Yield Curve and the Business Cycle in Conventional Times

Roman Šustek^1,2

¹School of Economics and Finance, Queen Mary University of London, London, UK.
²Centre for Macroeconomics, London School of Economics, London, UK.

DOI: 10.4236/jmf.2024.141004                

Abstract

This paper offers a structural interpretation of the “leading indicator” properties of the yield curve observed in conventional times of monetary policy. Low levels of nominal interest rates and inflation, but a steeper yield curve, typically precede economic expansions. According to the model, if investors use bond markets mainly to hedge risk, positive economic news are only weakly transmitted into real interest rates, but monetary policy transmits them into lower inflation and nominal rates. A steeper yield curve reflects both expected faster growth and higher uncertainty about the growth path. Importantly, the mechanism conforms with other important term structure properties.

Keywords

Term Structure of Interest Rates, Monetary Policy, Business Cycle, Recursive Preferences, Stochastic Volatility

1. Introduction

Inflation is back and with it the return of central banks to conventional monetary policy and a renewed attention of investors to bond markets. This paper offers a structural interpretation of yield curve dynamics over the business cycle—the “leading indicator” properties of the yield curve-that have been observed in times of conventional policy. In the data, the levels of nominal interest rates and inflation are typically negatively correlated with future output, while the long-short spread and expected excess returns (risk premia) are positively correlated with future output. That is, ahead of an expansion, nominal interest rates and inflation are low, while the yield curve is steep and expected excess returns on long-term bonds over short-term bonds are high. Accounting for these lead-lag dynamics of the yield curve is the main contribution of the paper relative to the literature. At the same time, however, emphasis is placed on the proposed mechanism to be consistent, in general equilibrium, with standard yield curve moments: the average yield and volatility curves; the decomposition of the term structure into level, slope, and curvature factors; a single factor driving excess returns on bonds of different maturities; and the statistical properties of these reduced-form factors and their correlations with macro variables.

In the model, the central bank follows a conventional monetary policy by controlling the interest rate on the shortest maturity in accordance with a Taylor rule. Preferences have the [1] form and the state space consists of four shocks (risk factors): a mean-reverting shock to the current level of productivity, common in real business cycle models; a persistent shock to the expected future growth rate of productivity a-lá [2] ; a Taylor rule shock; and a volatility shock. Risk prices depend endogenously on these processes. Interestingly, the correlations of expected excess returns with output growth at various leads and lags in the data suggest a dual role of the volatility shock: a positive volatility shock temporarily increases both the conditional variance and the conditional mean of future output growth. Consequently, volatility can be welfare neutral. The model is agnostic about the sources of this dual role and simply allows for it in the joint process for the shocks, a generalization of the consumption-volatility process of [2] . The model has a mapping into the [3] affine term structure model, whereby the reduced-form parameters of the [3] setup depend on the structural parameters of the model. Most of results can be derived analytically, providing a clear insight into the mechanism. For reasons discussed below, the model also allows for the presence of hand-to-mouth agents and nominal price rigidities in goods markets. The equilibrium bond prices, however, are not particularly sensitive to such frictions.

Starting with a flexible-price version of the parameterized model, in which hand-to-mouth agents do not play any role and the endogenous comovement between output and inflation is induced only by the Taylor rule, the notable properties of the equilibrium are as follows: 1) Only the expected growth factor has a price of risk substantially different from zero; 2) The time variation in the risk premium attached to this factor is driven by the volatility factor, which itself has a price of risk close to zero due to its near welfare neutrality; 3) The pricing kernel depends essentially only on expected inflation and the Epstein-Zin part pricing risk to lifetime utilities, with the standard intertemporal smoothing motive almost absent.¹ These properties make the model consistent with the standard yield curve moments and, at the same time, offer a simple interpretation of the yield curve lead-lag dynamics: Low levels of nominal interest rates and a steeper yield curve observed in the data ahead of an economic expansion reflect news about higher future output growth, which is only weakly transmitted into the real interest rate by intertemporal smoothing, but which the Taylor rule transmits into lower inflation. If the positive news about output growth is contained in the volatility factor, a steeper yield curve also reflects higher expected excess returns due to elevated uncertainty about the (persistent) future growth path.

In more detail, to carry a significant price of risk, a shock has to be either persistent or large in size (have a large conditional variance). The expected growth factor has a persistent effect on bond investors’ expected consumption and lifetime utilities and thus has a significant price of risk. However, for this mechanism to generate positive term premia on long-term nominal bonds in equilibrium, the elasticity of intertemporal substitution of the stand-in investor has to be sufficiently high. This is different from models which have an exogenous joint consumption-inflation process (or at least contain some sources of exogenous covariance between the two variables).² There are two reasons for this. First, a high elasticity of intertemporal substitution is required for a negative covariance between consumption growth and inflation, which is endogenously induced by the Taylor rule. Second, if the elasticity was low, a persistent decline in expected future consumption growth would significantly reduce the real interest rate through the intertemporal smoothing motive. This would increase bond prices, making long-term nominal bonds a hedge, despite the negative effect on bond prices of higher inflation.³ The empirical lead-lag dynamics of the yield curve impose yet another constraint on the elasticity of intertemporal substitution to be high, by requiring a subdued response of the real interest rate to news about output growth.⁴

Bond prices in the model thus predominantly reflect attitudes to risk interacting with monetary policy, rather then intertemporal smoothing motives. In other words, from the perspective of the estimated model, bond prices imply that investors require only a small compensation to postpone consumption by an extra period, when investment payoffs appear to be certain. However, when faced with risky payoffs, the compensation for bearing risk has to be large. Nominal bonds are risky because of the negative comovement between inflation and real economic activity, which is induced by conventional monetary policy summarized by the Taylor rule.

A high elasticity of intertemporal substitution is not unusual in structural models of the yield curve. For instance, [10] and [6] , who assume an exogenous consumption-inflation process, require the elasticity of intertemporal substitution to be around five and two, respectively.⁵ The endogeneity of the consumption-inflation process in this paper, as well as matching the lead-lag dynamics (not typically taken into account by the literature), require the elasticity to be even higher, between eight and ten. The real pricing kernel then effectively depends only on the Epstein-Zin part pricing risk to lifetime utilities. This part is sufficiently volatile to satisfy the Hansen-Jagannathan bound without requiring unrealistically volatile consumption. The high elasticity of intertemporal substitution inferred from the yield curve, however, appears to fly in the face of the literature represented by e.g. [12] . This literature points out that consumption of many households is irresponsive to changes in interest rates but responds strongly to changes in current income. To check the robustness of the results against such empirical evidence, the model allows for the presence of hand-to-mouth households, as well as for sticky prices, which provide an additional source of endogenous comovement between output and inflation that determines bond prices. Although nominal price rigidities and hand-to-mouth agents improve the quantitative properties of the model in relation to the data, they do not materially change the equilibrium pricing kernel and, thus, the main results. This is because the New-Keynesian Philips Curve (NKPC) transmits, in a quantitatively meaningful way, only temporary shocks. While the impact of such shocks on macro variables is sizable, it is short-lived and its overall effect on equilibrium risk prices is small. The size of the hand-to-mouth population, in line with other macro models, amplifies the transmission of policy shocks. But for empirically relevant fractions of such households in the population, the resulting amplification does not overturn the main results.

The practical relevance of the model lies in providing further support to long-run growth shocks, in combination with monetary policy, as the main risk factor for bond prices. The additional support comes from showing that such shocks can not only account for the average yield curve, as already shown by the literature [6] [13] , but also for its lead-lag dynamics. For instance, if the current geopolitical situation leads to subdued long-run growth and persistently higher inflation, then it is exactly the kind of shock that fits the long-run growth factor in the model.

Affine term structure models [3] [14] have a long tradition in the study of monetary policy.⁶ The term structure of interest rates has been also studied within structural monetary models by, e.g. [24] [25] [26] and [27] , as well as [28] , and [29] .⁷ Relative to this literature, the primary focus of this paper is on the cyclical lead-lag dynamics of the nominal term structure. A lead-lag behavior of various asset prices has been studied by [36] . But their model abstracts from the nominal side of the economy. In relation to the reduced-form affine term structure models, the model-of course-cannot compete with that literature in terms of its empirical performance. For instance, the results suggest that the model misses factors behind movements in risk premia that are unrelated to the average business cycle.⁸

Finally, a large literature studies the real effects of uncertainty shocks [37] . This paper is not concerned with the channels of transmission from uncertainty to real activity. While in the model (under sticky prices) output responds endogenously to volatility, most of the interaction between volatility and output comes from the exogenous process, which, in the asset pricing tradition [2] [36] , is inferred from asset prices. This reveals that certain types of volatility shocks are related to the average business cycle and precede output.⁹

The paper is structured as follows. Section 2 lists basic stylized facts about the nominal yield curve. Section 3 describes the model and explains the mechanism. Section 4 reports quantitative findings. Section 5 concludes. Online material contains an Appendix.

2. Stylized Facts about the Term Structure

This section lists selected stylized facts about the nominal yield curve and its relationship to the macroeconomy that inform the construction and calibration of the model in the next sections. Most of the stylized facts are well known, a few less so. Where relevant, I note examples of studies that have previously documented various versions of these empirical regularities, possibly in different samples. Before proceeding, some notation and terminology are introduced.

To start, one period in both the data and the model refers to a quarter. It is convenient to work with continuously compounded yields, returns, and growth rates. These variables are then reported in percent per annum. Let $q_t^{\left(n\right)}$ be the period-t price of a zero-coupon default-free bond that matures and pays one dollar in n periods. Continuously compounded yields can be inferred from a discounting formula $q_t^{\left(n\right)}=\exp \left(-n{i}_t^{\left(n\right)}\right)$ , implying $i_t^{\left(n\right)}=\left(-1/n\right)\log q_t^{\left(n\right)}$ . Realized returns on holding a n-period bond for one period are defined as $r_{t+1}^{\left(n\right)}\equiv \log q_{t+1}^{\left(n-1\right)}-\log q_t^{\left(n\right)}$ . Excess returns are then computed as $r_{X\mathrm{,}t+1}^{\left(n\right)}\equiv r_{t+1}^{\left(n\right)}-i_t$ , where $i_t=i_t^{\left(1\right)}$ is the short rate. Expected excess returns are given by $E_t{r}_{X\mathrm{,}t+1}^{\left(n\right)}$ , where the expectation operator is with respect to information up to and including period t. Expected excess return quantifies the risk compensation, required ex-ante, for holding the n-period bond for one period and is estimated from standard forecasting regressions.

The focus is on the period of conventional monetary policy 1961-2008. The stylized facts are presented for the period as a whole in order to capture the large long-run swings in inflation and interest rates and a sufficient number of business cycles. Nonetheless, splitting the sample into the two commonly studied regimes, 1961-1979 and 1985-2008, produces qualitatively similar facts. The period of the zero-lower bound and quantitative easing is excluded as this period represents a major departure from conventional monetary policy and, as such, requires separate attention and different modeling approach. The maturities included are 3 months and 1 to 7 years (the stylized facts are similar for the period 1971-2008, for which the maturities are available up to 10 years).¹⁰ The stylized facts taken into account are as follows:

1) Average yield and volatility curves. The yield curve slopes up on average; see the top-left panel of Figure 1. The volatility curve is fairly flat-the volatility at the long end is almost as high as the volatility at the short end; see the top-right panel of Figure 1.

2) Level, slope, and return factors. Two principal components (PCs) account for over 99% of the total variance of yields across maturities, with the 1^st PC accounting for about 97% and the 2^nd PC for a little over 2.5%. The 1^st PC works like a “level factor”, shifting all yields more or less in parallel; the 2^nd PC works like a “slope factor”, increasing the spread between the long and short rates [42] [43] .¹¹ See the bottom-left panel of Figure 1. A single PC accounts for essentially all variance (99%) of excess returns across maturities. The effect of this “return factor” on excess returns increases with maturity [4] . See the bottom-right panel of Figure 1.

3) Properties of the level factor. The level factor is close to a random walk and is unrelated to the variation in excess returns [44] . The upper panel of Table 1 shows the estimate of a VAR (1) matrix for the first five PCs of yields. It shows that the level factor is highly persistent, with statistically insignificant interactions with the other PCs.¹² (Granger causality tests, not reported, confirm that the level factor neither forecasts nor is forecastable by any other PCs.) The lower panel shows that forecasting excess returns with the level factor has R² approximately equal to zero.¹³ The level factor, however, is strongly positively correlated with inflation [15] ; in the sample considered here, the correlation is 0.71.¹⁴

4) Properties of the slope and return factors. The slope factor is statistically related to the return factor [47] [48] . The results of the forecasting regressions for the return factor (the lower panel of Table 1) report R² equal to 0.08 when the slope factor is used as a regressor, with a statistically significant coefficient. If I let the return holding period be the more conventional one year, the R² raises

Figure 1. Top panel: U.S. average yield and volatility curves for 1961-2008. Bottom panel: loadings on the PCs of yields and excess returns. For yields, the contribution of the PCs is: 1^st PC = 97.2%, 2^nd PC = 2.6%, 3^rd PC = 0.2%. For excess returns, the first PC accounts for 99% of the total variance.

Table 1. Time series and forecasting properties of principal components of yields.

Notes: The VAR (1) matrix is for a regression of a vector of the first five principal components of yields in period t + 1 on the same vector in period t. In the forecasting regressions, the dependent variable is the first principal component of excess returns (the return factor), the independent variables are a constant and the principal components of yields specified in the table. The holding period is one quarter. In both tables, numbers in bold represent statistically significant estimates at 5% confidence level. PC1 is the first principal component of yields, PC2 is the second principal component of yields, and so on. The period is 1961-2008.

to the typical value of about 0.2. As a direct consequence, the slope factor and expected (fitted) excess returns are closely related.¹⁵

5) Yield curve and the business cycle. Yields exhibit a negative lead with respect to the growth rate of real GDP, whereas the slope of the yield curve and expected excess returns exhibit a positive lead [36] [50] [51] [52] .¹⁶ Specifically, Figure 2 plots $\text{corr}\left(x_{t+j}\mathrm{,}{g}_t\right)$ , $j=-6,\cdots ,0,\cdots ,6$ , where x is the variable of interest and g is the continuously compounded growth rate of real GDP, either quarter-on-quarter or centered year-on-year. The figure shows that the short rate has a strong negative lead, the long (7-year) rate has a weak negative lead, and the inflation rate has a negative lead similar to that of the short rate. Also, interest rates and inflation are negatively correlated with output growth contemporaneously.¹⁷ The negative lead in yields occurs due to the level factor; the slope factor exhibits a positive lead, similar to that of the expected excess return.^18,19

3. The Model

To avoid having to introduce new notation and equations, it is convenient to

Figure 2. Yield curve and the business cycle. Cross-correlations with the growth rate of real GDP, 1961-2008. Bars are for a quarter-on-quarter growth rate of real GDP, the solid line is for a centered year-on-year growth rate. The correlations are $\text{corr}\left(x_{t+j}\mathrm{,}{g}_t\right)$ , $j=-\mathrm{6,}\cdots \mathrm{,0,}\cdots 6$ , where x is the variable of interest and g is the growth rate of real GDP.

present the model in its full form that allows for sticky prices and hand-to-mouth agents. It is based on a stripped-down version of a two-agent New-Keynesian model studied by [55] . The flexible-price version used for the headline results is a special case of the general setup and this is pointed out where relevant. In the flexible-price version, hand-to-mouth agents play no role, as will become clear below.

