Equilibrium Yield Curves and the Interest Rate Lower Bound*

Abstract:

1 Introduction

Since the onset of the 2007–2009 global financial crisis, many central banks have been forced to implement monetary policy under the effective lower bound (ELB) constraint on the short-term nominal interest rate. In particular, central banks have implemented various “unconventional" policies such as those aimed at providing more transparency in the likely path of the policy rate (so-called forward guidance policies) and purchases of long-term government bonds, to stimulate the economy through their effects on the term structure of interest rates. Accordingly, if we are to explain the dynamics of interest rates and evaluate the efficacy of these unconventional policies, we should have a coherent understanding of the interaction between the macroeconomy, the term structure of interest rates, and monetary policy at the ELB. While many studies have contributed to this effort, the effect of the ELB on term premiums—a crucial component of the term structure—appears to be particularly understudied within an explicit macroeconomic framework.¹

In this paper, we study how the ELB constraint jointly affects the macroeconomy, the term structure of interest rates, and monetary policy in a structural general equilibrium (DSGE) model. Our model is a variant of a standard New Keynesian model featuring nominal price rigidities in the product market and an interest rate feedback rule. Explicit microfoundations set our model apart from the existing term structure models with the ELB—which are mostly statistical models with limited economic structure. We offer two contributions. First, we clarify how the ELB constraint can affect the dynamics of term premiums using a stylized model that can be solved in closed form. Second, we analyze a fully nonlinear model which augments the stylized model with additional features to match key properties of the macroeconomy and the term structure of interest rates in the United States.

Our main finding from the stylized model is that the ELB constraint generates state dependency in term premiums through two forces affecting term premiums—macroeconomic uncertainty and the sensitivity of interest rates to macroeconomic fluctuations. On the one hand, macroeconomic uncertainty is higher when the policy rate is constrained by the ELB than when it is not, as the ELB constraint prevents the central bank from counteracting the effects of exogenous shocks to demand on output and inflation. This increased macroeconomic uncertainty at and near the ELB amplifies the absolute size of term premiums. On the other hand, the sensitivity of interest rates to macroeconomic fluctuations can be smaller when the policy rate is at or near the ELB than when it is not, as the central bank faces restrictions in adjusting its policy rate in the near term. This reduced sensitivity of interest rates at and near the ELB can compress the absolute size of term premiums. While the compression effect of the ELB may be widely understood, the amplification effect, which arises from the endogenous response of the macroeconomy to the ELB constraint, can only be uncovered through an equilibrium macro-finance model such as ours.

We further show that the two forces can be conflicting, and which of these dominates depends on the state of the economy and yield maturity. When the economy is in a deep recession and the policy rate is expected to be at the ELB for a long time, the second compression effect typically dominates the first amplification effect, and the size of the term premium is smaller than when the policy rate is comfortably above the ELB. When the economy is in a milder recession and the policy rate is expected to be at the ELB for a shorter time, the first effect is likely to dominate the second and the size of term premiums is larger than when the policy rate is comfortably above the ELB. The compressing effect on term premiums induced by the reduced sensitivity of interest rates is stronger for shorter-maturity yields that are more strongly affected by the presence of the ELB than for longer-maturity yields. Overall, our analysis shows that the ELB constraint can add a unique source of variation in term premiums, which suggests that term premiums may warrant more attention in a severe recession involving the ELB than under normal circumstances.

We next extend the stylized model in several directions to improve its ability to quantitatively match key features of output, inflation, and the term structure of interest rates in the U.S. First, we introduce recursive utility. Second, we add policy inertia in the monetary policy rule. Third, we add productivity shocks with stochastic volatility, which makes the conditional moments of our model while the policy rate is above the ELB consistent with those in the data. In particular, yield curves are on average upward sloping, and term premiums are positive, large, and volatile while the policy rate is above the ELB in our model. In addition, our yield curve is steeper when the policy rate is constrained at the ELB than otherwise, as in the data. Fourth, we include stochastic volatility in the demand shock process. By allowing the volatility of demand shocks to increase as demand falls, we generate term premiums that are on average negative at the ELB. This is because the volatility of demand shocks is heightened around the ELB where demand is particularly depressed, magnifying negative term premiums that arise from a given level of demand shock volatility. Negative term premiums are consistent with survey-based estimates of term premiums—as well as term premium estimates based on some available term structure models—largely from the period in which the federal funds rate was at the ELB. Based on a moment-matching exercise for the first ELB episode, we find that the compression effect of the ELB on term premiums was strong.

Our quantitative model is used to interpret the term structure dynamics during the first ELB episode in the U.S. following the financial crisis. In particular, we show how shorter-maturity term premiums were compressed as the expected ELB duration spiked upon the introduction of forward guidance, and later became negative as the liftoff approached. We also provide one of the first analyses on how to interpret term structure dynamics during the Covid-19 crisis—the latest period when the ELB was binding—using a structural macro-finance model. There, we find that in order to explain the increase in term premiums, it is important to consider not only negative demand and supply shocks, but also the possibility of a notable increase in the volatility of the supply shocks.

A key benefit of structural models is that they allow counterfactual policy experiments that are robust to the Lucas critique. In our final exercise, we use the quantitative model to study how forward guidance—the central bank’s announcement to keep the policy rate at the ELB for longer than previously expected—affects the macroeconomy and the term structure of interest rates. We find that this forward guidance not only reduces the expected short-rate path but also the absolute size of term premiums as a result of the strengthened compression effect. In our benchmark specification in which average term premiums are negative at the ELB, this result means that forward guidance increases term premiums, partially offsetting the declines in the expected short-rate path. However, this result also implies that if average term premiums are positive even at the ELB, forward guidance reduces term premiums, amplifying the decline in the expected short-rate path.

One caveat in our analysis is that our model does not have an explicit role for the central bank balance sheet, an ingredient that is likely to be important in understanding the effects of unconventional policy such as quantitative easing (QE). Introducing such an ingredient into our model can be an important extension. However, we believe that, because unconventional policy was typically undertaken while the policy rate was constrained at the ELB, clarifying how the ELB constraint affects the term structure of interest rates is an essential first step toward a comprehensive understanding of unconventional monetary policy. Moreover, if balance sheet policies work mainly by signaling to the private sector about the central bank’s commitment to an accommodative policy stance and thus affecting the expected path of the policy rate, as suggested by studies such as Woodford, 2012, the analysis of forward guidance provided in our paper may capture part of the effects of balance sheet policies.

After staying at the ELB for a few years since the pandemic crisis, interest rates have recently risen notably above the ELB in the U.S. However, measures of the natural rate of interest (or, the “r-star") remain historically low possibly due to various structural factors, suggesting that the long-run trend of low interest rates may not change significantly. Thus, the ELB is likely to remain a relevant constraint for future monetary policy.

Literature Review:Our contribution is to present the first structural analysis of the yield curve and term premiums using a DSGE model with an occasionally binding ELB constraint that is calibrated to match key features of U.S. data. Naturally, our work is related to several strands of the literature. First, our paper is closely related to papers analyzing the implications of the ELB on the macroeconomy in fully nonlinear New Keynesian models with an occasionally binding ELB constraint. Examples are Fernández-Villaverde et al., 2015, Gavin et al., 2015, Gust et al., 2017, and Nakata, 2017, among many others. The key difference between these papers and ours is our focus on the term structure of interest rates. Reflecting this difference, our model includes some features absent in these papers, such as recursive preferences and stochastic volatilities in exogenous shocks.

