19)−θp (cid:18) P∗(cid:19)−θp (cid:18) P (cid:19)−θp F = t dj =(1−α ) t +α Λ−θp t−1 F . p,t P p P p p,t−1,t P p,t−1 0 t t t Wage dispersion (cid:90) 1(cid:18) W (k) (cid:19)−θw (cid:18) W∗(cid:19)−θw (cid:18) W (cid:19)−θw F = t dk=(1−α ) t +α Λ−θw t−1 F . w,t W w W w w,t−1,t W w,t−1 0 t t t 38Notice that Bc,(n) is the real price of the real bond, while B$,(n) is the nominal price of the nominal t t bond. 42

Wage aggregator (cid:18) W (cid:19)1−θw (cid:90) 1(cid:18) W (k) (cid:19)1−θw (cid:18) W∗(cid:19)1−θw (cid:18) P (cid:19)1−θw(cid:18) W (cid:19)1−θw t = t dk=(1−α ) t +α Λ1−θw t−1 t−1 , P P w P w w,t−1,t P P t 0 t t t t−1 Price setting (cid:18) P∗(cid:19) µ W t H = p tG , P p,t A P p,t t t t (cid:34) (cid:18) Y (cid:19)(cid:18) P (cid:19)−θp (cid:35) H = 1+α E M$ Λ1−θp t+1 t H , p,t p t t,t+1 p,t,t+1 Y P p,t+1 t t+1 (cid:34) (cid:18) Y (cid:19)(cid:18) P (cid:19)−θp(cid:18) W (cid:19)(cid:18) A (cid:19) (cid:35) and G = 1+α E M$ Λ−θp t+1 t t+1 t G . p,t p t t,t+1 p,t,t+1 Y P W A p,t+1 t t+1 t t+1 Price aggregator (cid:18) P∗(cid:19)1−θp (cid:18) P (cid:19)1−θ 1 = (1−α ) t +α Λ1−θp t−1 . p P p p,t−1,t P t t Aggregate labor supply and demand Y Ns =F Nd, Nd = tF . t w,t t t A p,t t Pricing kernel M = (cid:34) β (cid:18) C h,t+1 (cid:19)−ϕ (cid:35) 1 1 − − ϕ γ (cid:18) 1 (cid:19)1− 1 1 − − ϕ γ , t,t+1 C R h,t Q,t+1 R Q,t+1 = (1−ν t )R Ch,t+1 +ν t R LI∗,t+1 , R Ch,t+1 = C h,t+1 S + Ch S ,t Ch,t+1, R LI∗,t+1 = LI t ∗ +1 S + LI S ∗ L ,t I∗,t+1, ν = ν¯S LI∗,t . t ν¯S LI∗,t −S Ch,t Real and nominal bond yields (cid:16) (cid:17) (cid:104) (cid:16) (cid:17)(cid:105) (cid:16) (cid:17) (cid:104) (cid:16) (cid:17)(cid:105) exp −nr(n) = E M exp −(n−1)r(n−1) , exp −ni(n) =E M$ exp −(n−1)i(n−1) . t t t,t+1 t+1 t t t,t+1 t+1 Indexation logΛ = π(cid:63), logΛ =g +π(cid:63). p,t,t+1 t w,t,t+1 a t Policy rule i =ρi +(1−ρ)[¯ı+ı (π −π(cid:63) )+ı (x −x )]+u . t t−1 π t t−1 x t ss t Goods market clearing Y =C . t t 43

Habit C = C −b C , h,t t h t−1 Flexible price and wage economy Cf = Cf −b Cf , h,t t h t−1 Yf = Cf, t t (cid:16) (cid:17)ω(cid:16) (cid:17)ϕ A1+ω Yf Cf = t . t h,t µ µ κ p w t Output steady state G 1 = µ µ κ (Y )ω(C −b C )ϕ w,ss, p w t ss ss h ss H w,ss (cid:16) (cid:17)ω(cid:16) (cid:17)ϕ 1 = µ µ κ Yf Cf −b Cf , p w t ss ss h ss Y = C , ss ss x = y −yf . ss ss ss B.6 Expected Excess Bond Returns and Risk Premia Risk differences between short- and long-term bonds, and between real and nominal bonds are analyzed in terms of differences in their expected returns, risk premia, or implied yields. The link between these measures is presented in this section. It allows us to decompose and quantify the compensations for real and nominal risks in real and nominal bond yields. In particular, real term and inflation risk premia are useful to decompose bond yields into compensations for real and nominal