The model has a convenient log-normal form that allows a straightforward, easy-to-interpret, mapping into the [3] affine term structure model. The New-Keynesian part is standard. The less standard features are the Epstein-Zin preferences and the state space. A fraction $1-\lambda$ of households are referred to as “bond investors”; the remaining fraction $\lambda$ are referred to as “hand-to-mouth” households who are excluded from financial markets.²⁰ Within the two types, agents are identical. The only input into production is labor. Profits (dividends) of monopolistically competitive firms are split between the two types in a fixed proportion. That is, there is no trade in the claims on profits between the two types. In this sense the claims represent illiquid assets, such as unincorporated business, making the hand-to-mouth agents the “rich” hand-to-mouths of [56] .

Where applicable, the notation from Section 2 carries over and interest rates, inflation rates, growth rates, and rates of return are, as before, continuously compounded. I adopt the convention that hats denote percentage or percentage point deviations from steady state and variables without a time subscript denote the steady state. The model allows for a deterministic trend. “Steady state” therefore refers to a balanced growth path. Up to a constant, ${\hat{y}}_t=\log y_t-gt$ , ${\hat{c}}_{Bt}=\log c_{Bt}-gt$ , ${\hat{c}}_{Ht}=\log c_{Ht}-gt$ , and ${\hat{w}}_t=\log w_t-gt$ , where $y_t$ is output, $c_{Bt}$ is consumption of the bond investor, $c_{Ht}$ is consumption of the hand-to-mouth household, $w_t$ is the real wage rate, and g is the growth rate of the deterministic trend, driven by productivity. The variables can be rewritten in terms of their growth rates as $g_{y,t+1}=\log y_{t+1}-\log y_t=g+\left({\hat{y}}_{t+1}-{\hat{y}}_t\right)$ and similarly for the growth rates of $c_{Bt}$ , $c_{Ht}$ , and $w_t$ . The steady state of labor, inflation, and interest rates is a constant. To economize on space, throughout the paper the details of various derivations are relegated to the Appendix.

3.1. Preferences, Technology, Monetary Policy

Bond investors have [1] preferences

$U_t={\left[\left(1-\beta \right)c_{Bt}^{\rho }+\beta {\mu }_t{\left(U_{t+1}\right)}^{\rho }\right]}^{1/\rho }\mathrm{,}$ (1)

where $\beta \in \left(\mathrm{0,1}\right)$ is a discount factor, $U_t$ is the lifetime utility from period t on, and ${\mu }_t\left(U_{t+1}\right)$ is period-t certainty equivalent of stochastic lifetime utilities from t + 1 on. Further, $\rho \le 1$ controls the elasticity of intertemporal substitution, given by $1/\left(1-\rho \right)$ . The certainty equivalent is based on expected utility

${\mu }_t\left(U_{t+1}\right)={\left[E_t\left(U_{t+1}^{\alpha }\right)\right]}^{1/\alpha },$ (2)

where $E_t$ is the expectation operator based on period-t state variables. The parameter $\alpha \le 1$ controls the coefficient of relative risk aversion, given by $1-\alpha$ . Implicitly, labor supply of bond investors is assumed to be inelastic.²¹

Nominal zero-coupon bonds of different maturities are available in zero net supply. The real pricing kernel is equal to the representative investor’s stochastic discount factor

$m_{t+1}=\beta {\left(\frac{c_{B\mathrm{,}t+1}}{c_{Bt}}\right)}^{\rho -1}{\left(\frac{U_{t+1}}{{\mu }_t\left(U_{t+1}\right)}\right)}^{\alpha -\rho }\mathrm{.}$ (3)

The nominal pricing kernel is given by $m_{t+1}^{\mathrm{<mi>\; \$\; </mi>}}\equiv m_{t+1}\exp \left(-{\pi }_{t+1}\right)$ , where ${\pi }_{t+1}$ is a continuously compounded inflation rate between t and t + 1. In the real pricing kernel, if $\alpha =\rho$ , $m_{t+1}$ becomes the standard marginal rate of intertemporal substitution for CRRA time-additive preferences. In that case, only consumption growth between t to t+1 affects asset prices. If $\alpha \ne \rho$ , the pricing kernel also depends on lifetime consumption streams, embedded in the lifetime utilities. A common assumption in the literature, which is also imposed here, is $\alpha -\rho <0$ . In this case, a higher $U_{t+1}$ is considered a good news by the investor and reduces the pricing kernel. In addition, it is assumed that $\alpha <0$ . The budget constraint of the bond investor is given by

$b_{t+1}+c_{Bt}=\frac{1+i_{t-1}}{1+{\pi }_t}{b}_t+w_t{l}_B+\frac{1-\epsilon }{1-\lambda }{d}_t\mathrm{,}$

where $b_{t+1}$ denotes holdings of a one-period nominal bond between periods t and t + 1, $w_t{l}_B$ is labor income, $d_t$ is aggregate dividends, and $\left(1-\epsilon \right)$ is the share of the dividends claimed by bond investors. As bonds are in zero net supply and bond investors are all alike, bonds are not traded in equilibrium. Bonds of longer maturities can be priced by arbitrage, once the equilibrium nominal pricing kernel is determined. Leaving long-term bonds out of the budget constraint is thus inconsequential for the equilibrium.²²

The per-period utility function of the hand-to-mouth household takes the standard form in the New-Keynesian literature, $\log c_{Ht}-\omega \left(l_{Ht}^{1+\eta }\right)/\left(1+\eta \right)$ . Here, $l_{Ht}$ is labor, $\omega \ge 0$ is a weight on disutility from labor, and $\eta \ge 0$ is the Frish elasticity. Like in the case of the bond investor, this utility function could be embedded in the Epstein-Zin form. However, as the decision problem of the hand-to-mouth household is static, such a formulation would be inconsequential for the equilibrium.²³ The budget constraint of the hand-to-mouth household is

$c_{Ht}=w_t{l}_{Ht}+\frac{\epsilon }{\lambda }{d}_t$

and the optimal labor supply is characterized by the first-order condition $\log w_t=\log c_{Ht}+\eta \log l_{Ht}$ .

Goods market clearing requires $y_t=\left(1-\lambda \right)c_{Bt}+\lambda c_{Ht}$ . Output is given by the production function $\log y_t=gt+z_t+\log l_t$ , where $z_t$ is a log-deviation of productivity from the deterministic trend and $l_t$ is aggregate labor. Dividends are determined as a residual from output, once labor is paid: $d_t=y_t-w_t{l}_t$ . The business sector has the usual setup with sticky prices, leading to the standard NKPC. When log-linearized around a zero inflation steady state (a common assumption) the NKPC takes the well-known convenient form, ${\pi }_t=\beta E_t{\pi }_{t+1}+\Phi {\hat{\upsilon }}_t$ , where ${\hat{\upsilon }}_t={\hat{w}}_t-z_t$ is the log-deviation of the marginal cost from steady state and $\Phi \equiv \left(1-\zeta \right)\left(1-\beta \zeta \right)/\zeta$ , with $\zeta$ being the Calvo parameter [57] .²⁴ Substituting for ${\hat{\upsilon }}_t$ yields the NKPC in terms of output

${\pi }_t=\beta E_t{\pi }_{t+1}+\Omega \left({\hat{y}}_t-z_t\right)\mathrm{,}$ (4)

where

$\Omega =\frac{\Phi }{\epsilon }\left[\frac{w}{z}+\eta \frac{c_H}{z{l}_H}+\epsilon \left(1-\frac{w}{z}\right)\right]\mathrm{.}$

This is derived by combining the first-order condition for labor, the hand-to-mouth agent’s budget constraint, the production function, and the equation for dividends (see the Appendix for the derivation).²⁵ When prices are flexible, $\zeta =0$ , $\Phi =\Omega =\infty$ , and ${\hat{y}}_t=z_t$ .

The model is closed with a Taylor rule

$i_t=i+{\nu }_{\pi }\left({\pi }_t-{\pi }^{\mathrm{<mo>\; *\; </mo>}}\right)+{\nu }_y\left(E_t{g}_{y\mathrm{,}t+1}-g\right)+{\xi }_t\mathrm{,}$ (5)

where ${\pi }^{\mathrm{<mo>\; *\; </mo>}}$ is an inflation target and ${\xi }_t$ is a shock. The standard restrictions on the parameters apply: ${\nu }_{\pi }>1$ and ${\nu }_y>0$ .²⁶

3.2. Exogenous Processes

Two shocks, the productivity shock ( $z_t$ ) and the Taylor rule shock ( ${\xi }_t$ ), have already been introduced and are standard in the macro literature. There are two additional shocks, $s_t$ and $v_t$ , taken from the finance literature, whose role is explained below. The following stationary Gaussian processes are adopted for the four shocks

$\underset{x_{t+1}}{\underset{︸}{\left(\begin{array}{c}{z}_{t+1}\\ s_{t+1}\\ {\xi }_{t+1}\end{array}\right)}}=\underset{A}{\underset{︸}{\left(\begin{array}{ccc}{\varphi }_z& 1& 0\\ 0& {\varphi }_s& 0\\ 0& 0& {\varphi }_{\xi }\end{array}\right)}}\underset{x_t}{\underset{︸}{\left(\begin{array}{c}{z}_t\\ s_t\\ {\xi }_t\end{array}\right)}}+\underset{a}{\underset{︸}{\left(\begin{array}{c}{a}_z\\ a_s\\ 0\end{array}\right)}}\left(v_t-v\right)+v_t^{1/2}B{\omega }_{t+1},$ (6)

$v_{t+1}=v+\theta \left(v_t-v\right)+b{\omega }_{t+1}\mathrm{.}$ (7)

Here, ${\varphi }_z\mathrm{,}{\varphi }_s\mathrm{,}{\varphi }_{\xi }\mathrm{,}\theta \in \left[\mathrm{0,1}\right)$ , $v>0$ , and $a_z\mathrm{,}{a}_s\ge 0$ . Further, $B\ge 0$ is a 3 × 4 matrix with positive entries only at $B_{11}\mathrm{,}{B}_{22}$ , and $B_{33}$ , and $b\ge 0$ is a 1 × 4 vector with a positive entry only at $b_4$ . Consequently, $B{b}^{{\rm T}}=0$ . Finally, ${\omega }_t~N\left(\mathrm{0,}I\right)$ is a 4 × 1 vector of innovations. At a certain point in the derivations below (at the point of evaluating the real pricing kernel, which depends on consumption growth), it will be convenient to work with the state space (6)-(7) written as

$\underset{\Delta x_{t+1}}{\underset{︸}{\left(\begin{array}{c}\Delta z_{t+1}\\ \Delta s_{t+1}\\ \Delta {\xi }_{t+1}\end{array}\right)}}=\underset{A_d}{\underset{︸}{\left(\begin{array}{ccc}{\varphi }_z-1& 1& 0\\ 0& {\varphi }_s-1& 0\\ 0& 0& {\varphi }_{\xi }-1\end{array}\right)}}\underset{x_t}{\underset{︸}{\left(\begin{array}{c}{z}_t\\ s_t\\ {\xi }_t\end{array}\right)}}+\underset{a}{\underset{︸}{\left(\begin{array}{c}{a}_z\\ a_s\\ 0\end{array}\right)}}\left(v_t-v\right)+v_t^{1/2}B{\omega }_{t+1}\mathrm{,}$ (8)

$\Delta v_{t+1}={\theta }_d\left(v_t-v\right)+b{\omega }_{t+1}\mathrm{,}$ (9)

which is obtained by simply subtracting $x_t$ and $v_t$ from both sides of Equations (6) and (7), respectively. Here, ${\theta }_d\equiv \theta -1$ . The joint process (6)-(7), or equivalently (8)-(9), belongs in the class of stochastic volatility in the mean processes and conforms with the setup of the [3] affine term structure model.

The shock $v_t$ affects the conditional volatility of $x_{t+1}$ (or equivalently $\Delta x_{t+1}$ ), through B, as well as its conditional mean, through a. The shock is thus both a volatility shock and a news shock about future productivity. This specification is motivated by the Stylized Fact 5. In the model, $v_t$ makes the second moments of the pricing kernel time varying and thus generates time-varying risk premia. The parameter a controls the extent to which the time-variation in risk premia, and thus expected excess returns, precedes the time variation in productivity growth, and thus in output growth. The lead-lag dynamics and risk premia, however, are not independent phenomena, and risk premia in equilibrium also depend on the parameter a.^27,28

The shock $s_t$ is a shock to the conditional mean of $z_{t+1}$ (or equivalently $\Delta z_{t+1}$ ). As such, it is a pure news shock about future productivity, similar to the shock to consumption and dividends in [2] . In contrast, $z_t$ is a mean reversing shock to the current productivity level, typical for RBC models. Unlike the $s_t$ shock, which can generate persistent changes in the growth rate, it leads to a growth rate that is dominated by purely temporary changes.²⁹

3.3. Equilibrium

This section describes the conditions characterizing the equilibrium, with the actual solutions reported and discussed in the next section.