Second, our work is related to the literature on term structure models with the ELB. Examples are Ichiue and Ueno, 2007, Kim and Singleton, 2012, Krippner, 2012, Bauer and Rudebusch, 2016, Christensen and Rudebusch, 2016, Wu and Xia, 2016, and Kim and Priebsch, 2024. As noted above, the existing models are largely reduced form in the sense that they impose very limited economic structure, such as lack of arbitrage opportunities.² In contrast, our general equilibrium model features explicit microfoundations and includes a description of monetary policy, allowing us to give economic interpretations to the term structure of interest rates.

Third, our paper extends the literature on equilibrium term structure models to incorporate the ELB constraint. In particular, our paper is closely related to papers analyzing the term structure of interest rates in a production economy. Examples are Rudebusch and Swanson, 2008; Rudebusch and Swanson, 2012, Andreasen, 2012a; Andreasen, 2012b, Van Binsbergen et al., 2012, Dew-Becker, 2014, Kung, 2015, Lopez et al., 2015, Andreasen et al., 2018, Swanson, 2019, Bretscher et al., 2020, Campbell et al., 2020, Hsu et al., 2021, Nguyen, 2022, Pflueger and Rinaldi, 2022, and Pflueger, 2025.³ Our main departure from these models is to incorporate an occasionally binding ELB constraint. Even though these papers have made substantial progress in fitting macro and yield data jointly, they usually find it difficult to generate endogenous volatility in term premiums. In our model, endogenous volatility in term premiums can arise naturally due to the ELB constraint.

Fourth, our paper belongs to the nascent literature on equilibrium asset pricing models with the ELB constraint, such as Datta et al., 2021. Within this literature, papers on equilibrium term structure models with the ELB are particularly scarce. Branger et al., 2015 and Katagiri, 2022 introduce the ELB constraint into an endowment economy model. In contrast, we study a production economy with a realistic description for monetary policy. Sakurai, 2016 develops a New Keynesian bond pricing model with the ELB constraint, but his pricing kernel is reduced form and the discussion on term premiums is brief. The pricing kernel in our model builds explicitly on household behavior. Most closely related to our paper are contemporaneous studies by Bilal, 2017 and Gourio and Ngo, 2020 that examine asset-pricing implications of a DSGE model with the ELB constraint.⁴ While they focus on the correlation between stock and bond returns, we focus on the term structure of interest rates and analyze term premiums in more detail. Specifically, we characterize the state dependency of term premiums in closed form, and our quantitative model, which features time-varying volatility, produces term premiums that are significantly negative at the ELB, while remaining positive above the ELB.

Finally, since the periods of QE policy implementation largely coincide with the ELB episodes, our work complements studies that analyze QE policy and its impact on the term structure. Examples include Chen et al., 2012, Gertler and Karadi, 2011, King, 2019, and Sims and Wu, 2021. In contrast to these papers, our model does not incorporate QE. However, we use a fully nonlinear DSGE model with an occasionally binding ELB constraint to study the implications of the ELB on the term structure in greater depth.

The paper is structured as follows. Section 2 presents the stylized model. We then extend it to a quantitative model to match certain features of U.S. data in Section 3. We use this model to interpret term structure dynamics during the two ELB episodes in Section 4. Section 5 conducts monetary policy experiments at the ELB based on the model. Section 6 concludes.

2 Equilibrium Yield Curves in the Stylized Model

We start out by analyzing the dynamics of equilibrium yields and term premiums in a term structure model that builds on a stylized New Keynesian core subject to the ELB constraint. A key feature of the model is that the variables follow a 3-state Markov chain, which makes it very tractable. In fact, our model is simple enough to be solved in closed form despite the nonlinearity arising from the occasionally binding ELB constraint. The goal of this section is to describe how the ELB affects macroeconomic variables and the yield curve in a transparent way. We build a more elaborate quantitative model in Section 3, where we further explore how the mechanisms highlighted in this section can help us better understand the data.

2.1 Model Specification

The Macro Block: As in the canonical New Keynesian model, we assume the economy consists of a representative household, monopolistically competitive firms that face nominal price rigidities, and a central bank which sets monetary policy based on a Taylor-type rule. The agents’ decisions constitute the standard “macro block", which is a log-linear system except for the nonlinearity due to the ELB constraint.⁵ Specifically, the macro block consists of the following three equations: \[\begin{equation} \hat{y}_{t}=\mathbb E_t\left[\hat{y}_{t+1} \right]-\frac{1}{\gamma}\left(r_t^{(1)}-\mathbb E_t[\hat{\pi}_{t+1}]-\bar{r}\right)+\delta_{t}\tag{1} \end{equation}\] \[\begin{equation} \hat{\pi}_{t}=\kappa \hat{y}_{t}\tag{2} \end{equation}\] \[\begin{equation} r_t^{(1)}=\max[0, \bar{r}+\phi\hat{\pi}_t],\tag{3} \end{equation}\] where \(y_t\) is (log) output, \(\pi_t\) is the inflation rate, and \(r_t^{(1)}\) is the 1-period nominal interest rate (the policy rate or the short rate). \(\bar{x}\) denotes the deterministic steady state of \(x\), and \(\hat{x}\equiv x-\bar{x}\) indicates the deviation of \(x\) from its steady state. Equation (1) is the log-linearized Euler equation and \(\delta_t\) is an exogenous shock to be specified below. The exact origin of \(\delta_t\) is not important for the discussion in this section, and we interpret \(\delta_t\) as a generic “demand" shock that moves output and inflation in the same direction and occasionally pushes the economy to the ELB. However, we revisit the microfoundation of (1) in more detail when we formulate the quantitative model in Section 3. Equation (2) is a log-linearized static Phillips curve. We choose this specification to obtain tractable closed-form solutions, following studies such as Bilbiie, 2019.⁶ Equation (3) is the monetary policy rule (with \(\phi>1\)) subject to the ELB constraint. The ELB is set to zero for simplicity (zero lower bound).

The Asset Pricing Block: We assume there exists another representative agent called the “investor", who receives funds from the household and invests in default-free nominal bonds of arbitrary maturity that are in zero net supply. The investor takes macroeconomic dynamics determined by the macro block as given, and prices bonds based on the following recursive equation: \[\begin{equation} Q_t^{(n)} = \mathbf{E}_t[ M_{t+1} Q_{t+1}^{(n-1)}],\tag{4} \end{equation}\] where \(Q_t^{(n)}\) (\(n\geq1\)) is the price of an \(n\)-period zero-coupon nominal bond that pays $1 at maturity, and \(Q_t^{(0)}=1\). \(M_t\) is the nominal pricing kernel. We assume \(m_{t+1}\equiv\ln M_{t+1}\) has the following form consistent with standard power utility: \[\begin{equation} m_{t+1}=-\bar{r}-\gamma\delta_t-\gamma(\hat{y}_{t+1}-\hat{y}_t)-\hat{\pi}_{t+1}.\tag{5} \end{equation}\] Note (5) and (4) reduce to (1) when linearized for \(n=1\). Thus, we can interpret the pricing equation as a result of the investor’s optimal strategy which coincides with the household’s consumption-saving decision, except that the investor takes risk premiums into account whereas the household does not. A similar setup is proposed by Bianchi et al., 2022, though they do not impose the ELB constraint.⁷ Such a separation of households and investors could be justified if the latter can make more sophisticated investment decisions. Alternatively, we can view the framework as a tractable approximation of the fully nonlinear model with a single representative agent, which allows us to highlight its key implications in closed form by abstracting from nonlinearities that are not crucial for our purposes. We do not make this separation in the quantitative model in Section 3; the household’s and investor’s Euler equations for bonds (and hence their attitudes towards risk) are consistent with each other.