risks, respectively. The model determinants of these premia are analyzed in Section 4 of the paper. One-period gross bond returns are R(cid:96),(n) ≡ B t (cid:96) + ,( 1 n−1) , for (cid:96) = {c,$}. Real and nominal gross risk-free t,t+1 Bt (cid:96),(n) rates are Rc ≡ exp(r ) and R$ ≡ exp(i ), respectively. One-period expected excess returns relative to f,t t f,t t the risk-free rate are E (cid:104) XR(cid:96),(n) (cid:105) = E (cid:104) R(cid:96),(n) (cid:105) −R(cid:96) , and Sharpe ratios are SR(cid:96),(n) ≡ Et (cid:104) XR t (cid:96) , , t ( + n) 1 (cid:105) , t t,t+1 t t,t+1 f,t t σt (cid:16) XR t (cid:96) , , t ( + n) 1 (cid:17) (cid:104) (cid:105) (cid:16) (cid:17) for (cid:96) = {c,$}. In equilibrium, E XR(cid:96),(n) = −R(cid:96) cov M(cid:96) ,XR(cid:96),(n) , where Mc ≡ M . t t,t+1 f,t t t,t+1 t,t+1 t,t+1 t,t+1 Expected excess bond returns capture the compensation for macroeconomic risk in long-term bonds. This compensation depends on the correlation between bond returns and the marginal utility of consumption. The one-period real term premium of an n-period (real) bond is defined as (cid:104) (cid:105) rTP(n) ≡ logE Rc,(n) −logRc . (37) t t t,t+1 f,t Appendix C shows that this premium and the average spread r(n)−r can be approximated as39 t t rTP(n) =cov (cid:16) m ,(n−1)r(n−1) (cid:17) , and E (cid:104) r(n)−r (cid:105) =J.I.(n)+ 1 n (cid:88) −2 E (cid:104) rTP(n−s) (cid:105) , (38) t t t,t+1 t+1 t t r n t+s s=0 39As shown in the appendix, this derivation relies on the the assumption of joint normality for the logpricingkernelandbondyields. Thisisusedonlyforillustrationpurposes,sincetheeconomicmodelissolved using a second-order perturbation method, which does not imply log-normality. Similar approximations areusedthroughoutthepaperforillustrationpurposesonly. Equation(37)isusedforthecomputationof real term premia in the quantitative analysis. 44

respectively, where m ≡ logM , and J.I. denotes Jensen’s inequality terms not important for the t,t+1 t,t+1 analysis. Therealtermpremiumcapturesthecorrelationbetweenthemarginalutilityofconsumptionand the bond one-period return. This return depends on the bond yield at the end of the period. A positive correlation between marginal utility and the bond yield implies low bond real returns during periods of highmarginalutilityand,therefore,positiveexpectedexcessbondreturns. Theunconditionalyieldspread can be seen as an average of one-period real term premia during the life of the bond. The one-period inflation risk premium πTP(n) is the difference in (log) real return for investing in an t n-period nominal bond over an n-period real bond for one-period. That is, (cid:104) (cid:105) (cid:104) (cid:105) πTP(n) ≡ logE R$,(n)P /P −logE Rc,(n) , (39) t t t,t+1 t t+1 t t,t+1 Appendix C shows that this premium and the average spread i(n)−r(n) can be approximated as t t πTP(n) =cov (cid:32) m , (cid:88) n π (cid:33) , andE (cid:104) i(n)−r(n) (cid:105) =E[π ]+J.I.