3.3.1. Sharing Rules

As bond investors are all alike, in equilibrium $b_t=0$ and bond investors consume their entire income. The budget constraints of the two types, the equation for dividends, the production function, and the first-order condition for labor yield “sharing rules” (consumption claims on output) for the two agents. See the Appendix. For bond investors:

${\hat{c}}_{Bt}=z_t+\underset{{\Phi }_B}{\underset{︸}{\left[1-\frac{w}{z}\frac{\lambda }{1-\lambda }\left(\frac{1-\epsilon }{\epsilon }\left(1+\eta \right)-\frac{1-\lambda }{\lambda }\eta \right)\right]}}\left({\hat{y}}_t-z_t\right),$ (10)

which relates the bond investor’s consumption to aggregate output in a way that depends on the fraction $\lambda$ of hand-to-mouth agents in the population. The larger is $\lambda$ , the smaller is ${\Phi }_B$ . This property reflects the aspect of sticky-price models that dividends and labor income move in opposite directions in response to shocks that affect ${\hat{y}}_t-z_t$ [57] . When $\lambda$ is large, the given share of aggregate dividends, $1-\epsilon$ , accruing to bond investors is divided among a smaller measure of them $1-\lambda$ , thus providing each of them with a stronger hedge against labor income fluctuations. The overall effect of $\lambda$ on ${\hat{c}}_{Bt}$ , however, depends also on the endogenous ${\hat{y}}_t$ , which in equilibrium is also affected by $\lambda$ .

The sharing rule for hand-to-mouth agents is

${\hat{c}}_{Ht}=z_t+\underset{{\Phi }_H}{\underset{︸}{\left[1+\frac{w}{z}\left(\frac{1-\epsilon }{\epsilon }\left(1+\eta \right)-\frac{1-\lambda }{\lambda }\eta \right)\right]}}\left({\hat{y}}_t-z_t\right),$ (11)

where ${\Phi }_H$ depends positively on $\lambda$ . For a given $\epsilon$ , a sufficiently large $\lambda$ makes consumption of hand-to-mouth households more volatile than consumption of bond investors.³⁰ Observe that under flexible prices (i.e., ${\hat{y}}_t=z_t$ ), the sharing rules are reduced to ${\hat{c}}_{Bt}={\hat{c}}_{Ht}=z_t$ .

3.3.2. A system in Output and Inflation

Bond investors satisfy the Euler equation for the one-period nominal bond. Two conditions then characterize equilibrium processes for output and inflation. One condition is the NKPC (4), the other is a combination of the Taylor rule and the Euler equation for the one-period bond, $\exp \left(-i_t\right)=E_t\left[m_{t+1}\exp \left(-{\pi }_{t+1}\right)\right]$ , with $m_{t+1}$ given by (3) and ${\hat{c}}_{Bt}$ given by (10). This condition will be referred to as the ‘bond market equilibrium condition’, as it relates bond investors to the central bank. Hand-to-mouths affect the equilibrium through $\lambda$ affecting the sharing rule for ${\hat{c}}_{Bt}$ and thus the pricing kernel. Assuming for the moment that $i_t$ , $\log m_{t+1}$ , and ${\pi }_{t+1}$ are jointly normally distributed (verified later on), we can expand the Euler equation and write the bond market equilibrium condition as

$i+{\nu }_{\pi }\left({\pi }_t-{\pi }^{\mathrm{<mo>\; *\; </mo>}}\right)+{\nu }_y\left(E_t{g}_{y\mathrm{,}t+1}-g\right)+{\xi }_t=-E_t\log m_{t+1}+E_t{\pi }_{t+1}+m_t^{\left(2\right)}\mathrm{,}$ (12)

where $m_t^{\left(2\right)}\equiv -0.5va{r}_t\log m_{t+1}-0.5va{r}_t{\pi }_{t+1}+co{v}_t\left(\log m_{t+1}\mathrm{,}{\pi }_{t+1}\right)$ subsumes the second moments of the nominal pricing kernel. It is shown below that $\log m_{t+1}$ is linear in ${\hat{c}}_{Bt}$ and thus, by (10), in ${\hat{y}}_t$ .

Given the log-linear/log-normal form of the model, we can consider equilibrium functions of the state space

${\hat{y}}_t=y+y_x^{{\rm T}}{x}_t+y_v{v}_t\mathrm{,}$ (13)

${\pi }_t=\pi +{\pi }_x^{{\rm T}}{x}_t+{\pi }_v{v}_t\mathrm{,}$ (14)

where ( $y\mathrm{,}{y}_x^{{\rm T}}\mathrm{,}{y}_v\mathrm{,}\pi \mathrm{,}{\pi }_x^{{\rm T}}\mathrm{,}{\pi }_v$ ) are endogenous coefficients, commensurate to the state variables. The functions (13) and (14) solve the two functional equations (4) and (12) and the equilibrium coefficients are obtained by the method of undetermined coefficients.

The rest of this section describes how the pricing kernel is transformed into the [3] form, which provides a convenient form for solving for the equilibrium yield curve and establishes a close connection with affine term structure models.

3.3.3. The Real Pricing Kernel and the Value Function

The Epstein-Zin pricing kernel depends on endogenous lifetime utilities. Starting with (3), the real pricing kernel can be expressed in a log form

$\begin{array}{c}\log m_{t+1}=\log \beta +\left(\rho -1\right)g_{c\mathrm{,}t+1}+\left(\alpha -\rho \right)\{\left(g_{c\mathrm{,}t+1}+\log u_{t+1}\right)\\ -\log {\mu }_t\left[\exp \left(g_{c\mathrm{,}t+1}\right)u_{t+1}\right]\}\mathrm{,}\end{array}$ (15)

where $u_{t+1}\equiv U_{t+1}/c_{B\mathrm{,}t+1}$ is a scaled lifetime utility, which is constant on the balanced growth path. Further, $\log {\mu }_t\left[\exp \left(g_{c\mathrm{,}t+1}\right)u_{t+1}\right]={\alpha }^{-1}\log E_t\left[\exp \alpha \left(g_{c\mathrm{,}t+1}+\log u_{t+1}\right)\right]$ , which follows from the homogeneity of degree one of the certainty equivalent (2); see the Appendix. If $\rho =1$ , the standard margin depending on short-term consumption growth is eliminated from the pricing kernel; if $\alpha =\rho$ , the part depending on lifetime utilities is eliminated.

The rest of this subsection evaluates $g_{c\mathrm{,}t+1}$ and $u_{t+1}$ in the pricing kernel (15) to make the kernel depend only on state variables and innovations. The coefficients of the resulting pricing kernel are functions of the coefficients of the output process ( $y\mathrm{,}{y}_x^{{\rm T}}\mathrm{,}{y}_v$ ).

Given the linear relationship (10) between ${\hat{c}}_{Bt}$ and ${\hat{y}}_t$ , the growth rate $g_{c\mathrm{,}t+1}$ can be written as $g_{c\mathrm{,}t+1}=g+{\Phi }_B\left(g_{y\mathrm{,}t+1}-g\right)+\left(1-{\Phi }_B\right)\Delta z_{t+1}$ , which, using (13), can be further expanded as $g_{c\mathrm{,}t+1}=g+{\Phi }_B\left(y_x^{{\rm T}}\Delta x_{t+1}+y_v\Delta v_{t+1}\right)+\left(1-{\Phi }_B\right)\Delta z_{t+1}$ or

$g_{c\mathrm{,}t+1}=g+c_x^{{\rm T}}\Delta x_{t+1}+c_v\Delta v_{t+1}\mathrm{,}$ (16)

where

$c_x^{{\rm T}}\equiv {\Phi }_B{y}_x^{{\rm T}}+\left(1-{\Phi }_B\right)e_z^{{\rm T}}\mathrm{,}\text{and}{c}_v\equiv {\Phi }_B{y}_v\mathrm{.}$ (17)

Further, $e_z^{{\rm T}}\equiv \left[100\right]$ , and $\Delta x_{t+1}$ and $\Delta v_{t+1}$ are given by (8) and (9), respectively.

The log utilities in the pricing kernel (15) must satisfy the recursive Equation (1). Adopting the [59] approximation

$\log u_t\approx {\kappa }_0+{\kappa }_1{\alpha }^{-1}\log E_t\left[\exp \alpha \left(g_{c\mathrm{,}t+1}+\log u_{t+1}\right)\right]\mathrm{.}$ (18)

Here ${\kappa }_0\equiv {\rho }^{-1}\log \left[\left(1-\beta \right)+\beta \exp \left(\rho \mu \right)\right]-{\kappa }_1\mu$ and ${\kappa }_1\equiv \left[\beta \exp \left(\rho \mu \right)\right]/\left[\left(1-\beta \right)+\beta \exp \left(\rho \mu \right)\right]\in \left(\mathrm{0,1}\right)$ works like a discount factor. Further, $\mu \equiv \log \left(\exp \left(g\right)u\right)$ is the steady-state value of the log certainty equivalent, with u denoting a steady-state (balanced growth path) scaled utility.³¹ The functional Equation (18), which by (16) and (17) depends on ( $y\mathrm{,}{y}_x^{{\rm T}}\mathrm{,}{y}_v$ ), admits a linear solution

$\log u_t=u+u_x^{{\rm T}}{x}_t+u_v{v}_t\mathrm{,}$ (19)

where ( $u,u_x^{{\rm T}},u_v$ ) are endogenous coefficients that solve (18) and depend on ( $y\mathrm{,}{y}_x^{{\rm T}}\mathrm{,}{y}_v$ ); see the next section for the solution.

3.3.4. The Duffie-Kan Pricing Kernel

The value function (19), the equation for consumption growth (16), and the stochastic processes (8) and (9) allow to express the real pricing kernel (15) only in terms of the state variables and innovations

$\log m_{t+1}=\delta +{\delta }_x^{{\rm T}}{x}_t+{\delta }_v{v}_t+{\lambda }_x^{{\rm T}}{v}_t^{1/2}{\omega }_{t+1}+{\lambda }_v^{{\rm T}}{\omega }_{t+1}\mathrm{,}$ (20)

where ( $\delta \mathrm{,}{\delta }_x^{{\rm T}}\mathrm{,}{\delta }_v$ ) are factor loadings and ( ${\lambda }_x^{{\rm T}}\mathrm{,}{\lambda }_v^{{\rm T}}$ ) are prices of risk, commensurate to the state variables and shocks (see the Appendix for derivation). The factor loadings and prices of risk, reported in the next section, depend on ( $y\mathrm{,}{y}_x^{{\rm T}}\mathrm{,}{y}_v$ ). Equation (20) takes the form of the pricing kernel in the [3] affine term structure model. The key difference is that here the factor loadings and prices of risk are not free parameters, but depend on the deep parameters of the model.

The equilibrium nominal pricing kernel is: $\log m_{t+1}^{\mathrm{<mi>\; \$\; </mi>}}=\log m_{t+1}-\left(\pi +{\pi }_x^{{\rm T}}{x}_{t+1}+{\pi }_v{v}_{t+1}\right)$ , where ( $\pi \mathrm{,}{\pi }_x^{{\rm T}}\mathrm{,}{\pi }_v$ ) are the equilibrium coefficients of the inflation process. It also preserves the [3] form

$\log m_{t+1}^{\mathrm{<mi>\; \$\; </mi>}}={\delta }^{\mathrm{<mi>\; \$\; </mi>}}+{\delta }_x^{\mathrm{<mi>\; \$\; </mi>}{\rm T}}{x}_t+{\delta }_v^{\mathrm{<mi>\; \$\; </mi>}}{v}_t+{\lambda }_x^{\mathrm{<mi>\; \$\; </mi>}{\rm T}}{v}_t^{1/2}{\omega }_{t+1}+{\lambda }_v^{\mathrm{<mi>\; \$\; </mi>}{\rm T}}{\omega }_{t+1}\mathrm{,}$ (21)

where the coefficients are

${\delta }^{\mathrm{<mi>\; \$\; </mi>}}=\delta -\pi +{\pi }_x^{{\rm T}}av-{\pi }_v\left(1-\theta \right)v\mathrm{,}$

${\delta }_x^{\mathrm{<mi>\; \$\; </mi>}{\rm T}}={\delta }_x^{{\rm T}}-{\pi }_x^{{\rm T}}A\mathrm{,}$

${\delta }_v^{\mathrm{<mi>\; \$\; </mi>}}={\delta }_v-{\pi }_x^{{\rm T}}a-{\pi }_v\theta \mathrm{,}$

${\lambda }_x^{\mathrm{<mi>\; \$\; </mi>}{\rm T}}={\lambda }_x^{{\rm T}}-{\pi }_x^{{\rm T}}B\mathrm{,}$

${\lambda }_v^{\mathrm{<mi>\; \$\; </mi>}{\rm T}}={\lambda }_v^{{\rm T}}-{\pi }_{v}b\mathrm{.}$

Note that as $\log m_{t+1}$ , ${\pi }_t$ , $y_t$ are linear functions of the normally distributed factors, they are normally distributed too, confirming the earlier conjecture.

3.4. Inspecting the Coefficients

Before moving on to the quantitative results, I list the coefficients of the processes for lifetime utility, the real pricing kernel, inflation, and output and point out their most important properties to provide insight into the quantitative findings. The coefficients of each of these processes have a recursive structure. First, the loadings on $x_t$ are determined, independently of the constant and the loading on $v_t$ . Second, the loading on $v_t$ is determined. It depends on the loadings on $x_t$ but not on the constant. Finally, the constant is determined and it depends on both the loadings on $x_t$ and $v_t$ . The loadings on $x_t$ are related only to conditional expectations; the loadings on $v_t$ reflect both conditional expectations and conditional second moments. I only discuss the loadings on $x_t$ and $v_t$ , which affect the dynamics, relegating constants to footnotes.

3.4.1. Lifetime Utility

Lifetime utility is used to evaluate the real pricing kernel. Recall that $\log u_t$ is the log of lifetime utility scaled by current consumption. It can therefore either increase or decline, in response to a positive consumption shock, depending on whether the shock affects more the lifetime utility or current consumption. Positive mean reversing shocks to the level of consumption reduce $\log u_t$ , whereas the opposite is true for persistent positive shocks to the consumption growth rate. For the following set of expressions, take ( $y\mathrm{,}{y}_x^{{\rm T}}\mathrm{,}{y}_v$ ) as given. These expressions characterize the solution to the bond market equilibrium condition (12); or to the flexible-price version of the model, i.e., the special case of $y_x^{{\rm T}}=\left[100\right]$ and $y_v=0$ .

Before proceeding, recall that $\alpha -\rho <0$ and $\alpha <0$ , and that $c_x^{{\rm T}}$ and $c_v$ are related to $y_x^{{\rm T}}$ and $y_v$ through (17) and, through ${\Phi }_B$ , depend on the fraction of hand-to-mouths in the population.