The continuously compounded yield to maturity of the \(n\)-period bond is \(r_t^{(n)} = -\frac{1}{n}\ln Q_t^{(n)}\). The term premium of this bond is defined in the standard way as the difference between the yield and its “risk-neutral” counterpart \(r_t^{(n)\mathbb Q}\): \[\begin{equation} tp_t^{(n)} \equiv r_t^{(n)}-r_t^{(n)\mathbb Q},\tag{6} \end{equation}\] where \(r_t^{(n)\mathbb Q}=-\frac{1}{n}\ln Q_t^{(n)\mathbb Q}\). \(Q_t^{(n)\mathbb Q}\) is the risk-neutral price of an \(n\)-period zero-coupon nominal bond that is computed analogously to (4) by replacing \(M_{t+1}\) with the 1-period rate; \(Q_t^{(n)\mathbb Q} = \exp(-r_t^{(1)})\mathbf{E}_t[ Q_{t+1}^{(n-1)\mathbb Q}]\), where \(Q_t^{(0)\mathbb Q}=1\).⁸

While we can solve for yields and term premiums of arbitrary maturity, we focus on the 2-period maturity (\(n=2\)) in the rest of this section. We further approximate (4) to obtain a simple expression for the 2-period yield: \[\begin{equation} r_t^{(2)}=-\frac{1}{2}\left[ \mathbb E_t[m_{t+1}-r_{t+1}^{(1)}]+\frac{1}{2}\mathbb V_t[m_{t+1}-r_{t+1}^{(1)}] \right].\tag{7} \end{equation}\] Then, the 2-period term premium follows from (6): \[\begin{equation} tp_t^{(2)}=\frac{1}{2}\mathbf{C}\text{ov}_t[m_{t+1},r_{t+1}^{(1)}],\tag{8} \end{equation}\] where \(\mathbb V_t[\cdot]\) and \(\mathbf{C}\text{ov}_t[\cdot]\) denote the conditional variance and covariance, respectively.⁹

State Space and Equilibrium: Finally, we specify \(\delta_t\) as a 3-state Markov chain with the state space \(\{\delta_H,\delta_M,\delta_L\}=\{\delta,0,-\delta\}\), an initial distribution, and transition matrix: \[\begin{bmatrix} p_{HH} & p_{HM} & p_{HL}\\ p_{MH} & p_{MM} & p_{ML}\\ p_{LH} & p_{LM} & p_{LL} \end{bmatrix} = \begin{bmatrix} p & 1-p & 0\\ \frac{p(1-p)}{2} & 1-p(1-p) & \frac{p(1-p)}{2}\\ 0 & 1-p & p \end{bmatrix},\] where \(p_{ij}\equiv \Pr\left(\delta_{t+1}=\delta_{j}|\delta_{t}=\delta_{i}\right)\). This specification resembles a discretized AR(1) process in that \(\delta_t\) is stationary, has an unconditional mean of zero with a first-order autocorrelation of \(p\), and is conditionally homoskedastic.¹⁰ We exclude a jump from \(H\) to \(L\) or vice versa in one step to obtain tractable closed-form solutions. While a 2-state process is more common in equilibrium models with the ELB (e.g., Eggertsson and Woodford, 2003, Bilbiie, 2019, and Nakata and Schmidt, 2019), we need at least three states to demonstrate state dependency of term premiums. Given the specification of \(\delta_t\), the equilibrium is characterized by the vector \(\{y_s,\pi_s,r_s^{(1)},r_s^{(2)},tp_s^{(2)}\}_{s\in\{H,M,L\}}\) that solves the system of equations (1), (2), (3), (5), (7), and (8).

To capture the key implications of a more realistic setup with continuous states, we further impose the following assumptions on the state space: \[\begin{equation} \bar{r}+\phi\hat{\pi}_H> 0, \quad \bar{r}+\phi\hat{\pi}_L< 0, \quad \bar{r}+\phi\hat{\pi}_M=0.\tag{9} \end{equation}\] The first two assumptions are fairly standard in 2-state models with the ELB, and simply say that the state space covers the two states of interest; state \(H\), in which the economy is strictly above the ELB, and state \(L\), in which the ELB is strictly binding. The last assumption means that the size of the decline in the policy rate in state \(M\) is such that the policy rate is exactly at the ELB in \(M\).¹¹ The main results hold without this assumption, but the specific choice of the intermediate state simplifies the problem considerably.

2.2 Results

We summarize the main results in a few propositions based on closed-form solutions, but it is still useful to first characterize the equilibrium visually. For this purpose, we assign standard parameter values as shown in Table 1:

Table 1: Parameter Values for the Stylized Model

The top panels of Figure 1 plot the equilibrium policy rate, output and inflation (deviation from their steady states for the latter two) as functions of the demand shock \(\delta_t\). In addition to the functions for the model with the ELB (blue lines with circle markers), we also show the corresponding functions for the model without the ELB (light blue lines with plus markers) for comparison.

Figure 1: Equilibrium Functions of the 3-State Model
* Blue lines with circle markers indicate variables of the model with the ELB constraint, and light blue lines with plus markers indicate variables of the model without the ELB constraint.

Accessible version

The policy rate, output and inflation all decrease as \(\delta_t\) decreases. Weaker demand for the final good by the households leads to lower marginal costs and inflation. According to our interest rate feedback rule, the policy rate declines in response to lower inflation (we set the coefficient on output to zero), partially offsetting the contractionary effects of the negative demand shock. The levels of all three variables from the model with the ELB are lower than those from the model without the ELB. This is because, even at state \(H\), in which the policy rate is above the ELB, the anticipation of becoming constrained by the ELB in the future reduces today’s inflation and policy rate by lowering inflation expectations (see the discussion of Nakata and Schmidt, 2019, among others). For a sufficiently small realization of \(\delta_t\) (\(\delta_M\) and \(\delta_L\)), the policy rate hits the ELB. When the policy rate is constrained at the ELB, the contractionary effects of a decrease in \(\delta_t\) cannot be offset by a decline in the policy rate. As a result, a further decrease in demand leads to larger falls in inflation and output when the policy rate is at the ELB than when it is not. As is evident from the panels, the equilibrium variables are linear with respect to \(\delta_t\) when the model does not include the ELB. This implies that the variables are homoskedastic. In contrast, the variables are nonlinear with respect to \(\delta_t\) when the model incorporates the ELB, which implies heteroskedasticity. We explicitly characterize these features of output, inflation and the policy rate in the proposition below:

In particular, Proposition 1.2 highlights how the conditional volatilities of output and inflation increase as the economy approaches the ELB, while the volatility of the policy rate decreases. The volatility of the policy rate decreases naturally due to the ELB constraint. However, precisely because the policy rate cannot be adjusted to mitigate the effects of shocks, the volatilities of output and inflation increase. This contrasting behavior of volatilities induced by the ELB constraint is key to understanding how the ELB affects term structure dynamics, and in particular, understanding how term premiums can become time-varying, as we show next.¹²

The bottom left and middle panels of Figure 1 show the equilibrium 2-period nominal yield and its spread with the policy rate. A couple features are worth highlighting when the model is not subject to the ELB constraint. First, the 2-period yield increases with \(\delta_t\) similar to the policy rate. Second, the yield spread decreases with \(\delta_t\), i.e., the yield spread is countercyclical. These two features also imply that the 2-period yield is less sensitive to changes in \(\delta_t\), reflecting the natural consequence of stationarity that is built into the model, i.e., the term structure of yield volatility is downward sloping. These are all well-documented features of U.S. Treasury yield data.¹³

When the model is subject to the ELB constraint, the 2-period yield does not hit the ELB at \(\delta_M\), contrary to the policy rate; the ELB leads to a positive yield spread at \(\delta_M\). This is because the yield accounts for the possibility of moving to state \(H\) in the future. Interestingly, the yield spread is no longer linear in \(\delta_t\); it is increasing in \(\delta_t\) for \(\delta_t\in\{\delta_L,\delta_M\}\), showing \(cyclicality\) in part of the state space when the ELB is binding. This also implies that yield volatility is no longer necessarily downward sloping but potentially upward sloping, especially near or at the ELB.