(n)+ 1 (cid:88) n E (cid:104) πTP(n−s) (cid:105) , (40) t t t,t+1 t+s t t t π n t+s s=1 s=0 The inflation risk premium is then an expected return compensation in nominal bonds for the correlation between the marginal utility of consumption and inflation. If this correlation is positive, the expected real returns of nominal bonds are higher than for real bonds: during periods of high marginal utility, high inflation has a negative impact on nominal bond returns. The unconditional spread between nominal and real rates captures average inflation and inflation risk premia. C Bond Risk Premia C.1 Real Term and Inflation Risk Premia Consider the no arbitrage equation for the n-period real bond: B t c,(n) = e−nrt (n) =E t (cid:104) M t,t+1 B t c + ,( 1 n−1) (cid:105) =E t (cid:20) emt,t+1−(n−1)r t ( + n− 1 1) (cid:21) , where m ≡logM . Assuming normality and homoskedasticity for the log-pricing kernel and bond t,t+1 t,t+1 yields, it follows that (cid:20) (cid:21) e−nrt (n) = E t [emt,t+1]E t e−(n−1)r t ( + n− 1 1) e−covt(mt,t+1,(n−1)r t ( + n− 1 1)). The equation above also implies 1 (cid:16) (cid:17) (cid:104) (cid:105) nr(n) =r − var (n−1)r(n−1) +rTP(n)+E (n−1)r(n−1) . t t 2 t t+1 t t t+1 Solving for the last term iteratively and applying unconditional expectations, we get (cid:16) (cid:17) rTP(n) =cov logM ,(n−1)r(n−1) . (41) t t t,t+1 t+1 Consider the inflation risk premium in equation (43) for n=1, πTP(1) =cov (m ,π )=i −r +logE [exp(−π )]. (42) t t t,t+1 t+1 t t t t,t+1 45

In general, the inflation risk premium in equation (43) can be written in terms of bond yields as πTP t (n) = n(i t (n)−r t (n))+logE t (cid:20) e (cid:16) −(n−1)i( t n + − 1 1)(cid:17)(cid:21) −logE t (cid:20) e (cid:16) −(n−1)r t ( + n− 1 1)(cid:17)(cid:21) (cid:16) (cid:17) + logE [e(−πt,t+1)]+cov (n−1)i(n−1),π . t t t+1 t+1 From equation (42), the recursive bond pricing equation e−ni( t n) =e−itE t (cid:20) e−(n−1)i t (n + − 1 1) (cid:21) e−covt (cid:16) m$ t,t+1 ,(n−1)i t (n + − 1 1)(cid:17) , where m$ ≡logM$ , and a similar equation for the comparable real bond yield, it follows that t,t+1 t,t+1 (cid:16) (cid:17) (cid:16) (cid:17) πTP(n) = πTP(1)+cov m$ ,(n−1)i(n−1) −cov m ,(n−1)i(n−1) t t t t,t+1 t+1 t t,t+1 t+1 (cid:16) (cid:17) +cov π ,(n−1)i(n−1) t t+1 t+1 (cid:16) (cid:16) (cid:17)(cid:17) = πTP(1)+cov m$ ,(n−1) i(n−1)−r(n−1) , t t t,t+1 t+1 t+1 where the second equality follows from m = m$ +π . Realizing that under log-normality and t,t+1 t,t+1 t+1 homoskedasticity assumptions the nominal-real bond spread is (n−1) (cid:16) i(n−1)−r(n−1) (cid:17) = n (cid:88) −1 E [π ]− 1 var (cid:32)n (cid:88) −1 π (cid:33) −cov (cid:32)n (cid:88) −1 m , n (cid:88) −1 π (cid:33) . t+1 t+1 t t+s 2 t t+s t t,t+s