The coefficients of the value function are given by

$u_x^{{\rm T}}={\kappa }_1{c}_x^{{\rm T}}{A}_d{\left(I-{\kappa }_{1}A\right)}^{-1}\mathrm{,}$

$u_v=\frac{{\kappa }_1}{1-{\kappa }_1\theta }\left[{\left(c_x+u_x\right)}^{{\rm T}}a+c_v{\theta }_d+\frac{\alpha }{2}{\left(c_x+u_x\right)}^{{\rm T}}B{B}^{{\rm T}}\left(c_x+u_x\right)\right]\mathrm{.}$

The coefficient $u_x^{{\rm T}}$ is an infinite discounted sum of expected future consumption, conditional on a unit of $x_t$ . Thus, even shocks that affect only future consumption (not current consumption) affect $u_x^{{\rm T}}$ . In $u_v$ , the linear part within the square brackets captures expected lifetime utility from consumption from next period on, while the quadratic part reflects uncertainty about lifetime utility from consumption from next period on, both being conditional on a unit of $v_t$ . The linear part is present in $u_v$ due to $v_t$ being a news shock about future productivity (and due to a general equilibrium effect of $v_t$ on consumption, the $c_v$ term, in the version with the NKPC). The quadratic part is present due to $v_t$ being a volatility shock. Observe that the two parts can potentially offset each other (as $\alpha <0$ ), making $u_v$ equal to zero. Volatility in the model is thus potentially a “welfare-neutral” risk factor. Observe also that $u_x^{{\rm T}}$ and $u_v$ increase in absolute value with the persistence of the respective shocks, summarized by the eigenvalues of A and the size of $\theta$ .^32,33

3.4.2. Real Pricing Kernel

The real pricing kernel enters the bond market equilibrium condition (12). Its coefficients depend on the coefficients of lifetime utility and are given by

$\delta =\log \beta +\left(\rho -1\right)\left(g-c_x^{{\rm T}}av-c_v{\theta }_{d}v\right)-\left(\alpha -\rho \right)\frac{\alpha }{2}{\left(c_v+u_v\right)}^{2}b{b}^{{\rm T}}\mathrm{,}$

${\delta }_x^{{\rm T}}=\left(\rho -1\right)c_x^{{\rm T}}{A}_d\mathrm{,}$

${\delta }_v=\left(\rho -1\right)\left(c_x^{{\rm T}}a+c_v{\theta }_d\right)-\left(\alpha -\rho \right)\frac{\alpha }{2}{\left(c_x+u_x\right)}^{{\rm T}}B{B}^{{\rm T}}\left(c_x+u_x\right)\mathrm{,}$

${\lambda }_x^{{\rm T}}=\left(\rho -1\right)c_x^{{\rm T}}B+\left(\alpha -\rho \right){\left(c_x+u_x\right)}^{{\rm T}}B\mathrm{,}$

${\lambda }_v^{{\rm T}}=\left(\rho -1\right)c_{v}b+\left(\alpha -\rho \right)\left(c_v+u_v\right)b\mathrm{.}$

The pricing kernel has two parts: the standard part depending on short-term consumption growth, the terms pre-multiplied by $\left(\rho -1\right)$ , and a part depending on lifetime utilities, the terms pre-multiplied by $\left(\alpha -\rho \right)$ .³⁴ To focus on the second part, consider the limiting case of $\rho =1$ (infinite elasticity of substitution), so that the short-term part drops out. Under this restriction, ${\delta }_x^{{\rm T}}$ is eliminated from the pricing kernel. The quadratic terms in the factor loadings $\delta$ and ${\delta }_v$ are related to the certainty equivalent (pertaining to its constant and time-varying margins, respectively). If $v_t$ increases, the certainty equivalent, under the restriction $\alpha <0$ , unambiguously declines, reducing ${\delta }_v$ .³⁵ The prices of risk, ${\lambda }_x^{{\rm T}}$ and ${\lambda }_v^{{\rm T}}$ , determine the impact of the innovations to $x_{t+1}$ and $v_{t+1}$ , respectively, on the pricing kernel.

Because of the dependence of the risk prices on $u_x^{{\rm T}}$ and $u_v$ , the more persistent is a given shock, the larger is its price, in absolute value. In addition, the risk prices are scaled by the variance of the respective innovations (B and b). The larger is the conditional variance of a given shock, the larger is its price.

3.4.3. Inflation and Implications for Term Premia and the Lead-Lag Dynamics

The coefficients of the inflation process, obtained from the equilibrium equation (12), using the real pricing kernel (20), for a given ( $y\mathrm{,}{y}_x^{{\rm T}}\mathrm{,}{y}_v$ ), are:

${\pi }_x^{{\rm T}}=-\left({\nu }_y{y}_x^{{\rm T}}{A}_d+e_{\xi }^{{\rm T}}+{\delta }_x^{{\rm T}}\right){\left({\nu }_{\pi }I-A\right)}^{-1}\mathrm{,}$ (22)

${\pi }_v=\frac{1}{{\nu }_{\pi }-\theta }\left(-{\nu }_y\left(y_x^{{\rm T}}a+y_v{\theta }_d\right)-{\delta }_v+{\pi }_x^{{\rm T}}a-\frac{1}{2}{\lambda }_x^{{\rm T}}{\lambda }_x-\frac{1}{2}{\pi }_x^{{\rm T}}B{B}^{{\rm T}}{\pi }_x+{\lambda }_x^{{\rm T}}{B}^{{\rm T}}{\pi }_x\right)\mathrm{,}$ (23)

where $e_{\xi }^{{\rm T}}\equiv \left[001\right]$ . The effect summarized by ${\pi }_x^{{\rm T}}$ is standard [60] . It is a solution to the expectations part (i.e., $m_t^{\left(2\right)}=0$ ) of the difference equation in inflation (12), conditional on $x_t$ . Note that ${\nu }_y>0$ translates positive shocks to output growth (captured by $y_x^{{\rm T}}{A}_d$ ) to negative shocks to inflation. In contrast, ${\delta }_x^{{\rm T}}$ does the opposite, unless $\rho =1$ . The horse race between these two effects plays an important role in the determination of term premia and would not arise in settings with exogenous inflation [5] [6] .

In ${\pi }_v$ , the linear terms are expectations terms similar to those in ${\pi }_x$ . They come from the effect of $v_t$ on output growth in the Taylor rule (the first term) and on the conditional mean of the nominal pricing kernel (the second and third term). The quadratic terms result from the effect of $v_t$ on the second moments of the nominal pricing kernel (the terms in $m_t^{\left(2\right)}$ in Equation (12)). The variance term of the real pricing kernel, ${\lambda }_x^{{\rm T}}{\lambda }_x$ , reduces inflation when uncertainty rises. This effect on inflation can be interpreted as the effect of precautionary saving, similar to [40] .³⁶ The term ${\lambda }_x^{{\rm T}}{B}^{{\rm T}}{\pi }_x$ reflects covariance between inflation and the real pricing kernel, induced by variation in $x_t$ . If the elements, corresponding to a given element of $x_t$ , in both ${\lambda }_x^{{\rm T}}$ and ${\pi }_x$ are negative, then the covariance is positive. This corresponds to a situation of low inflation when the marginal value of real income is low (good times for the investor), so that a given nominal payoff in such a state translates into a high real payoff. This covariance plays an important role in the determination of term premia derived below.³⁷

The second moments of the pricing kernel impose restrictions on term premia and the lead-lag dynamics of nominal interest rates and inflation in relation to output growth. Observe that the three quadratic terms in ${\pi }_v$ can be rewritten as $-0.5{\left({\lambda }_x-B^{{\rm T}}{\pi }_x\right)}^{{\rm T}}\left({\lambda }_x-B^{{\rm T}}{\pi }_x\right)$ . Their joint effect on inflation is thus unambiguously non-positive but the magnitude depends on the counteracting effects of the variance and covariance terms (precautionary savings v.s. term premia effects). The larger is the relative contribution of ${\pi }_x$ to the covariance term, the smaller is the joint effect of the second moments on inflation. In the limit, it can be zero. This creates the following potential tension: the larger is the contribution of the negative covariance between output growth and inflation to term premia, the more likely is the negative lead of inflation (and nominal interest rates) due to the expectations part of the pricing kernel (the news shock role of $v_t$ ), rather than its second moments (the volatility shock role of $v_t$ ).³⁸

3.4.4. Output

To solve the NKPC, take ( $\pi \mathrm{,}{\pi }_x^{{\rm T}}\mathrm{,}{\pi }_v$ ) as given. Solving Equation (4) for the output process yields

$y_x^{{\rm T}}=\frac{1}{\Omega }{\pi }_x^{{\rm T}}\left(I-\beta A\right)+e_z^{{\rm T}}\mathrm{,}$ (24)

$y_v=\frac{1}{\Omega }\left[{\pi }_v\left(1-\beta \theta \right)-\beta {\pi }_x^{{\rm T}}a\right]\mathrm{.}$ (25)

Observe again the recursive structure: $y_x^{{\rm T}}$ depends only on ${\pi }_x^{{\rm T}}$ , whereas $y_v$ depends on both ${\pi }_v$ and ${\pi }_x^{{\rm T}}$ .³⁹ As the NKPC does not depend on the share of hand-to-mouth agents in the economy, these agents affect the coefficients of the output process only in general equilibrium, through ${\pi }_x^{{\rm T}}$ and ${\pi }_v$ . Observe from (24) that the more persistent is a given shock, the closer the corresponding element of $\left(I-\beta A\right)$ is to zero and thus, for a given ${\pi }_x^{{\rm T}}$ , the smaller is the transmission of the shock to output through the NKPC. For highly persistent shocks, the model with the NKPC behaves almost like a flexible-price model. In (25), the situation regarding the effect of the persistence of $v_t$ is more involved, as the general equilibrium effect of $v_t$ on output operates through both ${\pi }_v$ and ${\pi }_x^{{\rm T}}$ . Thus, even for $\theta$ close to one, $v_t$ can propagate through the NKPC due to the second term in (25). Under flexible prices, $\Omega =\infty$ and $y_x^{{\rm T}}=e_z^{{\rm T}}=\left[100\right]$ , $y_v=0$ .

3.4.5. The System of Equilibrium Coefficients

Substituting for the coefficients of the value function and the real pricing kernel, the joint system of the equilibrium coefficients (22)-(25), pinned down by the functional Equations (4) and (12), is linear in the unknowns and recursive. Observe that Equations (22) and (24) can be solved for ${\pi }_x^{{\rm T}}$ and $y_x^{{\rm T}}$ . Given this solution, Equations (23) and (25) can then be solved for ${\pi }_v$ and $y_v$ . (The coefficients $\pi$ and y are obtained in the last step.) The response of the economy to the volatility shock thus depends on how the economy responds to the $x_t$ shocks.⁴⁰

The rigidities in the real economy affect the equilibrium coefficients in two ways. First, the fraction of the hand-to-mouth households ( $\lambda$ ) enters the coefficients (22) and (23) of the inflation process through the sharing rule entering the real pricing kernel. Second, the Calvo parameter ( $\zeta$ ) enters the coefficients (24) and (25) of the output process. The effects of the rigidities are, however, interlinked: if prices are flexible ( $y_t=z_t$ ), the fraction of hand-to-mouths in the population has no effect on the pricing kernel, as follows from (10).

3.5. Yield Curve and Risk Premia

The yield curve for zero-coupon bonds can be derived from a set of no-arbitrage conditions. Assume that the log price of a n-maturity bond is linear in the state space

$-\log q_t^{\left(n\right)}={\gamma }^{\left(n\right)}+{\gamma }_x^{\left(n\right){\rm T}}{x}_t+{\gamma }_v^{\left(n\right)}{v}_t\mathrm{.}$ (26)

Using the relationship between bond prices and interest rates, $-\log q_t^{\left(n\right)}=n{i}_t^{\left(n\right)}$ , interest rates are given by

$i_t^{\left(n\right)}=\frac{1}{n}\left({\gamma }^{\left(n\right)}+{\gamma }_x^{\left(n\right){\rm T}}{x}_t+{\gamma }_v^{\left(n\right)}{v}_t\right)\mathrm{,}$ (27)

where $i_t^{\left(1\right)}=i_t$ is the short rate.

Bond prices have to satisfy the no-arbitrage condition $q_t^{\left(n\right)}=E_t\left(m_{t+1}^{\mathrm{<mi>\; \$\; </mi>}}{q}_{t+1}^{\left(n-1\right)}\right)$ , starting with $q_{t+1}^{\left(0\right)}=1$ . Recall that $\log m_{t+1}^{\mathrm{<mi>\; \$\; </mi>}}=\log m_{t+1}-{\pi }_{t+1}$ , so that one could also write $q_t^{\left(n\right)}=E_t\left[m_{t+1}{q}_{t+1}^{\left(n-1\right)}\exp \left(-{\pi }_{t+1}\right)\right]$ and think of the no-arbitrage condition in terms of the real pricing kernel and a real payoff. Substituting the guess (26) in both sides of the no-arbitrage condition gives a recursive system

${\gamma }_x^{\left(n\right){\rm T}}=-{\delta }_x^{\mathrm{<mi>\; \$\; </mi>}{\rm T}}+{\gamma }_x^{\left(n-1\right){\rm T}}A\mathrm{,}$ (28)

${\gamma }_v^{\left(n\right)}=-\left({\delta }_v^{\mathrm{<mi>\; \$\; </mi>}}-{\gamma }_x^{\left(n-1\right){\rm T}}a\right)-\frac{1}{2}\left({\lambda }_x^{\mathrm{<mi>\; \$\; </mi>}{\rm T}}-{\gamma }_x^{\left(n-1\right){\rm T}}B\right){\left({\lambda }_x^{\mathrm{<mi>\; \$\; </mi>}{\rm T}}-{\gamma }_x^{\left(n-1\right){\rm T}}B\right)}^{{\rm T}}+{\gamma }_v^{\left(n-1\right)}\theta \mathrm{,}$ (29)

$\begin{array}{c}{\gamma }^{\left(n\right)}=-\left[{\delta }^{\mathrm{<mi>\; \$\; </mi>}}+{\gamma }_x^{\left(n-1\right){\rm T}}av-{\gamma }_v^{\left(n-1\right)}\left(1-\theta \right)v\right]\\ -\frac{1}{2}\left({\lambda }_v^{\mathrm{<mi>\; \$\; </mi>}{\rm T}}-{\gamma }_v^{\left(n-1\right)}b\right){\left({\lambda }_v^{\mathrm{<mi>\; \$\; </mi>}{\rm T}}-{\gamma }_v^{\left(n-1\right)}b\right)}^{{\rm T}}+{\gamma }^{\left(n-1\right)}\mathrm{,}\end{array}$ (30)

where in each equation the respective recursive coefficient at $\left(n-1\right)$ is listed as last on the right-hand side. The system can be solved from the initial conditions $\gamma =0$ , ${\gamma }_x^{{\rm T}}=0$ , and ${\gamma }_v=0$ (i.e., $q_t^{\left(0\right)}=1$ ). Observe that, here again, ${\gamma }_x^{\left(n\right){\rm T}}$ is determined first, followed by ${\gamma }_v^{\left(n\right)}$ , and finally by ${\gamma }^{\left(n\right)}$ .