The bottom right panel plots the 2-period nominal term premiums. Here, we highlight two features. First, term premiums are (weakly) negative regardless of the level of \(\delta_t\) for both models with and without the ELB. Second, the ELB introduces a unique state dependency (or time variation) in the term premium, which is constant in the model without the ELB constraint. Specifically, at \(\delta_H\), where the economy is away from the ELB but still close enough to bind in the near future, the term premium is more negative (its absolute value is larger) compared to what it would be without the ELB. In other words, the term premium is amplified by the ELB. However, at \(\delta_L\), where the economy is at the ELB and is expected to remain there in the next period, the term premium is zero, and hence smaller in absolute value compared to what it would be without the ELB. In other words, the term premium is compressed by the ELB. Note that the ELB affects term premiums even when the policy rate is away from the ELB because it binds stochastically in our model. We summarize these results in the following two propositions. The first proposition derives explicit expressions of the term premiums:

This proposition shows that term premiums are indeed negative, except at \(\delta_L\), where they are zero in the model with the ELB. Also, it is evident that the time variation of the term premium comes from the heteroskedasticity of the macro variables, which we showed was induced by the ELB. The next proposition characterizes the amplification and compression effects of the ELB on term premiums:

According to this proposition, the ELB unambiguously amplifies the term premium in state \(H\) and compresses the term premium in state \(L\). Whether the ELB amplifies or compresses the term premium in state \(M\) depends on the difference in output between states \(M\) and \(L\).

To better understand the force behind the dynamics of term premiums just described, it is useful to decompose the 2-period term premium into its macroeconomic determinants. From (8): \[\begin{equation} \begin{split} tp_t^{(2)} &= \frac{1}{2}\mathbf{C}\text{ov}_t[m_{t+1},r_{t+1}^{(1)}] \\ & \propto -\mathbf{C}\text{ov}_t(\gamma\Delta y_{t+1},r_{t+1}^{(1)})-\mathbf{C}\text{ov}_t(\pi_{t+1},r_{t+1}^{(1)}) \\ & = -\sum_{x\in\{\Delta y,\pi\}} \gamma_x\times\underbrace{\sigma_{t}[x]}_\text{\parbox{2cm}{\centering macro\\[-0.4cm] uncertainty}}\times \underbrace{\sigma_{t}[r^{(1)}] \times \rho_{t}[x,r^{(1)}]}_\text{\parbox{2cm}{\centering interest rate\\[-0.4cm] sensitivity}}, \end{split}\tag{21} \end{equation}\] where \(\sigma_{t}[x]\) and \(\rho_{t}[x,y]\) denote the 1-period-ahead conditional standard deviation of variable \(x\) and conditional correlation of variables \(x\) and \(y\), respectively. Also, \(\gamma_{\Delta y}=\gamma\) and \(\gamma_\pi=1\). Since inflation is proportional to output in our model with a static Phillips curve, the decomposition further simplifies to: \[\begin{equation} tp_t^{(2)} \propto -(\gamma+\kappa)\sigma_{t}[y]\sigma_{t}[r^{(1)}]\rho_{t}[y,r^{(1)}].\tag{22} \end{equation}\] Thus, term premiums fluctuate based on two time-varying components—macroeconomic uncertainty and the sensitivity of interest rates to macroeconomic fluctuations. The latter can further be decomposed into interest rate volatility and the correlation between interest rates and a macro variable (output growth or inflation).¹⁴

According to this decomposition, the negative term premium shown in Proposition 2 follows from the fact that the correlations between the policy rate and inflation/output growth are both positive for a given \(\delta_t\).¹⁵ Intuitively, the correlations reflect the risk exposure of bonds. Suppose we expect a negative shock to \(\delta_t\) next period. A negative \(\delta_t\) shock implies a decrease in output growth and inflation next period. According to the monetary policy rule, the expected decline in output and inflation means that the policy rate is expected to fall, or, equivalently, the price of the 2-period bond next period—which becomes the price of a 1-period bond—is expected to rise. Thus, the 2-period bond price next period is expected to rise when output is expected to fall, implying that the 2-period bond works as a hedge and the term premium must be negative.

We can also understand the state dependency of term premiums through this decomposition by analyzing how macroeconomic uncertainty and the sensitivity of interest rates to macroeconomic fluctuations vary with \(\delta_t\).

First, consider state \(H\). In this state, the policy rate is above the ELB, but output volatility is higher compared to the model without the ELB due to the possibility of the ELB binding in the future (Proposition 1.3).¹⁶ The same proposition also implies that policy-rate volatility is also higher with the ELB, since the policy rate is still sufficiently away from the ELB that it can aggressively counter a recession that leads to the ELB. Relatedly, output and the policy rate are perfectly correlated, i.e., \(\rho_{t}[y,r^{(1)}]=\rho_{m,H}[y,r^{(1)}]=1\) for both models \(m\in\{U,E\}\). Thus, both macro uncertainty and interest rate sensitivity are higher when the ELB is present. This leads to an unambiguous increase in the term premium (in absolute terms).

Next, consider state \(M\). On the one hand, output volatility in \(M\) is even higher than in state \(H\) in the model with the ELB, and is thus higher than that in \(M\) in the model without the ELB, as monetary policy is at the threshold of the ELB with less room to mitigate the effects of shocks (Proposition 1.2 and 1.3). On the other hand, interest rate volatility in \(M\) is lower than in state \(H\) (Proposition 1.2) in the model with the ELB, precisely because it is more likely that the policy rate will remain at the ELB in the future. We can also show that interest rate sensitivity can decline accordingly, and more than offset the increase in macro uncertainty, leading to the ambiguous effect of the ELB on the term premium in this state.¹⁷

Finally, consider state \(L\). Output volatility is the highest in this state in the model with the ELB (Proposition 1.2). However, since the policy rate is at the ELB and expected to stay there in the next period, interest rate sensitivity is zero. Hence, the term premium is zero, which is smaller (in absolute terms) compared to the term premium in state \(L\) in the model without the ELB.

To better understand the robustness of our results, we conduct a few additional exercises. First, in Appendix B, we analyze a version of the model with government spending shocks instead of the \(\delta_t\) shock, which is an interesting case where consumption dynamics decouple from output dynamics. We show that even in this model, the ELB constraint can amplify and compress term premiums. Second, in Appendix C, we confirm that the results carry over to a version of the stylized model in which the states are continuous and the macro block is nonlinear with a forward-looking Phillips curve. We also analyze the effects of the ELB constraint across yield maturities in more detail, where we show the relative magnitudes of the amplification and compression effects vary across maturities.