t+s s=1 s=1 s=1 s=1 Since the variance and covariance terms are constant, it follows that (cid:32) n−1 (cid:33) πTP(n) =cov m , (cid:88) π . t t t,t+1 t+s s=1 Computingtheunconditionalexpectationofthenominal-realbondspreadaboveandreplacingthecovariance terms for the one-period inflation risk premia, we get E [exp(−π )] 1 πTP(n) ≡ log t t,t+n −log =cov (m ,π ), (43) t B$,(n) B(n) t t,t+n t,t+n t t C.2 Understanding the Mechanism Tounderstandthemechanismofwhatdrivestherealandnominaltermstructures,wederivetheloglinear analyticalsolutionofthemodelwithouthabit.40 Forthemodelwithouthabit,allvariablescanbeexpressed as a loglinear function of the state variables, ∆a and z . t t The labor-only linear production technology in equation (28) implies that aggregate consumption is Nd C =ApZ t , t t tF F p,t w,t where the difference-stationary shocks a ≡ logAp and the trend-stationary shocks z ≡ logZ follow t t t t the processes in equations (30), and F and F are distortions generated by price and wage rigidities, p,t w,t respectively. It can be shown that a first-order approximation of the distortions implies F ≈ 1 and p,t F ≈1. We use this approximation for simplicity. It implies that Ns =Nd =N . w,t t t t 40The analytical solution of the model with habit is too complicated to illustrate the intuitions. 46

Notice that when prices and wages are perfectly flexible, consumption growth becomes (cid:18) (cid:19) (cid:20) (cid:18) (cid:19)(cid:18) (cid:19)(cid:21) 1+ω 1−ϕ 1 ∆c =∆a + ∆z , and Q =C 1− . t t ω+ϕ t t t 1+ω µ µ p w Thatis,thedividendofthewealthportfolioisproportionaltoconsumptionand,then,thereturnonwealth is a “levered” claim on the return on the consumption claim. Consider the recursive preferences on consumption and labor in equation (16) and its associated real pricing kernel in equation (21). Under the change of variable v˜≡ (1−ϕ)−1log(V /C ), these preferences t t can be written as (cid:20) (cid:18) (cid:19) (cid:21) (1−ϕ)v˜ t =log (1−β) 1− 1 1 + − ω ϕ e(ω+ϕ)nt−(1−ϕ)zt +βe (1 1 − +ω ψ)logEt [exp((1−γ)(v˜t+1+∆ct+1))] . A log-linear approximation of this term implies v˜ = constant+η n +η z +η E [v˜ +∆c ] t n t z t vc t t+1 t+1 ∞ = constant+ (cid:88) ηs E [η ∆c +η n +η z ], (44) vc t vc t+1+s n t+1+s z t+1+s s=0 where η , η , and η are appropriate approximation constants, and the second equality follows from n z vc V1/(1−ϕ) solving the first equation recursively. The term t+1 in the pricing kernel can be written Et (cid:104) V t ( + 1− 1 γ)/(1−ϕ)(cid:105)1/(1−γ) in log-form as 1 v˜ +∆c − logE [exp((1−γ)(v˜ +∆c ))]. t+1 t+1 1−γ t t+1 t+1 Replacing equation (44), and realizing that πTP(2) =cov (m ,r ), and r =constant+ϕE [∆c ], t t t,t+1 t+1 t t t+1 we can write the real pricing kernel as ∞ logM =logβ−ϕ∆c −(γ−ϕ) (cid:88) ηs (E −E )[∆c +η n +η z ], (45) t,t+1 t+1 vc t+1 t t+1+s n t+1+s z t+1+s