3.5.1. The Economic Interpretation of the Yield Curve Coefficients

To gain economic insight into the implications of the recursive system (28)-(30) for the yield curve, consider first Equation (28). Substituting for ${\delta }_x^{\mathrm{<mi>\; \$\; </mi>}{\rm T}}$ and solving the equation forward by recursive substitutions gives a closed-form solution

${\gamma }_x^{\left(n\right){\rm T}}=-\left(\rho -1\right)c_x^{{\rm T}}{A}_d{\Pi }_n+{\pi }_x^{{\rm T}}A{\Pi }_n\mathrm{,}$ (31)

where ${\Pi }_n={\left(I-A\right)}^{-1}\left(I-A^{n+1}\right)$ , which depends positively on the persistence of the $x_t$ process. The loading ${\gamma }_x^{\left(n\right){\rm T}}$ is a pure expectations hypothesis term (corresponding to the solution to a sequence of simple Fisher equations), where $c_x^{{\rm T}}{A}_d{\Pi }_n$ is expected consumption growth between t and $t+n$ and ${\pi }_x^{{\rm T}}A{\Pi }_n$ is expected inflation between t and $t+n$ , conditional on a unit of $x_t$ . Higher expected consumption growth or inflation thus increase the nominal interest rate on the n-period bond, consistent with the Fisher relationship (recall that $\rho \le 1$ ).

In the expression (29) for ${\gamma }_v^{\left(n\right)}$ , the linear terms after the equality sign are expectations terms. In addition to expectations about consumption growth and inflation (embedded in ${\gamma }_x^{\left(n-1\right){\rm T}}$ and $-{\delta }_v^{\mathrm{<mi>\; \$\; </mi>}}$ ), the terms include expectations about the certainty equivalent (see the expression for ${\delta }_v$ derived in Section 3.4.2). As in the case of $x_t$ , higher expected consumption growth or inflation increase the interest rate (through both ${\gamma }_x^{\left(n-1\right){\rm T}}$ and $-{\delta }_v^{\mathrm{<mi>\; \$\; </mi>}}$ ), in line with the Fisher relationship. The effect of the certainty equivalent is also positive. When $v_t$ increases, the agent is willing to accept a lower certain price today for the bond, increasing the interest rate.

The quadratic term in (29) comprises of a variance term for the nominal pricing kernel, $-0.5{\lambda }_x^{\mathrm{<mi>\; \$\; </mi>}{\rm T}}{\lambda }_x^{\mathrm{<mi>\; \$\; </mi>}}$ , Jensen’s inequality term, $-0.5{\gamma }_x^{\left(n-1\right){\rm T}}B{B}^{{\rm T}}{\gamma }_x^{\left(n-1\right)}$ , and a risk premium term, ${\lambda }_x^{\mathrm{<mi>\; \$\; </mi>}{\rm T}}{B}^{{\rm T}}{\gamma }_x^{\left(n-1\right)}$ , which is the covariance between the price of risk and the yield of a $\left(n-1\right)$ -period bond. The term premium on the entire bond is determined by a sequence of these terms in recursive forward substitutions of Equation (29). Observe that all three quadratic terms pertain to $x_t$ , even though they are a part of the coefficient loading onto $v_t$ in the interest rate Equation (27). The response of the n-period yield to $v_t$ working through the second moments thus depends on the properties of the response of the $\left(n-1\right)$ -period yield and the nominal pricing kernel to $x_t$ . If a given element of $x_t$ has its corresponding element in ${\lambda }_x^{\mathrm{<mi>\; \$\; </mi>}{\rm T}}$ negative, then for the risk premium associated with this factor to be positive, we need the respective element in ${\gamma }_x^{\left(n-1\right)}$ to be also negative. That is, the yield must be low (the nominal bond price must be high) in “good times” for the investor, when the marginal value of nominal income is low.

Finally, note that the parameter a, which controls the lead-lag relationship between volatility and productivity growth, shows up in the expectations part of ${\gamma }_v^{\left(n\right)}$ , as well as in the term premium part of ${\gamma }^{\left(n\right)}$ (through both ${\gamma }_x^{\left(n-1\right)}$ and the presence of $u_v$ in ${\lambda }_v^{\mathrm{<mi>\; \$\; </mi>}{\rm T}}$ ). It thus affects not only the responses of interest rates to $v_t$ due to the expectations hypothesis but also steady-state term premia. The lead-lag dynamics and term premia are thus interconnected.

3.5.2. Term Premia and Intertemporal Substitution

From (31) follows that the yield is low (the price is high) when a given element of $x_t$ is associated with either low expected consumption growth or low expected inflation. Thus, to get a positive risk premium, we need these expectations to prevail in times when the same $x_t$ implies a low marginal value of nominal income (good times for the investor). From the expression for ${\lambda }_x^{{\rm T}}$ follows that this is the case when either current consumption growth or expected future consumption growth are high. The latter effect, however, is inconsistent with a low yield brought about by low expected consumption growth due to the same $x_t$ . From ${\lambda }_x^{\mathrm{<mi>\; \$\; </mi>}{\rm T}}={\lambda }_x^{{\rm T}}-{\pi }_x^{{\rm T}}B$ follows that a low marginal value of nominal income also occurs when the $x_t$ implies high current inflation. However, to the extent that inflation is positively autocorrelated, high current inflation is inconsistent with a low yield brought about by low expected inflation due to the same $x_t$ .

A combination of ${\gamma }_x^{\left(n-1\right)}$ and ${\lambda }_x^{\mathrm{<mi>\; \$\; </mi>}{\rm T}}$ that does work is if the effect of expected consumption growth on ${\gamma }_x^{\left(n-1\right)}$ is attenuated by $\rho$ sufficiently close to one-see Equation (31)-and ${\gamma }_x^{\left(n-1\right)}$ thus predominantly reflects inflation expectations. Then, if ${\pi }_x$ is negative and $u_x^{{\rm T}}$ is positive and sufficiently large, we could have both ${\gamma }_x^{\left(n-1\right)}$ and ${\lambda }_x^{\mathrm{<mi>\; \$\; </mi>}{\rm T}}$ negative (the former due to a negative ${\pi }_x$ , the latter through the presence of a sufficiently large $u_x^{{\rm T}}$ in ${\lambda }_x^{{\rm T}}$ ; see Subsection 3.4.2 and recall that $\alpha <0$ ). From the solution for $u_x^{{\rm T}}$ in Section 3.4.1 follows that $u_x^{{\rm T}}$ is positive and large for persistent shocks to consumption growth. From equation (22) and the solution for ${\delta }_x^{{\rm T}}$ in Section 3.4.2 follows that ${\pi }_x$ is negative if the respective element of $x_t$ increases expected output growth, the Taylor rule weight on output growth is positive, and $\rho$ , again, is sufficiently close to one. $\rho$ sufficiently close to one is thus necessary for both ${\gamma }_x^{\left(n-1\right)}$ and ${\pi }_x$ being negative. Like $u_x^{{\rm T}}$ , both ${\gamma }_x^{\left(n-1\right)}$ and ${\pi }_x$ increase in absolute value with the persistence of the shock.

In sum, the above combination describes a situation when the yield is low (the bond price is high) due to low inflation expectations (showing up in ${\gamma }_x^{\left(n-1\right)}$ ) and, at the same time, the marginal value of income is low due to high expected future consumption growth (showing up in ${\lambda }_x^{{\rm T}}$ ), with these expectations not being significantly reflected in bond prices (due to a high $\rho$ ; i.e., not showing up in ${\gamma }_x^{\left(n-1\right)}$ ).⁴¹

3.5.3. Time Variation in Expected Excess Returns

The above principles that determine term premia also determine expected excess returns. Following the definition from Section 2, one-period excess return on a n-period bond is given by $r_{X\mathrm{,}t+1}^{\left(n\right)}\equiv \left(\log q_{t+1}^{\left(n-1\right)}-\log q_t^{\left(n\right)}\right)-i_t$ . Using the equilibrium functions for $\log q_{t+1}^{\left(n-1\right)}$ , $\log q_t^{\left(n\right)}$ , and $i_t$ derived above, and taking expectations, gives the expected excess return on the $n$ -period bond

$E_t{r}_{X\mathrm{,}t+1}^{\left(n\right)}={\nu }^{\left(n-1\right)}+\left({\gamma }_x^{\left(n-1\right){\rm T}}B{\lambda }_x^{\mathrm{<mi>\; \$\; </mi>}}-\frac{1}{2}{\gamma }_x^{\left(n-1\right){\rm T}}B{B}^{{\rm T}}{\gamma }_x^{\left(n-1\right)}\right)v_t\mathrm{,}$ (32)

where ${\nu }^{\left(n-1\right)}\equiv {\gamma }_v^{\left(n-1\right)}b{\lambda }_v^{\mathrm{<mi>\; \$\; </mi>}}-0.5{\gamma }_v^{\left(n-1\right)}b{b}^{{\rm T}}{\gamma }_v^{\left(n-1\right){\rm T}}$ ; see the Appendix for derivation. The first term in the parentheses is the covariance term determining term premia, discussed above, while the second term is the Jensen’s inequality term, which is small. The covariance term clearly affects the extent to which $E_t{r}_{X\mathrm{,}t+1}^{\left(n\right)}$ responds to $v_t$ . In contrast, the covariance term ${\gamma }_v^{\left(n-1\right)}b{\lambda }_v^{\mathrm{<mi>\; \$\; </mi>}}$ , contained in ${\nu }^{\left(n-1\right)}$ , affects the mean (steady-state) excess return, but not its variation. It also affects the mean of term premia; see Equation (30). The parameter a controls the lead-lag relationship between volatility and productivity growth, and thus between expected excess returns and output growth. However, it also affects steady-state expected excess returns through the terms in ${\nu }^{\left(n-1\right)}$ .

4. Quantitative Analysis

Having explained the mechanism, this section: i) evaluates if the model is quantitatively consistent with the stylized facts summarised in Section 2 and ii) shows that the resulting asset pricing structure coexists with a large fraction of the population behaving like hand-to-mouths in an environment with nominal price rigidities.

4.1. Calibration

As a benchmark, consider the solution to the bond market equilibrium condition (12), given $y_x^{{\rm T}}=\left[100\right]$ and $y_v=0$ . This is a flexible-price version of the model, denoted by ${\mathcal{M}}_1$ . Recall that hand-to-mouth agents do not affect the pricing kernel under flexible prices.

The following parameters are shared across the flexible- and sticky-price specifications: $g=2/400$ , $i=5.55/400$ , and ${\pi }^{\mathrm{<mo>\; *\; </mo>}}=3.9/400$ . They are chosen to be consistent with the sample averages, 1961-2008. Further, $\omega =0.65$ is chosen on the grounds of the average labor share in NIPA.⁴² Conditional on ${\mathcal{M}}_1$ , the remaining 15 parameters are pinned down by minimizing the distance between the model and the data of 15 equally weighted calibration targets, listed in Table 2. The parameters thus calibrated are: $\beta$ , $\rho$ , $\alpha$ (preferences), ${\nu }_{\pi }$ , ${\nu }_y$ (Taylor rule), and ${\varphi }_z$ , ${\varphi }_s$ , ${\varphi }_{\xi }$ , $a_z$ , $a_s$ , $\theta$ , $B_{11}$ , $B_{22}$ , $B_{33}$ , $b_4$ (stochastic processes). The resulting parameter values are reported in the first column of Table 2. The largest discrepancy between the model and data moments is in the volatility of the expected excess return on the long bond. This is discussed in further detail in Section 4.3.

A noteworthy feature of the resulting parameterization is that $\rho =0.9$ , as anticipated by the discussion in Section 3.5. This implies the elasticity of intertemporal substitution equal to 10. The risk aversion parameter is −28.⁴³

The estimates of the utility function imply the following behavior of bond investors: when faced with payoffs that appear to be certain (in real terms), only a small increase in real interest rates in sufficient to convince investors to postpone consumption by an extra period. However, when faced with uncertain payoffs, the compensation for the investment has to be large. Consequently, asset prices mainly reflect hedging motives of investors, rather than

Table 2. Calibration.

Notes. Model nomenclature: ${\mathcal{M}}_1$ = flexible prices, ${\mathcal{M}}_2$ = sticky prices. Parameters that are shared across the models: $g=2/400$ , $i=5.55/400$ , ${\pi }^{\mathrm{<mo>\; *\; </mo>}}=3.9/400$ , which are chosen to be consistent with the sample averages, 1961-2008; and $\omega =0.65$ , which reflects the average labor share in NIPA. Conditional on these parameters (and the parameters of the high street in model ${\mathcal{M}}_2$ ), the parameters in the table are determined by minimizing the distance between the model and the data of the 15 equally weighted calibration targets, which are the averages for 1961-2008. For the long bond, $N=28$ stands for a 7-year bond (28 quarters).

intertem poral substitution.

The Taylor rule parameters are within the bounds found in the literature. The Taylor rule shock is highly persistent, thus resembling the inflation target shock of, e.g. [61] rather than a transitory policy disturbance (the role of transitory policy shocks is explored later).⁴⁴ The shock to the conditional mean of productivity growth is also highly persistent, in line with [2] . However, the persistence of the volatility shock (0.8) is much lower than in their model, where it takes a value close to one. This is because, unlike in their paper, the calibration here takes into account the lead-lag pattern of expected excess returns. To capture this dynamics, the autocorrelation of the volatility shock cannot be too high. The persistence of the shock to the level of productivity is a little lower but close to the RBC literature. Both elements of a are positive, with $a_z$ being two orders of magnitude larger than $a_s$ . Finally, while the volatility shock is substantially less persistent than the other shocks, it has the largest conditional standard deviation.

In the version with sticky prices ( ${\mathcal{M}}_2$ ), $\lambda =0.41$ , $\epsilon =0.478$ , and $\eta =1$ , which are chosen to reproduce Table 1 in [62] , the [12] case. Recall that the parameters of the hand-to-mouth population affect the part of the pricing kernel related to shocks other than $z_t$ . The Calvo parameter is chosen to make Ω in the NKPC (4) achieve the standard value in the literature. This yields the value of the Calvo parameter close to 0.7, which is also standard. The remaining parameters are calibrated following the same strategy as for ${\mathcal{M}}_1$ . The resulting values are reported in the second column of Table 2 and are in general similar to ${\mathcal{M}}_1$ , with the exception of $B_{11}$ .