In sum, whether term premiums increase or decrease due to the ELB depends on two forces—macro uncertainty and interest rate sensitivity—which can conflict with each other as the economy falls deeper into the ELB state. In the model, the ELB increases both macro uncertainty and interest rate sensitivity at \(\delta_H\). For this case, the ELB leads to a more negative term premium (amplification). By contrast, when the demand shock is sufficiently negative at \(\delta_L\), the increase in macro uncertainty is more than offset by the decrease in interest rate sensitivity, and the size of the term premium decreases and approaches zero (compression).¹⁸

3 Equilibrium Yield Curves in the Quantitative Model

We now turn to the analysis of a quantitative model that incorporates additional features into the stylized model so as to better capture key features of U.S. data. First, we formulate the model and calibrate its parameters to match some key features of output, inflation, yield data and term premium estimates at and above the ELB (this section). The exercise offers a coherent interpretation of macro and term structure dynamics of the past two decades including the recent U.S. ELB episodes through the lens of our model (Section 4). Second, we examine how alternative monetary policy strategies affect the dynamics of yield curves and their decomposition into the expected short-rate path and term premiums at the ELB (Section 5).

3.1 Model Specification

The quantitative model extends the stylized model to a standard New Keynesian model with continuous state variables, a forward-looking Phillips curve and a more elaborate monetary policy rule with policy inertia. Importantly, the quantitative model introduces Epstein-Zin preferences and a richer shock structure—a shock to the preference for holding default-free bonds and a shock to productivity, both exhibiting countercyclical volatility. While these additional features help the model to jointly fit the macroeconomic and term structure data significantly better, we still keep the model relatively parsimonious such that its mechanism remains transparent.

Household:The representative household’s value function \(V_t\) takes the following recursive form proposed by Epstein and Zin, 1989 (EZ preferences):¹⁹

\[\begin{equation} V_t=\begin{cases} U_t(C_t,N_t) +\beta \left\{ \mathbf{E}_{t}\left[ {V}_{t+1}^{1-\tilde{\gamma} }\right]\right\} ^{\frac{1}{1-\tilde{\gamma} }} \qquad \qquad \text{ for } U_t\geq 0\\ U_t(C_t,N_t) -\beta \left\{ \mathbf{E}_{t}\left[ (-V_{t+1})^{1-\tilde{\gamma} }\right]\right\} ^{\frac{1}{1-\tilde{\gamma} }} \quad \; \text{ for } U_t\leq 0.\\ \end{cases}\tag{23} \end{equation}\]

\(\beta\) is the time discount rate. \(\tilde{\gamma}\) is a measure of risk aversion. Note that a larger \(\tilde{\gamma}\) implies higher risk aversion when \(U\geq 0\) and lower risk aversion when \(U\leq 0\). \(\tilde{\gamma}=0\) implies power utility. The period utility function \(U_t(C_t,N_t)\) takes a standard form: \[\begin{equation} U_t(C_t,N_t)=\frac{{C}_{t}^{1-\chi_c }}{1-\chi_c } - Z_t^{1-\chi_c}\frac{N_{t}^{1+\chi_{n} }}{1+\chi_{n} },\tag{24} \end{equation}\]

where \(\chi_{c}>0\) is the inverse elasticity of intertemporal substitution and \(\chi_{n}>0\) is the inverse Frisch elasticity. \(Z_t\) is a deterministic trend in total factor productivity (TFP) that grows at a rate of \(\zeta\) (i.e., \(\zeta = \frac{Z_{t}}{Z_{t-1}})\). The scaling of labor disutility by \(Z_t\) ensures the existence of a balanced growth path.

\(C_t=\left(\int^1_0 C_t(i)^{\frac{\theta-1}{\theta}}di \right)^{\frac{\theta}{\theta-1}}\) is the household’s aggregate consumption of final goods based on a CES aggregator of intermediate goods, where \(\theta>1\) is the elasticity of demand across the intermediate goods. \(N_t=\int_0^1N_t(i)di\) denotes the household’s total supply of labor, which is the integral of labor \(N_t(i)\) supplied to each intermediate good producer \(i\) in a perfectly competitive labor market given nominal wage \(W_t\).

The household faces the following sequence of budget constraints: \[\begin{equation} P_tC_{t} +\exp(-\delta_t)\mathbf{E}_t\left[\widetilde{M}_{t+1} \mathcal{W}_{t+1} \right] \leq W_{t}N_{t} + \mathcal{W}_{t} + \Xi_t + T_t,\tag{25} \end{equation}\] where the aggregate price level of the consumption basket \(P_t\equiv \left(\int^1_0 P_t(i)^{1-\theta}di \right)^{\frac{1}{1-\theta}}\) is implied by the household’s cost minimization problem. \(\Xi_t\) is firms’ profits rebated back to the household, \(T_t\) denotes lump-sum government taxes and/or transfers and \(\mathcal{W}_{t}\) is the household’s wealth portfolio consisting of default-free bonds of arbitrary maturity chosen by the end of period \(t\). Given this flow of income, the household maximizes (23) by choosing \(C_t\), \(N_t\) and bond holdings next period \(\mathcal{W}_{t+1}\). \(\widetilde{M}_{t+1}\) is the nominal pricing kernel based on EZ preferences: \(\widetilde{M}_{t+1}=\beta\left( \frac{U_{c,t+1}}{U_{c,t}}\right)\left[ \frac{V_{t+1}}{\left[ \mathbf{E}_{t}\left[ V_{t+1}^{1-\tilde{\gamma} }\right] \right] ^{\frac{1}{1-\tilde{\gamma} }}}\right] ^{-\tilde{\gamma} } \frac{1}{\Pi_{t+1}}\) (if \(U_t\geq 0\)), where \(\Pi_{t+1}\equiv\frac{P_{t+1}}{P_t}\) is the gross inflation rate.

In contrast to the stylized model in which we assumed the household was risk-neutral and the investor was risk-averse, here there is no such discrepancy; a representative risk-averse household makes an integrated optimal consumption-investment decision. This internal consistency may be theoretically appealing and is in line with many equilibrium asset pricing models.

We assume the household has a particular preference for holding bonds, which is represented by the time-varying parameter \(\delta_t\). A positive shock to \(\delta_t\) can be interpreted as a decrease in preference for holding bonds over other (risky) assets priced by \(\widetilde{M}_{t+1}\), and thus we call it a “risk preference" shock. As we confirm below, this shock represents a “demand" shock similar to the shock in the stylized model. Such a shock is also considered to be an important driver of business cycles in the macroeconomic literature.²⁰