s=1 and the one-period real term premium in a 2-period bond as rTP(2) = −ϕ2cov (∆c ,E [∆c ]) t t t+1 t+1 t+2 ∞ − (γ−ϕ)ϕ (cid:88) ηs cov (E [∆c +η n +η z ],E [∆c ]) . (46) vc t t+1 t+1+s n t+1+s z t+1+s t+1 t+2 s=1 The real pricing kernel also can be written in terms of the return on wealth R as Q,t M = (cid:34) β (cid:18) C t+1 (cid:19)−ϕ (cid:35) 1 1 − − ϕ γ (cid:20) 1 (cid:21)1− 1 1 − − ϕ γ , where Q =C (cid:20) 1− (cid:18) 1−ϕ (cid:19) κ (cid:18) N t ϕ+ω(cid:19)(cid:21) t,t+1 C t R Q,t+1 t t 1+ω 0 Z t 1−ϕ is the dividend associated to the wealth portfolio. The log-pricing kernel can be written as (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) 1−γ 1−γ ϕ−γ m = logβ−ϕ ∆c + r . t,t+1 1−ϕ 1−ϕ t+1 1−ϕ q,t+1 47

The log-return on wealth, r can be approximated as q,t+1 (cid:18) 1−ϕ (cid:19) (1−ϕ)2 r =η¯ +η p +∆q −p , where ∆q =∆c − (ω+ϕ)κ¯∆n + κ¯∆z q,t+1 q q q,t+1 t+1 q,t t t 1+ω t 1+ω t is the wealth-dividend ratio for appropriate approximation constants η¯ , η , and κ¯. q q Assume that labor follows the process n = n¯+n ∆a +n z , where n¯, n , and n are determined t a t z t a z in equilibrium. From this process, the consumption growth processes ∆c = ∆a +∆n , and the not t t arbitragepricingequation1=E [exp(m +r )],itcanbeshownthatthewealth-dividendratiocan t t,t+1 q,t+1 be approximated as p =p¯ +p ∆a +p z , q,t q q,a t q,z t where (cid:18) (cid:19)(cid:20) (cid:18) (cid:18) (cid:19)(cid:19)(cid:21) 1−ϕ ω+ϕ p = φ −(1−φ )n 1−κ¯ , q,a 1−η φ a a a 1+ω q a (cid:20) (cid:18) (cid:19)(cid:21) (1−φ )(1−ϕ) 1−ϕ−(ω+ϕ)n and p = − z 1+n +κ¯ z . q,z 1−η φ z 1+ω q z C.2.1 The real consol bond Consider the real consol bond that pays one unit of consumption every period. The price of this bond can be written recursively as Bc,∞ =E (cid:2) M (cid:0) 1+Bc,∞(cid:1)(cid:3) . t t t,t+1 t+1 Its one-period log-return can be written as (cid:18)1+exp(pc )(cid:19) rc =log ∞,t+1 ≈η¯c +ηc pc −pc , ∞,t+1 exp(pc ) ∞ ∞ ∞,t+1 ∞,t ∞,t where pc ≡logBc,∞, and η¯c , and ηc <1, are appropriate approximation constants. From the pricing equation ∞ 1 ,t =E (cid:2) ex t p(m + ∞ rc ) (cid:3) ∞ ,itcanbeshownthatthelog-bondpricefollowsthelinearfunction t t,t+1 ∞,t+1 pc =p¯c +pc ∆a +pc z ∞,t ∞ ∞,a t ∞,z t where ϕ[(1−φ )n −φ ] (1−φ )(1+n )ϕ pc = a a a , and pc = z z . ∞,a 1−η φ ∞,z 1−η φ ∞ a ∞ z C.2.2 Inflation dynamics Considertheinterest-ratepolicyrulesaysthatthecurrentinterestratedependsonthelaggedinterestrate as follows i =¯ı+ı (π −π∗)+ı (x −x +u , t π t x t ss t where the response to the lagged interest rate i is ρ = 0. Under nominal rigidities, the output gap is t−1 given by log(µ µ ) x =y −yf =n −nf + w p , t t t t t ω+ϕ where nf denotes labor under no price and wage rigidities. The output gap can be written as t (cid:18) (cid:19) 1−ϕ x =x¯+n ∆a + n − z , t a t z ω+ϕ t 48

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