4.2. Properties of the Equilibrium Pricing Kernel

Table 3 reports the quantitative properties of the equilibrium pricing kernel, and its determinants, to connect the quantitative results with the discussion in the previous sections and help interpret the results that follow. Starting with ${\mathcal{M}}_1$ , there are only small differences between the real and nominal pricing kernels in terms of risk prices, with the resulting nominal risk prices being determined predominantly by the real kernel. Further, the only factor that is significantly priced is $s_t$ and the time-variation in the risk premium attached to this factor is driven by another factor, $v_t$ , which itself has a price of risk equal to zero. Including the variance of expected excess returns among the calibration moments drives ${\lambda }_v^{\mathrm{<mi>\; \$\; </mi>}}$ down to zero, thus making $v_t$ close to welfare neutral, with ${\lambda }_v$ being almost zero (more on this in the next section). Such a parsimonious asset pricing structure is akin to the reduced-form model of [4] . Also, in accordance with their paper, the priced factor is closely related to the reduced-form level factor, as shown in Table 4, while the factor driving the time-variation in risk

Table 3. Equilibrium pricing kernel.

Notes. Model nomenclature: ${\mathcal{M}}_1$ = flexible prices; ${\mathcal{M}}_2$ = sticky prices. The order of the factors in the above vectors is: $z_t$ , $s_t$ , ${\xi }_t$ , $v_t$ , with volatility, where applicable, reported separately. The nominal pricing kernel is related to the real pricing kernel as: ${\delta }_x^{\mathrm{<mi>\; \$\; </mi>}{\rm T}}={\delta }_x^{{\rm T}}-{\pi }_x^{{\rm T}}A$ , and ${\delta }_v^{\mathrm{<mi>\; \$\; </mi>}}={\delta }_v-{\pi }_x^{{\rm T}}a-{\pi }_v\theta$ for the factor loadings; and as ${\lambda }_x^{\mathrm{<mi>\; \$\; </mi>}{\rm T}}={\lambda }_x^{{\rm T}}-{\pi }_x^{{\rm T}}B$ and ${\lambda }_v^{\mathrm{<mi>\; \$\; </mi>}{\rm T}}={\lambda }_v^{{\rm T}}-{\pi }_{v}b$ for the prices of risk. The standard deviations of the shocks are: in ${\mathcal{M}}_1$ , $B_{11}=0.0053$ , $B_{22}=0.002$ , $B_{33}=0.000232$ , $b_4=0.23$ ; in ${\mathcal{M}}_2$ , $B_{11}=0.001$ , $B_{22}=0.002$ , $B_{33}=0.000232$ , $b_4=0.23$ .

premia is correlated with the reduced-form slope factor.⁴⁵

The significant price of risk of $s_t$ is due to the large value of this factor’s corresponding element in $u_x^{{\rm T}}$ , reflecting the fact that this shock persistently shifts the expected future growth rate of output. Observe also that the loading on $s_t$ in the equilibrium inflation process is negative, as required for a positive term premium attached to $s_t$ . Turning to ${\mathcal{M}}_2$ , the presence of the NKPC does not have a material effect on the pricing kernel. If anything, it strengthens the result that only $s_t$ is priced by reducing the conditional variance of $z_t$ required to match the data, thus reducing the price of risk of $z_t$ . Further, despite the nominal rigidities, the Taylor rule shock is not significantly priced. Referring back to Section 3.4, this is because the NKPC transmits into output, in a quantitatively meaningful way, only shocks that are temporary. However, in order to match the yield curve moments listed Table 2, the Taylor rule shock has to be persistent.

Anticipating the findings below, observe that the equilibrium loading on $v_t$ in the inflation process is larger (in absolute value) in ${\mathcal{M}}_2$ than in ${\mathcal{M}}_1$ . Consequently, in ${\mathcal{M}}_2$ , volatility accounts for some short-run movements in output at the expense of the decline in the conditional standard deviation of the temporary shock $z_t$ , which in ${\mathcal{M}}_2$ is five times smaller than in ${\mathcal{M}}_1$ . The effect of volatility on output working through sticky prices is negative, in line with the uncertainty literature noted in the Introduction. The shock thus first reduces

Table 4. Principal components and structural shocks.

Notes. Model nomenclature: ${\mathcal{M}}_1$ = flexible prices; ${\mathcal{M}}_2$ = sticky prices.

output through nominal price rigidities, before spilling over into future productivity, as captured by the parameter a. While this has only marginal implications for the pricing kernel, it improves the model’s ability to account for the observed lead-lag patterns of inflation and interest rates.

In both ${\mathcal{M}}_1$ and ${\mathcal{M}}_2$ , the effect of $z_t$ on output ( $y_t$ ) is similar, equal to one (in ${\mathcal{M}}_1$ this is by construction, in ${\mathcal{M}}_2$ there is an additional effect of sticky prices on aggregate demand); see Table 3. The immediate effect of $s_t$ on output in ${\mathcal{M}}_1$ is zero. This is because $s_t$ is a news shock about future $z_t$ and only $z_t$ affects output. The news shock thus affects output only over time as the news starts materializing. In ${\mathcal{M}}_2$ , the news shock has also an immediate effect on output as the news affects aggregate demand and, through nominal price rigidities, also output.

Finally, the resulting pricing kernel satisfies the Hansen-Jagannathan bound. The Sharpe ratio in the data is 0.29 for the 1-year bond and 0.13 for the 7-year bond. The ratio of the unconditional standard deviation of the pricing kernel to the mean is 0.46 in ${\mathcal{M}}_1$ and 0.45 in ${\mathcal{M}}_2$ .

4.3. The Model and the Stylized Facts

Stylized Facts 1. Figure 3 is the model counterpart to Figure 1. As in the data, the average yield curve is upward sloping and concave, with the term premium on mid and long bonds almost the same as in the data. The volatility curve shares with its empirical counterpart the key property that volatility is fairly flat across maturities. To the naked eye, there are no differences between ${\mathcal{M}}_1$ and ${\mathcal{M}}_2$ and the figure only contains plots for one model.

Stylized Facts 2. Figure 3 also shows that the loadings on the three most important PCs of yields are almost the same as in the data. Again, to the naked eye, there are no differences between ${\mathcal{M}}_1$ and ${\mathcal{M}}_2$ . The loadings on the single most important PC of excess returns in Figure 3 are, as in the data, upward

Figure 3. Model results: average yield and volatility curves and loadings on principal components. The results are nearly identical for the flexible ( ${\mathcal{M}}_1$ ) and sticky price ( ${\mathcal{M}}_2$ ) specifications. Only one set of curves is therefore plotted as separate plots for the two specifications would be almost indistinguishable.

sloping, but the value at the long end is lower than in the data. The loadings are again essentially the same for ${\mathcal{M}}_1$ and ${\mathcal{M}}_2$ . The PCs in the model also account for similar magnitudes of the total variance of yields across maturities as in the data (Table 4).

Stylized facts 3 and 4. Similarly to the data, the first PC of yields in the model is highly persistent and, as already reported in Table 2, strongly positively correlated with inflation. A direct consequence of the structure of the pricing kernel reported in Table 3 is that the time-variation in risk premia is related to the slope factor (the second PC of yields). As reported in Table 4, the correlation between $v_t$ and the slope factor is around 0.7 in both ${\mathcal{M}}_1$ and ${\mathcal{M}}_2$ . The level factor (the first PC of yields) is unrelated to movements in risk premia. Its correlation with $v_t$ is weak in both ${\mathcal{M}}_1$ and ${\mathcal{M}}_2$ .

Stylized facts 5. Figure 4 is the model counterpart to Figure 2. As in the data, the short rate and inflation are similarly negatively correlated with output growth, with the strongest negative correlation occurring at a quarter lead. In contrast, the slope factor and the expected excess return on the long bond are positively correlated with output growth, with the strongest positive correlation occurring at a quarter lead. These correlations, however, are stronger than in the data. As in the data, the level factor has a negative lead. However, the stronger positive correlations of risk premia than in the data imply that the long rate is roughly uncorrelated with the business cycle in the model, instead of exhibiting weak negative correlations observed in the data. The tight comovement of the slope factor and expected excess returns with output growth indicates that the parsimonious asset pricing structure misses factors driving the slope of the yield curve and risk premia unrelated to the business cycle. The endogenous response of output to volatility in ${\mathcal{M}}_2$ makes the lead-lag dynamics more pronounced than in ${\mathcal{M}}_1$ , thus bringing the model closer to the data.

Volatility of expected excess returns. As already noted in Section 4.1, the model is unable to match the volatility of expected excess returns on the long bond, while being consistent with the other 14 calibration targets. In the model, the (annualized) standard deviation of the expected excess return is 0.82%, whereas in the data it is around 4%. The explanation is as follows. First, the adopted calibration strategy drives ${\lambda }_v^{\mathrm{<mi>\; \$\; </mi>}}$ down to zero by essentially choosing $a_s$ so that $v_t$ is close to welfare neutral. Equation (32) would suggest that in such a case the variance of $v_t$ can be chosen to exactly match the variance of $E_t{r}_{X\mathrm{,}t+1}^{\left(n\right)}$ without affecting steady-state risk premia through the ${\nu }^{\left(n-1\right)}$ term. However, there is a second constraint on the variance of $v_t$ . As $v_t$ is tied to $z_{t+1}$ through the spillover vector a in the stochastic process, increasing the variance of $v_t$ affects the properties of output growth. The empirical properties of output growth thus place further restrictions on the stochastic properties of $v_t$ . This supports the earlier conjecture that the model misses factors driving the slope of the yield curve and expected excess returns that are unrelated to the business cycle. In other words, the stochastic properties of output growth imply that the specific volatility factor considered in the model accounts for 25% of the

Figure 4. Model results: yield curve and the business cycle. Cross-correlations with the growth rate of output. The correlations are $\text{corr}\left(x_{t+j}\mathrm{,}{g}_t\right)$ , $j=-6,\cdots ,0,\cdots ,6$ , where x is the variable of interest and g is the growth rate of output.

variance of expected excess returns, leaving 75% to factors unrelated to the business cycle. This is different from models such as [2] and [6] , where the volatility factor follows an autonomous process.⁴⁶

Principal components and the structural shocks. A final result to note, reported in Table 4, is the relationship between the three reduced-form PCs of yields, frequently used as risk factors in affine term structure models, and the structural shocks in the model. While all four shocks are to some extent correlated with all three PCs of yields, the strength of the relationship is markedly different for different shocks. The level factor is strongly related to $z_t$ , $s_t$ , and ${\xi }_t$ . The slope factor is related to $v_t$ and $v_t$ is also strongly correlated with the quantitatively small curvature factor.

4.4. Hand-to-Mouths and Intertemporal Substitution

Table 5 explores the effect of hand-to-mouth agents on the pricing kernel. Recall, that the share $\lambda$ of hand-to-mouths in the population has a direct effect on consumption of bond investors through ${\Phi }_B$ in the sharing rule (10) and

Table 5. The share of hand-to-mouth households and the pricing kernel.

Notes. Applies to the sticky-price version (model ${\mathcal{M}}_2$ ). The order of the factors in the equilibrium vectors is: $z_t$ , $s_t$ , ${\xi }_t$ , ${\mu }_t$ , $v_t$ , where ${\mu }_t$ is the temporary Taylor rule shocks and volatility, where applicable, is reported separately. The loadings pertaining to the Taylor rule shock are highlighted in bold. The autocorrelation of the temporary shock is 0.7. The standard deviations of the shocks are: $B_{11}=0.001$ , $B_{22}=0.002$ , $B_{33}=0.000232$ , $B_{44}=0.0025$ , $b_4=0.23$ .

general equilibrium effects working through the equilibrium responses of output to shocks other than $z_t$ , provided nominal prices are sticky. As the NKPC transmits only temporary shocks, whereas the yield-curve moments used in the calibration require the Taylor rule shock to be highly persistent, resembling an inflation target shock, for the purpose of this exercise I add a purely temporary shock ${\mu }_t$ in the Taylor rule. Its persistence is set equal to 0.7 and the conditional standard deviation to 0.0025.

[56] reports a fraction of rich hand-to-mouth households in the population between 30% and 50%. The baseline $\lambda =0.41$ is based on [62] , the [12] case in his terminology. In this case, consumption of hand-to-mouths responds 2.2 times as much to the temporary policy shock as consumption of bond investors. Table 5 explores values from 0.21, which (given the value of $\epsilon$ ) maximizes the hedge for the hand-to-mouths, to 0.91, a value well above any reasonable estimates in the literature. The table reports the loadings on the shocks in the equilibrium consumption process of the hand-to-mouths, the equilibrium nominal pricing kernel, and the steady-state risk premium on the 7-year bond. In line with the macro literature, the higher is $\lambda$ , the stronger is the response of consumption of hand-to-mouths to the temporary shock. The response increases exponentially. However, unless the value of $\lambda$ substantially exceeds the estimates in [56] , the effects on the pricing kernel are small. The same (to a lesser extent) applies to $v_t$ , the other temporary shock that is transmitted through the NKPC in a quantitatively significant way.⁴⁷

Finally, Figure 5 explores the consequences of a lower elasticity of intertemporal substitution of bond investors. Four values of $\rho$ are considered: $\rho =0.88$ (the baseline value), and three alternative values, $\rho =0.6,0.5,0.3$ . The baseline value corresponds to the elasticity of intertemporal substitution equal to 8.33; the alternative values to 2.5, 2, and 1.43, respectively. The figure demonstrates the effects of $\rho$ on the average yield curve and on the cross-correlations of expected excess returns (on the 7-year bond), inflation, and the short rate with output growth at various leads and lags. Lower values of $\rho$ lead to counterfactually positive cross-correlations of the short rate with future output growth, despite generally negative cross-correlations of the inflation rate with future output growth. This is because the real interest rate becomes strongly positively correlated with future output growth due to a strong intertemporal substitution effect: high expected future income growth induces bond investors to borrow, thus increasing the real rate in equilibrium. This, consequently, makes nominal bonds a hedge and leads to negative risk premia and a downward sloping average yield curve. Further, the long-short spread and expected excess returns become negatively correlated with future output growth. As discussed in Section 3.5, a negative correlation between inflation and output growth is not sufficient for positive term premia, as the cases of $\rho =0.5$ and $\rho =0.3$ demonstrate.