Intermediate Goods Producers:There is a continuum of monopolistically competitive intermediate goods producers indexed by \(i\in[0,1]\). Each producer faces costly price adjustments in the form proposed by Rotemberg, 1982. The expected discounted value of each producer \(i\)’s profit stream is: \[\begin{equation} \mathbf{E}_0\sum_{t=0}^{\infty} \widetilde{M}_{0,t}\left[ P_t(i)Y_t(i)-W_t N_t(i)-\frac{\varphi}{2} \left( \frac{P_t(i)}{P_{t-1}(i)\bar{\Pi}}-1 \right)^2 P_t Y_t \right],\tag{26} \end{equation}\] where \(\widetilde{M}_{0,0}=1\), \(\widetilde{M}_{0,t}\equiv\prod_{s=1}^{t}\widetilde{M}_{s-1,s}\), and \(\varphi>0\) measures the degree of costly price adjustment. We assume the producers index their prices on steady-state inflation \(\bar{\Pi}\) (or the central bank’s inflation target). Each producer maximizes (26) by choosing \(\{P_t(i),Y_t(i),N_t(i)\}\) subject to the demand and production functions: \[Y_t(i)=\left( \frac{P_t(i)}{P_t} \right)^{-\theta}Y_t,\]

\[Y_t(i)=\exp(a_t)Z_tN_t(i),\] where \(a_t\) and \(Z_t\) are two exogenous components of TFP. We assume \(a_t\) is stationary, while \(Z_t\) is a deterministic trend as described above. The details of the process \(a_t\) are described below. Monetary Policy:The central bank sets the nominal one-period interest rate, \(R_t^{(1)}\equiv\exp(r_t^{(1)})\), following a Taylor rule with an occasionally binding ELB constraint: \[\begin{equation} R_t^{(1)}=\max \left[ R_{ELB},R_t^{*} \right],\tag{27} \end{equation}\] where \(R_{ELB}\) is the ELB and the nominal shadow rate \(R_{t}^{*}\) follows: \[\begin{equation} R_{t}^{*}=\left(R_{t-1}^{*}\right) ^{\rho _{r}}\left(\bar{R} \left[ \frac{\Pi _{t}}{\bar{\Pi}}\right]^{\phi_{\pi}} \left[\frac{Y_{t}}{\bar{Y}Z_t}\right]^{\phi_y} \right)^{1-\rho _{r}},\tag{28} \end{equation}\] where \(\bar{R}\) and \(\bar{Y}\) denote the steady state of \(R_t^{(1)}\) and normalized output \(\hat{Y}_t\equiv\frac{Y_t}{Z_t}\), respectively.²¹ In the stylized model, \(\rho_{r}=\phi_y=0\), so that there was neither interest rate smoothing nor a response to the output gap. In the quantitative model, \(\rho_{r}, \phi_y>0\).

Exogenous Stochastic Processes: In the stylized model, \(a_{t}\) was constant and the only exogenous process was \(\delta_t\), which followed a 3-state Markov chain. Instead, for the quantitative model, we assume the state space is continuous and both variables follow an AR(1) process with time-varying volatility. In other words: \[\begin{equation} x_{t}=\rho_{x}x_{t-1}+\sigma_{x,t-1}\varepsilon_{x,t}, \qquad \varepsilon_{x,t}\sim\mathcal{N}(0,1),\tag{29} \end{equation}\] for each exogenous process \(x_t \in \{\delta_t,a_t\}\). The two processes can be interpreted as a “demand" and “supply" shock, respectively, and are standard in DSGE models fitted to macro variables. In our model, each process also serves an important role in generating plausible term structure dynamics at and above the ELB. In particular, (trend) stationarity of productivity is critical in obtaining an upward sloping term structure.²² Meanwhile, \(\delta_t\) can generate a negative term premium, as we saw in the stylized model.

We further specify time-varying volatility \(\sigma_{x,t}\) as a logistic function of \(x_t\) as follows: \[\begin{equation} \sigma_{x,t} = \frac{\theta_{1,x}}{1+\exp(\theta_{2,x}(x_{t}-\theta_{3,x}))}\bar{\sigma}_x,\tag{30} \end{equation}\] where \(\bar{\sigma}_x\) is the conditional standard deviation of \(x_t\) when \(x_t\) is at its average level of zero. The term multiplying \(\bar{\sigma}_x\) controls how much it is scaled depending on the deviation of \(x_{t}\) from its average. \(\theta_{1,x}>1\) sets the upper bound of the scaling and \(\theta_{2,x}\) controls the sensitivity of the scaling with respect to \(x_{t}\). Once \(\theta_{1,x}\) and \(\theta_{2,x}\) are chosen, \(\theta_{3,x}\) is set to guarantee that the scaling term equals 1 and hence \(\sigma_{x,t}=\bar{\sigma}_x\) when \(x=\bar{x}\).²³ Our specification is computationally convenient as it does not add additional state variables to the model. Note \(\sigma_{x,t}\) is a smooth monotonic function of \(x_t\) similar to \(\sigma_{x,t} =\exp(-\theta x_t)\), but the logistic form ensures that the volatility is bounded at the tails, which leads to better numerical behavior.

\(\sigma_{x,t}\) is increasing (decreasing) in \(x\) when \(\theta_{2,x}\) is negative (positive). We set \(\theta_{2,x}>0\) to capture countercyclical volatility of each shock, which is consistent with empirical evidence of economic uncertainty rising in recessions (Bloom, 2014 and references within). As shown below, this specification turns out to be useful in jointly fitting macroeconomic data and features of the term structure in the U.S., including the ELB periods.²⁴

Term Structure of Default-free Interest Rates:Consistent with the household’s optimization, we price the term structure of default-free interest rates using the pricing kernel that is adjusted for the risk preference shock. The pricing equation is the same as the one in the stylized model (4), except that we replace the pricing kernel with \(M_{t+1}=\exp(-\delta_t)\widetilde{M}_{t+1}\) for the nominal term structure.²⁵ The definition of yields and term premiums are also the same as in the stylized model.

Equilibrium Characterization and Solution Method: To close the model, we impose the goods market clearing condition \(Y_t=C_t\). Instead of the common assumption that the price adjustment cost entails a loss of aggregate resources, we assume that the cost is rebated back to the household, following studies such as Ascari and Rossi, 2012.²⁶ We also integrate the supply of intermediate goods to obtain \(Y_{t}=\exp(a_t) Z_t N_t\). Finally, we assume that the bonds are in zero net supply. The equilibrium is defined in the standard way and we relegate further details to Appendix D.

In solving models with an occasionally binding ELB constraint, standard local approximation methods such as perturbation can lead to significant fitting errors. Hence, we solve the model globally using a time-iteration method. This is important for our purposes since asset prices can be sensitive to the strong nonlinearity arising from the combination of highly risk averse agents, the ELB constraint, and time-varying volatility. The details of the solution method are described in Appendix E.

3.2 Nominal Yields and Term Premiums

As we discuss in detail below, we calibrate the model parameters based on U.S. data. However, before presenting the calibration, we discuss key features of nominal yields and term premium estimates that we are interested in fitting our model to.²⁷ The top-left panel of Figure 2 shows 3-month, 1-, 2-, and 5-year Treasury yields from January 1999 until June 2019.²⁸ We focus on maturities up to 5 years for two reasons. First, since the expected ELB duration throughout the (first) ELB episode did not exceed 3 years (Section 4), the effect of the ELB on yields beyond the 5-year maturity is likely to be small. Swanson and Williams, 2014 shows empirically that this was indeed the case. Second, our model abstracts from QE policy, which can be an important driver of the term structure since the financial crisis. Nevertheless, a series of empirical work such as Vissing-Jorgensen and Krishnamurthy, 2011 and D'Amico and King, 2013 suggests that the effects of QE are larger for longer-term yields, and in particular, yields beyond the 5-year maturity. Thus, we view our model’s abstraction from QE to be reasonable insofar as explaining the term structure up to 5 years, and useful for sharpening our discussion on how the ELB alone can affect yields and term premiums.²⁹

Nominal yields are procyclical. They are higher during the economic booms preceding the 2001 recession and the 2007–2009 recession than during those recessions and the subsequent periods of sluggish recovery. The slope of the yield curve is positive on average and countercyclical. Finally, as discussed further in Section 3.4, the term structure of yield volatility above the ELB is downward sloping, except for the shortest maturities. However, at the ELB, yield volatility is increasing in maturity.