Figure 5. Consequences of the elasticity of intertemporal substitution ( $1/\left(1-\rho \right)$ ). The cross-correlations are with respect to the growth rate of output.

5. Conclusions

The paper shows that a parsimonious pricing kernel goes a long way accounting for key stylized facts of the term structure, including its leading indicator properties over the business cycle. The joint macro and nominal yield curve data suggest that the stand-in bond investor cares mainly about hedging consumption-inflation risk, rather than intertemporal smoothing. That is, the data imply a high elasticity of intertemporal substitution but a low appetite for risk. Furthermore, the riskiness of only one factor-the conditional mean of output growth, is substantially priced by the equilibrium pricing kernel. The riskiness of this factor is time-varying due to time-varying volatility, but shocks to volatility are approximately welfare-neutral, thus themselves not contributing to risk premia. The negative covariance, induced in equilibrium by the Taylor rule, between inflation and nominal interest rates on one hand and the priced factor on the other makes nominal bonds risky. The equilibrium pricing kernel implies that low levels of interest rates observed in the data ahead of an economic expansion reflect news about higher future output growth, resulting in lower inflation. If the positive news is contained in the volatility factor, the associated increase in the long-short spread (a steeper yield curve) also reflects elevated uncertainty about the future growth path, leading to higher term premia. It is this dual role of the volatility factor that makes it approximately welfare neutral, thus carrying a zero price of risk.

The nominal nonneutrality embedded in the New-Keynesian Phillips Curve, as well as the size of the hand-to-mouth population, have quantitatively negligible effects on this basic result. This is because these rigidities, even if leading to sizable macro outcomes, have only short-term effects on consumption of bond investors and thus small effects on their lifetime utilities underpinning the equilibrium prices of risk.

Compared with the multiple sources of risk in many other term structure models, the structural model explored here may seem too simplistic. An advantage of its parsimony is that the mechanism is transparent and the model provides a simple bird’s eye interpretation of the joint macro and yield curve data, as summarized by the stylized facts. The lead-lag dynamics discipline the extent to which the model can account for the empirical volatility of expected excess returns. It suggests that about one quarter of the volatility of expected excess returns is tied to the business cycle. The remaining sources of the time variation in risk premia would appear unrelated to the average business cycle.

The proposed model also has a number of potential limitations. First, the model’s predictions are conditional on monetary policy following the Taylor rule. The model is thus not suitable for periods in which monetary policy is constrained by the zero lower bound and resorts to unconventional policy. The model is also not suitable for periods in which monetary policy independence is subordinated to fiscal policy. Second, the predictions of the model are conditional on the particular parameterization of the Taylor rule. The parameterization was chosen so that the model fits the historical data as well as possible. The estimated parameter values are within the estimates in the literature. However, if the parameters of the policy rule change (for instance, monetary policy starts to respond more to output and less to inflation), the empirical correlations may change too. Finally, as the empirical lead-lag correlations are not perfect, the interpretation proposed by the model is not applicable to all scenarios. The interpretation is conditional on output growth shocks being the main sources of aggregate fluctuations.

NOTES

¹Features (a) and (b) echo the properties of the reduced-form model of [4] . In accordance with [4] , the priced factor is correlated with the reduced-form level factor, while the factor driving movements in risk premia is correlated with the reduced-form slope factor.

²E.g., [5] [6] and [7] .

³Essentially, these adverse effects of a low elasticity of intertemporal substitution on the yield curve are different manifestations of the insights of [8] and [9] .

⁴A low elasticity of intertemporal substitution would generate a large enough increase in the real interest rate ahead of future output growth that would make nominal interest rates and future output growth, counterfactually, positively correlated and the term spread (excess returns) and future output growth, counterfactually, negatively correlated.

⁵This is higher than the median of the estimates in the literature, obtained typically from the responses of consumption growth to the real rate [11] .

⁶See e.g. [15] - [21] , and [22] . [23] provides a review of the literature.

⁷Predecessors to the above models either derive the pricing kernel from preferences but take the inflation-output (consumption) process as given [5] [6] [10] [30] , or derive the processes for output and inflation from a structural model but take the pricing kernel from an affine term structure model [31] [32] . Recent examples of the former approach are [7] and [33] . [13] and [34] solve for inflation, given a process for output; [35] do the opposite. [5] take into account the lead-lag correlations between output and inflation as a part of the estimated exogenous output-inflation process.

⁸ [7] point out shocks to the rate of time preference.

⁹Although, by its very nature, the model has no time-varying idiosynscratic uncertainty [38] [39] [40] , the volatility factor is a source of movements in the second moments of the pricing kernel, resembling time-varying precautionary saving.

¹⁰The data for yields of maturities of one year and above come from the Federal Reserve Board database on the nominal yield curve (the Gürkaynak-Sack-Wright dataset), with the 3-month T-bill rate taken from FRED. To compute realized returns, the required bond prices are obtained from the cross-sectional, date-specific, [41] curve that comes with the Gürkaynak-Sack-Wright dataset. The dataset is at daily frequency. Yields and log bond prices are converted to quarterly frequency by simple averaging (returns are then computed from the bond prices at quarterly frequency). Data for all other variables come from FRED.

¹¹A 3^rd PC, accounting for 0.2% of the total variance, works like a “curvature factor”, changing the shape of the yield curve.

¹²The persistence in the VAR is moreover likely underestimated due to a small sample bias [45] [46] .

¹³In the forecasting regressions, the dependent variable is the return factor, the independent variables are a constant and the PCs of yields specified in the table.

¹⁴I take as the reference inflation rate the 1^st PC (96% of the variance) of year-on-year inflation rates of the following price indexes: CPI, CPI less food and energy, PCE price index, PCE price index excluding food and energy, and the GDP deflator.

¹⁵Including the 3^rd PC raises the adjusted R² of the quarterly return regression from 0.08 to 0.11; including also the 4^th PC brings no further improvements in the fit. Including as a regressor the growth rate of real GDP, to allow for unspanned macro risk [49] , did not significantly change the results in the sample considered here (not reported in the table).

¹⁶ [53] demonstrate that the negative lead of nominal interest rates is crucial for understanding the leading business cycle behavior of residential investment when house purchases are financed with mortgages.

¹⁷As before, the inflation rate is the 1^st PC of the inflation rates for various indexes. [54] document such inflation dynamics for a number of countries.

¹⁸The expected excess return on the long bond is obtained from a [27] forecasting regression (i.e., from regressing excess return on the 7-year bond on a constant and the 7YR-3M spread). Essentially the same result is obtained if the slope factor is used as a regressor instead of the spread, or if the return factor capturing excess returns across maturities is used as the left-hand side variable.

¹⁹Some authors argue that risk premia should be counter-cyclical [49] . When the correlations are computed with respect to the HP-filtered cyclical component of the level of real GDP, the contemporaneous correlation for the expected excess return is −0.44, with correlations at leads −6 to −1 being 0.38, 0.31, 0.19, 0.04, −0.11, −0.30, while those at lags 1 to 6 being −0.53 −0.56 −0.54 −0.52 −0.48 −0.38. Risk premia in the sample are thus negatively correlated with current and past levels of output, in accordance with [49] .

²⁰Other terminology used in the literature is “savers” v.s. “spenders”, “unconstrained” v.s. “constrained”, or “participants” v.s. “nonparticipants”.

²¹This assumption simplifies the equilibrium pricing kernel, facilitating more straightforward insights into the results. An economic justification for this assumption could be the observation that most adjustments in aggregate employment and hours worked in the data occur in the lower half of the income distribution that likely characterizes hand-to-mouth households.

²²In other words, long-term bonds are redundant assets in this economy. The one-period bond is included since, as described below, its interest rate is set by the central bank in relation to inflation and, thus, the bond pins down the nominal side of the economy.

²³The per-period utility function of the bond investor embedded in Equation (1) has the same form as that of the hand-to-mouth household, but with a general elasticity of intertemporal substitution of consumption and the weight on disutility from labor equal to zero.

²⁴Log-linearizing the NKPC eliminates the upward pricing effect due to precautionary price setting [58] . This effect, however, is muted in the present model due to the volatility shock also affecting the conditional mean of productivity growth, not just its variance. To keep the analysis simple, I proceed with the log-linear version. Log-linearizing the NKPC around the zero inflation steady state reduces the stochastic discount factor in the NKPC only to $\beta$ . Given that $\beta$ is the same across agents, it renders irrelevant any discussion regarding which agent’s stochastic discount factor should be used to discount profits. In the calibrated model, the quarterly steady-state inflation rate $\pi$ is close to zero, equal to 0.00975.

²⁵When the steady state is normalized so that $w=z=1$ and bond investors are eliminated from the model ( $\lambda =1$ ), then $\epsilon =1$ (all dividends go to the hand-to-mouth agent) and $c_H=y=l_H$ . Consequently, Ω boils down to the standard expression in a representative-agent New-Keynesian model, $\Omega =\Phi \left(1+\eta \right)$ . As in [55] , I normalize the steady state so that $c_B=c_H$ , $l_B=l_H$ , $z=1$ , and $y=1$ . Further, $w=0.65$ , which reflects the labor share in NIPA and is consistent with the preference parameter $\omega =0.65$ .

²⁶Specifying the Taylor rule in terms of the output growth rate leads to a better fit of the model to macro and yield curve data than a specification in levels. Whether the current or expected growth rate is used has minuscule effects on the results, but the specification in terms of the expected growth rate is more convenient in terms of the state space. As in both the calibrated model and the data inflation is persistent, including into the Taylor rule also $E_t{\pi }_{t+1}$ has only small effects on the results. As in other models with Taylor rules, including ${\pi }_t$ is necessary for determinacy under flexible prices.

²⁷Strictly speaking, $v_t$ must be greater than zero and thus cannot be Gausian. However, as in [43] , it is possible to choose its variance so that the probability of $v_t$ being zero or negative is low enough and think of the Gausian assumption as a convenient approximation. In the numerical experiments, the incidence of $v_t\le 0$ is under 0.1%.

²⁸The implicit assumption in the above processes-that $v_t$ affects the conditional variance of all elements in $x_{t+1}$ -is adopted for parsimony. In a more general model, there could be a separate volatility variable for each element of $x_{t+1}$ .

²⁹The [2] process is a special case of (8)-(9), with ${\varphi }_z=1$ , ${\varphi }_s$ close to one, and $a_z=a_s=0$ . The specification used here can approximate their process arbitrarily well by letting ${\varphi }_z\to 1$ . I opt for the current specification as the lead-lag patterns in Figure 2 constitute dynamics for which the exact [2] process is too restrictive.

³⁰ [55] refers to this feature as “cyclical inequality”.

³¹See the Appendix for details.

³²The coefficient u has no effect on equilibrium allocations and prices; it only affects welfare and is given by $u=\frac{{\kappa }_0}{1-{\kappa }_1}+\frac{{\kappa }_1}{1-{\kappa }_1}\left[g-{\left(c_x+u_x\right)}^{{\rm T}}av-c_v{\theta }_{d}v+\left(1-\theta \right)u_{v}v+\frac{\alpha }{2}{\left(c_v+u_v\right)}^{2}b{b}^{{\rm T}}\right]$ .

³³The expression $c_x+u_x$ reflects the scaling of the lifetime utility at $t+1$ ; that is, $\frac{c_{B\mathrm{,}t+1}}{c_{Bt}}\frac{U_{t+1}}{c_{B\mathrm{,}t+1}}$ . Similarly for the expression $c_v+u_v$ . See the Appendix for details.

³⁴The second part, under the restriction $\alpha -\rho <0$ , is sometimes referred to in the literature as the “preference for an early resolution of uncertainty”. The standard pricing kernel for a time-additive CRRA utility function and constant volatility results under $\alpha =\rho$ and $b=c_v=a=0$ .

³⁵When risk increases, the agent is willing to accept lower certain income.

³⁶If a real one-period bond was priced by the real pricing kernel, the real interest rate would be given by $r_t=-\delta -{\delta }_x^{{\rm T}}{x}_t-{\delta }_v{v}_t-0.5{\lambda }_v^{{\rm T}}{\lambda }_v-0.5{\lambda }_x^{{\rm T}}{\lambda }_x{v}_t$ . When $v_t$ increases, the last term reduces the real rate, in line with the precautionary saving interpretation of the effect.

³⁷The third quadratic term in ${\pi }_v$ , ${\pi }_x^{{\rm T}}B{B}^{{\rm T}}{\pi }_x$ , is a Jensen's inequality term. This term is typically small.

³⁸Lastly, $\pi ={\left({\nu }_{\pi }-1\right)}^{-1}\left\{-i+{\nu }_{\pi }{\pi }^{\mathrm{<mo>\; *\; </mo>}}+{\nu }_y\left(y_x^{{\rm T}}a+y_v{\theta }_d\right)v-\delta -\left[{\pi }_x^{{\rm T}}a-\left(1-\theta \right){\pi }_v\right]v-\frac{1}{2}{\lambda }_v^{{\rm T}}{\lambda }_v-\frac{1}{2}b{b}^{{\rm T}}{\pi }_v^2+{\lambda }_v^{{\rm T}}{b}^{{\rm T}}{\pi }_v\right\}$ .

³⁹The constant is given by $y={\Omega }^{-1}\left[\pi \left(1-\beta \right)+\beta {\pi }_x^{{\rm T}}av-\beta {\pi }_{v}v\left(1-\theta \right)\right]$ .

⁴⁰This recursive property of the equilibrium is a direct consequence of the log-normality assumption for the shocks (i.e., only first and second moments matter) and the conditional variance of the shocks depending only on $v_t$ , not $x_t$ . Making the conditional variance depend on $x_t$ leads to a quadratic system with multiple solutions.

⁴¹This result does not mean that the expectations part of interest rates only reflects inflation expectations. It only states that such an effect has to sufficiently dominate the intertemporal substitution effect, reflecting expectations about consumption growth due to the same factor.

⁴²As already noted in Section 3.1, following [55] , I normalize the steady state so that $c_B=c_H$ , $l_B=l_H$ , $z=1$ , and $y=1$ . Under this normalization, $w=\omega =0.65$ . Finally, the normalization for v is $v=1$ .

⁴³Values of $\alpha$ similar to the one here are not unusual for Epstein-Zin preferences. For instance, in [6] , $\alpha =-20$ ; in [5] , $\alpha =-59$ . The value of $\rho$ has already been discussed in the context of the literature in the Introduction.