Figure 2: Nominal Yields and Term Premiums
* For term premiums: Survey—based on 3-month T-bill forecasts from Blue Chip surveys; KP—from the shadow-rate model of , AJM—from the linear-rational square-root model of ; Average—average of the three term premium measures. The solid vertical line indicates 2008:Q4 (start of the ELB period), and the shaded gray areas correspond to NBER recession periods. Data are at a quarterly frequency.

Accessible version

The top-right and bottom two panels show estimates of term premiums for 1-, 2- and 5-year bonds, respectively. Since term premiums are not clearly observed, we must derive them from either survey forecasts or statistical models. In this paper, we rely on three particular estimates, as shown in the figure. One is calculated using 3-month T-bill forecasts from Blue Chip surveys as a proxy for future short-rate expectations (blue dash-dotted lines). In terms of model-based estimates, the popular measures such as those by Kim and Wright, 2005 or Adrian et al., 2013 do not account for the ELB. Hence, we refer to two alternative measures that explicitly account for it. The first measure is based on the “shadow-rate" term structure model of Kim and Priebsch, 2024 (KP, blue dashed lines), which adds an ELB constraint to a standard no-arbitrage affine term structure model through a shadow rate specification of the short rate.³⁰ The second measure is from the linear-rational square-root model of Andreasen et al., 2025 (AJM, light-blue dash-dotted lines), which enforces the ELB and also allows for unspanned stochastic volatility.

Given the model- and survey-based measures broadly comove, we focus on a summary measure of term premiums for the rest of our analysis, which is a simple average of the three measures (dark-blue solid lines), computed for each maturity. We calibrate our quantitative model to fit this measure. Our construction of the measure is admittedly somewhat ad hoc, but to our knowledge, there are few term premium series that take into account the ELB, and we believe the three measures we consider are a good representation of plausible estimates. Averaging the three measures is a simple and transparent way to summarize that information.³¹ That said, we acknowledge that any term premium estimate is subject to measurement uncertainty; the estimates can diverge, and the levels can be quite different for longer maturities.

According to the average measure, term premiums were positive and generally increasing in maturity in the early part of the sample. Afterwards, term premiums appeared to trend downward, and following the financial crisis—when the ELB became a binding constraint—term premiums became smaller and even negative. Meanwhile, term premiums generally remained volatile both above and at the ELB, especially for longer maturities, but for shorter maturities, term premiums became more compressed towards zero, starting shortly after the ELB became binding, until the liftoff of the policy rate approached.

3.3 Calibration

The calibrated parameter values for the quantitative model are summarized in Table 2. For ease of exposition, we group the parameters into two broad categories; the first category includes parameters that are calibrated from external sources, and the second category includes parameters that are chosen to match selected moments of the data.

In terms of the parameters in the first category, the deterministic trend growth rate of productivity (\(\zeta\)) is set to 2.1 percent per annum, which is consistent with the average per capita output growth from 1999:Q1 to 2015:Q4. The inverse of the elasticity of intertemporal substitution (1/EIS or \(\chi_{c}\)) is set to 10, close to Rudebusch and Swanson, 2012. An EIS significantly below 1 is consistent with seminal macroeconomic studies (Hall, 1988), which has been confirmed more recently by Yogo, 2004 and Havránek, 2015.³² The inverse Frisch elasticity (\(\chi_{n}\)) of 2 is in line with existing studies.³³ The demand elasticity for intermediate goods (\(\theta\)) is also set to a standard value of \(6\). Moving on to the parameters for the monetary policy rule, the ELB rate (\(R_{ELB}-1\)) is an annualized 0.13 percent based on the middle of the lowest federal funds rate target range of 0 to 0.25 percent. The coefficient on inflation (\(\phi_\pi\)) is set to 4, which is the value estimated by Plante et al., 2018, and allows the model to fit the relatively stable inflation over the sample period.³⁴ The coefficient on output (\(\phi_y\)) is set to 0.5, well within the range of common values. The interest rate smoothing parameter (\(\rho_r\)) is fixed at 0.3. Introducing policy inertia makes ELB episodes in the model longer and thus more realistic, while also improving the fit to average yields at the ELB. That said, to retain some volatility in short-maturity term premiums at the ELB, we set the value to the lower end of the range used in the literature, as suggested by Rudebusch, 2006. Finally, we calibrate the persistence of the shock processes based on existing studies. The persistence of productivity (\(\rho_a\)) is set to 0.95, following Rudebusch and Swanson, 2012. The persistence of risk preference (\(\rho_\delta\)) is set to 0.85, following Gust et al., 2017. These values fall within the range typically used in the literature.

Given the externally calibrated parameters, the parameters in the second category are chosen to match the average and standard deviations of output, inflation, yields and term premiums both above and at the ELB. Since the number of targeted moments far exceeds the number of parameters (overidentified), it would be ideal to conduct a systematic optimization over a sufficiently wide parameter space. However, this remains quite challenging for our highly nonlinear model which includes both macro variables and yields with multiple maturities. Thus, as explained below, we determine the values through a coarse grid search over a parameter space guided by our understanding of the model mechanism as well as insights from previous studies.

Although, in general, there is no straightforward mapping between a parameter and a moment(s), some parameters predominantly affect certain moments. Based on this observation, the time discount rate accounting for trend productivity growth (\(\tilde{\beta}\equiv \beta\zeta^{-\chi_c}\)) is set primarily to match the average 1-year nominal yield above the ELB of 3.5 percent.³⁵ The steady-state inflation rate (\(\bar{\Pi}\)) is calibrated largely to match an average inflation rate of 2.3 percent. The firms’ price-adjustment-cost parameter (\(\varphi\)) is chosen primarily to match the volatility of inflation above and at the ELB, conditional on the other parameters. Consistent with studies such as Hills et al., 2019, a large \(\varphi\) implies a flat Phillips curve and plays a key role in capturing the lack of sizable deflation during the ELB period, as well as ensuring inflation does not become counterfactually volatile at the ELB.³⁶

The coefficient of relative risk aversion (\(\gamma\)) is set to 339, according to the formula provided by Swanson, 2018.³⁷ This choice is in line with other equilibrium term structure models which typically use values that are substantially larger than those suggested by microeconomic studies.³⁸ The high risk aversion—which we interpret as an aggregate measure that may also capture preferences toward ambiguity or early resolution of uncertainty in a concise way—helps the model generate sizable term premiums while fitting macro data.