⁴⁴An inflation target shock is isomorphic to the shock in the Taylor rule (5) and can be expressed in terms of that shock as ${\pi }_t^{\mathrm{<mo>\; *\; </mo>}}=-{\left({\nu }_{\pi }-1\right)}^{-1}{\xi }_t$ . A high persistence of a Taylor rule shock is typical for the term structure papers noted in the Introduction.

⁴⁵Unlike in [4] , the factor driving risk premia here is spanned by the yield curve (yields have nonzero loadings on this factor).

⁴⁶It would also appear that it is possible to increase the variance of expected excess returns by increasing ${\lambda }_x^{\mathrm{<mi>\; \$\; </mi>}{\rm T}}$ , for instance by increasing the absolute value of $\alpha$ . However, this makes the average yield curve counterfactually too steep by increasing the average term premia.

⁴⁷The loadings on the temporary policy shock in the output and inflation processes vary from −1.68 and −1.44, respectively, for $\lambda =0.21$ to −10.81 and −9.25 for $\lambda =0.91$ .

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Epstein, L.G. and Zin, S.E. (1989) Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset Returns: A Theoretical Framework. Econometrica, 57, 937-969. https://doi.org/10.2307/1913778
[2]	Bansal, R. and Yaron, A. (2004) Risks for the Long-Run: A Potential Resolution of Asset Pricing Puzzles. Journal of Finance, 59, 1481-1509. https://doi.org/10.1111/j.1540-6261.2004.00670.x
[3]	Duffie, D. and Kan, R. (1996) A Yield-Factor Model of Interest Rates. Mathematical Finance, 6, 379-406. https://doi.org/10.1111/j.1467-9965.1996.tb00123.x
[4]	Cochrane, J.H. and Piazzesi, M. (2008) Decomposing the Yield Curve. https://doi.org/10.2139/ssrn.1333274
[5]	Piazzesi, M. and Schneider, M. (2006) Equilibrium Bond Yields. In: Acemoglu, D., Rogoff, K. and Woodford, M., Eds., NBER Macroeconomics Annual, MIT Press, Cambridge, 389-472.
[6]	Bansal, R. and Shaliastovich, I. (2013) A Long-Run Risks Explanation of Predictability Puzzles in Bond and Currency Markets. Review of Financial Studies, 26, 1-33. https://doi.org/10.1093/rfs/hhs108
[7]	Creal, D.D. and Wu, J.C. (2020) Bond Risk Premia in Consumption-Based Models. Quantitative Economics, 11, 1461-1484. https://doi.org/10.3982/QE887
[8]	Campbell, J.Y. (1986) Bond and Stock Returns in a Simple Exchange Model. Quarterly Journal of Economics, 101, 785-803. https://doi.org/10.2307/1884178
[9]	Backus, D.K., Gregory, A.W. and Zin, S.E. (1989) Risk Premiums in the Term Structure: Evidence from Artificial Economies. Journal of Monetary Economics, 24, 371-399. https://doi.org/10.1016/0304-3932(89)90027-5
[10]	Eraker, B. (2008) Affine General Equilibrium Models. Management Science, 54, 2068-2080. https://doi.org/10.1287/mnsc.1070.0796
[11]	Havranek, M. (2015) Measuring Intertemporal Substitution: The Importance of Method Choices and Selective Reporting. Journal of the European Economic Association, 13, 1180-1204. https://doi.org/10.1111/jeea.12133
[12]	Kaplan, G., Moll, B. and Violante, G.L. (2018) Monetary Policy According to HANK. American Economic Review, 108, 697-743. https://doi.org/10.1257/aer.20160042
[13]	Gallmeyer, M.F., Hollifield, B., Palomino, F. and Zin, S.E. (2007) Arbitrage-Free Bond Pricing with Dynamic Macroeconomic Models. Federal Reserve Bank of St. Louis Review, July/August, 305-326. https://doi.org/10.3386/w13245
[14]	Dai, Q. and Singleton, K. (2000) Specification Analysis of Affine Term Structure Models. Journal of Finance, 55, 1943-1978. https://doi.org/10.1111/0022-1082.00278
[15]	Ang, A. and Piazzesi, M. (2003) A No-Arbitrage Vector Autoregression of Term Structure Dynamics with Macroeconomic and Latent Variables. Journal of Monetary Economics, 50, 745-787. https://doi.org/10.1016/S0304-3932(03)00032-1
[16]	Rudebusch, G.D., Swanson, E.T. and Wu, T. (2006) The Bond Yield “Conundrum” from the Macro-Finance Perspective. Working Paper 2006-16, Federal Reserve Bank of San Francisco, San Francisco. https://doi.org/10.2139/ssrn.1003033
[17]	Ang, A., Bekaert, G. and Wei, M. (2008) The Term Structure of Real Rates and Expected Inflation. Journal of Finance, 63, 797-849. https://doi.org/10.1111/j.1540-6261.2008.01332.x
[18]	Wright, J.H. (2011) Term Premia and Inflation Uncertainty: Empirical Evidence from an International Panel Dataset. American Economic Review, 101, 1514-34. https://doi.org/10.1257/aer.101.4.1514
[19]	Chernov, M. and Mueller, P. (2012) The Term Structure of Inflation Expectations. Journal of Financial Economics, 106, 367-394. https://doi.org/10.1016/j.jfineco.2012.06.004
[20]	Abrahams, M., Adrian, T., Crump, R.K., Moench, E. and Yu, R. (2016) Decomposing Real and Nominal Yield Curves. Journal of Monetary Economics, 84, 182-200. https://doi.org/10.1016/j.jmoneco.2016.10.006
[21]	Creal, D.D. and Wu, J.C. (2017) Monetary Policy Uncertainty and Economic Fluctuations. International Economic Review, 58, 1317-1354. https://doi.org/10.1111/iere.12253
[22]	Backus, D.K., Chernov, M., Zin, S. and Zviadadze, I. (2021) Monetary Policy Risk: Rules vs. Discretion. New York University, New York. https://doi.org/10.3386/w28983
[23]	Gurkaynak, R. and Wright, J.H. (2012) Macroeconomics and the Term Structure. Journal of Economic Literature, 50, 331-367. https://doi.org/10.1257/jel.50.2.331
[24]	Gurkaynak, R., Sack, B. and Swanson, E. (2005) The Sensitivity of Long-Term Interest Rates to Economic News: Evidence and Implications for Macroeconomic Models. American Economic Review, 95, 425-436. https://doi.org/10.1257/0002828053828446
[25]	Gallmeyer, M.F., Hollifield, B. and Zin, S.E. (2005) Taylor Rules, McCallum Rules and the Term Structure of Interest Rates. Journal of Monetary Economics, 52, 921-950. https://doi.org/10.1016/j.jmoneco.2005.07.010
[26]	Hordahl, P., Tristani, O. and Vestin, D. (2008) The Yield Curve and Macroeconomic Dynamics. Economic Journal, 118, 1937-1970. https://doi.org/10.1111/j.1468-0297.2008.02197.x
[27]	Doh, T. (2011) Yield Curve in an Estimated Nonlinear Macro Model. Journal of Economic Dynamics and Control, 35, 1229-1244. https://doi.org/10.1016/j.jedc.2011.03.003
[28]	Rudebusch, G.D. and Swanson, E.T. (2012) The Bond Premium in a DSGE Model with Long-Run Real and Nominal Risks. American Economic Journal: Macroeconomics, 4, 105-143. https://doi.org/10.1257/mac.4.1.105
[29]	Kung, H. (2015) Macroeconomic Linkages between Monetary Policy and the Term Structure of Interest Rates. Journal of Financial Economics, 115, 42-57. https://doi.org/10.1016/j.jfineco.2014.09.006
[30]	Wachter, J. (2006) A Consumption-Based Model of the Term Structure of Interest Rates. Journal of Financial Economics, 79, 365-399. https://doi.org/10.1016/j.jfineco.2005.02.004
[31]	Hordahl, P., Tristani, O. and Vestin, D. (2006) A Joint Econometric Model of Macroeconomics and Term-Structure Dynamics. Journal of Econometrics, 131, 405-444. https://doi.org/10.1016/j.jeconom.2005.01.012
[32]	Rudebusch, G.D. and Wu, T. (2008) A Macro-Finance Model of the Term Structure, Monetary Policy and the Economy. The Economic Journal, 118, 906-926. https://doi.org/10.1111/j.1468-0297.2008.02155.x
[33]	Gomez-Cram, R. and Yaron, A. (2021) How Important Are Inflation Expectations for the Nominal Yield Curve. Review of Financial Studies, 34, 985-1045. https://doi.org/10.1093/rfs/hhaa039
[34]	Song, D. (2017) Bond Market Exposures to Macroeconomic and Monetary Policy Risks. Review of Financial Studies, 30, 2762-2817. https://doi.org/10.1093/rfs/hhx039
[35]	Van Binsbergen, J.H., Fernandez-Villaverde, J., Koijen, R.S.J. and Rubio-Ramirez, J. (2012) The Term Structure of Interest Rates in a DSGE Model with Recursive Preferences. Journal of Monetary Economics, 59, 634-648. https://doi.org/10.1016/j.jmoneco.2012.09.002
[36]	Backus, D.K., Routledge, B.R. and Zin, S. (2010) The Cyclical Component of U.S. Asset Returns. New York University, New York.
[37]	Bloom, N. (2014) Fluctuations in Uncertainty. Journal of Economic Perspectives, 28, 153-176. https://doi.org/10.1257/jep.28.2.153
[38]	Werning, I. (2015) Incomplete Markets and Aggregate Demand. Working Paper 21448, NBER. https://doi.org/10.3386/w21448
[39]	Ravn, M. and Sterk, V. (2017) Job Uncertainty and Deep Recessions. Journal of Monetary Economics, 90, 125-141. https://doi.org/10.1016/j.jmoneco.2017.07.003
[40]	Den Haan, W.J., Rendahl, P. and Reigler, M. (2018) Unemployment (Fears) and Deflationary Spirals. Journal of the European Economic Association, 16, 1281-1349. https://doi.org/10.1093/jeea/jvx040
[41]	Nelson, C.R. and Siegel, A.F. (1987) Parsimonious Modelling of Yield Curves. Journal of Business, 60, 473-489. https://doi.org/10.1086/296409
[42]	Litterman, R. and Scheinkman, J. (1991) Common Factors Affecting Bond Returns. Journal of Fixed Income, 1, 54-61. https://doi.org/10.3905/jfi.1991.692347
[43]	Piazzesi, M. (2006) Affine Term Structure Models. In: Ait-Sahalia, Y. and Hansen, L.P., Eds., Handbook of Financial Econometrics, Elsevier, Amsterdam, 691-766.
[44]	Duffee, G.R. (2012) Forecasting Interest Rates. In: Timmermann, A. and Elliot, G., Eds., Handbook of Economic Forecasting, Elsevier, Amsterdam, 385-426. https://doi.org/10.1016/B978-0-444-53683-9.00007-4
[45]	Nicholls, D.F. and Pope, A.L. (1988) Bias in the Estimation of Multivariate Autoregressions. Australian and New Zealand Journal of Statistics, 30, 296-309. https://doi.org/10.1111/j.1467-842X.1988.tb00484.x
[46]	Shaman, P. and Stine, R.A. (1988) The Bias of Autoregressive Coefficient Estimators. Journal of the American Statistical Association, 83, 842-848. https://doi.org/10.1080/01621459.1988.10478672
[47]	Fama, E.F. and Bliss, R.R. (1987) The Information in Long-Maturity Forward Rates. American Economic Review, 77, 680-699.
[48]	Campbell, J.Y. and Shiller, R.J. (1991) Yield Spreads and Interest Rate Movements: A Bird’s Eye View. Review of Economic Studies, 58, 495-514. https://doi.org/10.2307/2298008
[49]	Ludvigson, S.C. and Ng, S. (2009) Macro Factors in Bond Risk Premia. Review of Financial Studies, 22, 5027-5067. https://doi.org/10.1093/rfs/hhp081
[50]	King, R.G. and Watson, M.W. (1996) Money, Prices, Interest Rates and the Business Cycle. Review of Economics and Statistics, 78, 35-53. https://doi.org/10.2307/2109846
[51]	Estrella, A. and Mishkin, F.S. (1998) Predicting U.S. Recessions: Financial Variables as Leading Indicators. Review of Economics and Statistics, 80, 45-61. https://doi.org/10.1162/003465398557320
[52]	Ang, A., Piazzesi, M. and Wei, M. (2006) What Does the Yield Curve Tell Us about GDP Growth? Journal of Econometrics, 131, 359-403. https://doi.org/10.1016/j.jeconom.2005.01.032
[53]	Kydland, F.E., Rupert, P. and Sustek, R. (2016) Housing Dynamics over the Business Cycle. International Economic Review, 57, 1149-1177. https://doi.org/10.1111/iere.12193
[54]	Wang, P. and Wen, Y. (2007) Inflation Dynamics: A Cross-Country Investigation. Journal of Monetary Economics, 54, 2004-2031. https://doi.org/10.1016/j.jmoneco.2006.07.007
[55]	Bilbiie, F.O. (2019) Monetary Policy and Heterogeneity: An Analytical Framework.
[56]	Kaplan, G. and Violante, G.L. (2014) A Model of the Consumption Response to Fiscal Stimulus Payments. Econometrica, 82, 1199-1239. https://doi.org/10.3982/ECTA10528
[57]	Galı, J. (2015) Monetary Policy, Inflation, and the Business Cycle: An Introduction to the New Keynesian Framework and Its Applications. 3rd Edition, Princeton University Press, Princeton.
[58]	Fernandez-Villaverde, J., Guerron-Quintana, P., Kuester, K. and Rubio-Ramirez, J. (2015) Fiscal Volatility Shocks and Economic Activity. American Economic Review, 105, 3352-3384. https://doi.org/10.1257/aer.20121236
[59]	Hansen, L.P., Heaton, J.C. and Li, N. (2008) Consumption Strikes Back? Measuring Long-Run Risk. Journal of Political Economy, 116, 260-302. https://doi.org/10.1086/588200
[60]	Cochrane, J. (2011) Determinacy and Identification with Taylor Rules. Journal of Political Economy, 119, 565-615. https://doi.org/10.1086/660817
[61]	Ireland, P.N. (2007) Changes in the Federal Reserve’s Inflation Target: Causes and Consequences. Journal of Money, Credit, and Banking, 39, 1851-1882. https://doi.org/10.1111/j.1538-4616.2007.00091.x
[62]	Bilbiie, F.O. (2020) The New Keynesian Cross. Journal of Monetary Economics, 114, 90-108. https://doi.org/10.1016/j.jmoneco.2019.03.003