The rest of the parameters characterize the (stochastic) volatility of productivity (\(a_t\)) and risk preferences (\(\delta_t\)). For the model to fit the data, we find that the above-ELB dynamics must be mostly due to variation in \(a_t\), while the variation in \(\delta_t\) plays an important role at the ELB. This is because, in the data, the slope of the yield curve and term premiums are positive on average above the ELB (Section 3.2), which requires \(a_t\) to be the dominant driver (Section 3.1, footnote 22), while term premiums are significantly lower and even negative at the ELB, which can be generated from moves in \(\delta_t\) (Section 2). As a result, when both \(a_t\) and \(\delta_t\) are at their means, the conditional volatility of \(a_t\) (\(\bar{\sigma}_a\)) is set to be significantly larger than the volatility of \(\delta_t\) (\(\bar{\sigma}_\delta\)). Meanwhile, stochastic volatility parameters for \(\delta_t\) (\(\theta_{1,\delta}\) and \(\theta_{2,\delta}\)) are chosen to generate meaningful volatility of \(\delta_t\) following large negative shocks to itself that takes the economy to the ELB.³⁹

The stochastic volatility parameters also serve a key role in fitting the volatility of term premiums, which is consistent with other related studies (Section 3.1). In particular, the parameters for \(a_t\) (\(\theta_{1,a}\) and \(\theta_{2,a}\)) are chosen primarily to match term premium volatility above the ELB, while \(\theta_{1,\delta}\) and \(\theta_{2,\delta}\) help match term premium volatility at the ELB. For additional discipline, we also check whether the degree of time variation in volatility has plausible implications for output and inflation dynamics, although we do not explicitly attempt to match them.

Table 2: Parameter Values for the Quantitative Model

Calibrated from External Sources

Calibrated to Match Moments

3.4 Results

3.4.1 Moments above and at the ELB

We now describe the ability of our model to fit the data by looking at selected moments of (detrended) output, inflation and the term structure of interest rates both above and at the ELB. The moments of interest are conditional averages and conditional standard deviations for “above the ELB” and “at the ELB” samples.⁴⁰ To compute moments for the data counterparts, we focus on the period until the end of the first ELB episode; we define the period above the ELB to be from 1999:Q1 to 2008:Q3, and the ELB period to be from 2008:Q4 to 2015:Q4, as the effective federal funds rate hit the ELB in the middle of December 2008.

Table 3: Macro and Yield Curve Moments above and at the ELB —Model versus Data—

* This table contains summary statistics for selected macroeconomic and term structure variables comparing data versus model counterparts. The sample period is 1999:Q1–2008:Q3 for data above the ELB, and 2008:Q4–2015:Q4 for data at the ELB. All variables are in annualized percentage terms except for output, which is not annualized. The “mean" reported for output at/below the ELB is the average deviation from trend (for data) or the model-implied trend (for the model) in percentage points. We do not report statistics above the ELB, as the mean roughly coincides with the trend.

A. Macro Variables (Mean)

B.1. Macro Variables (Volatility)

B.2. Macro Variables (Vol. of Vol.)

C. Nominal Yields (Mean)

D. Nominal Yields (Volatility)

E. Nominal Term Premium (Mean)

F. Nominal Term Premium (Volatility)

Table 3 presents the moments of the data and from the model. In each table, the first two columns show the moments when the policy rate is above the ELB and the next two columns show the moments when the policy rate is at the ELB. The fifth (final) column shows the conditional moments when the policy rate is below the ELB rate from a version of our model without the ELB constraint. As discussed earlier, our ELB episodes are largely driven by a fall in \(\delta_t\). However, this shock is also associated with an increase in the volatility of \(\delta_t\). The comparison of the final column with the fourth column helps us isolate the effects of the ELB on the model’s moments from the effects associated with the increased volatility of \(\delta_t\).

Following the discussion in Section 3.2, we focus on the nominal term structure of interest rates up to the 5-year maturity and use the average of term premium measures as the data counterpart for term premiums. However, we briefly discuss our model’s implications for the 10-year maturity as well as the real term structure in Appendix G.

Moments above the ELB: As shown in the first two columns of Table 3, panels A and B.1, the average inflation as well as volatilities of output and inflation in the data are in line with those from our model when the policy rate is above the ELB. In the data, yields are upward sloping and the term structure of yield volatility is downward sloping on average when the policy rate is above the ELB (panels C and D, first column). Our model successfully replicates these patterns qualitatively. Quantitatively, the average yield slope is shallower in the model, but the model-implied yield volatilities are broadly in line with the data (panels C and D, second column).

Panels E and F compare term premiums. The mean of the empirical estimates is positive and increases with maturity (Panel E, first column). The volatility of the empirical estimates also increases with maturity (Panel F, first column). Term premiums from our model are consistent with these patterns. Quantitatively, the mean term premiums from our model are somewhat smaller up to the 2-year maturity, but fairly close to the empirical counterparts at the 5-year maturity. While DSGE models tend to generate little variation in risk premiums, term premiums are significantly volatile in our model and within the ballpark of the empirical estimates. As discussed in the previous section, this is largely due to the introduction of stochastic volatility in \(a_t\). However, we also check whether the degree of time variation in volatility has reasonable implications for output and inflation dynamics. Indeed, the first two columns of Panel B.2 show that the model-implied volatility of historical volatility is roughly similar to the data, both for output and inflation.⁴¹ Overall, our model’s term premium dynamics appear quantitatively reasonable.

Moments at the ELB: We turn to the comparison of our model with the data at the ELB. According to the first and third columns of Panel A of Table 3, output and inflation are on average about 2.8 percent and 0.9 percentage points lower, respectively, at the ELB than above the ELB in the data. Our model broadly captures these declines in output and inflation observed at the ELB, although the drop in inflation is more pronounced (second and fourth columns). In the data, the volatility of output is higher at the ELB than above the ELB, while the volatility of inflation is only modestly higher at the ELB than above the ELB (Panel B, first and third columns). The output and inflation volatilities in our model are in line with the data (Panel B, second and fourth columns). The comparison of the fourth column with the fifth (final) column, where we show the moments from the model without the ELB, indicates that the existence of the ELB makes the recession more severe. Based on the model, the output gap and inflation were about 0.8 and 0.7 percentage points lower, respectively, due to the ELB constraint.⁴²

In the data, nominal yields are lower and the yield curve is steeper on average at the ELB than above the ELB (Panel C of Table 3, first and third columns). Moreover, the volatilities of yields are lower at the ELB than above the ELB across maturities (Panel D, first and third columns). While the yield volatility declines with maturity when the policy rate is above the ELB, the yield volatility increases with maturity when the policy rate is at the ELB. Our model captures these features reasonably well.⁴³

Panels E and F compare term premium moments. When the policy rate is constrained at the ELB, the empirical term premiums are on average significantly smaller than those above the ELB, and in fact, negative across maturities (Panel E, third column). Our model-implied averages (Panel E, fourth column) capture these features well. Turning to volatilities, the empirical term premiums remain volatile at the ELB (Panel F, third column), despite the significant difference in the average levels. Our model is also consistent with this pattern (Panel F, fourth column). While the 5-year volatility is somewhat smaller than the empirical estimate, volatilities for shorter maturities are close to their empirical counterparts. Similar to the case above the ELB, the third and fourth columns of Panel B.2 show that the model-implied volatility of historical volatility is roughly similar to the data, both for output and inflation. Hence, we conclude that the model-implied volatility of term premiums does not result from an implausible degree of stochastic volatility in \(\delta_t\) and \(a_t\). Term premium dynamics at the ELB are largely determined by two forces in our quantitative model. The first force is the countercyclical volatility of \(\delta_t\), as discussed in Section 3.3. With elevated demand volatility, the correlation between output growth and inflation becomes less negative and, as a result, term premiums fall. This force is present regardless of the ELB constraint. Indeed, term premiums are negative even in the model without the ELB constraint (Panel E, last column).

The second force is the ELB constraint. As discussed in detail in Section 2, the ELB constraint can either increase or decrease the size of term premiums depending on how much the ELB decreases the sensitivity of interest rates to macroeconomic fluctuations and how much it increases macroeconomic uncertainty. According to the fourth and final columns of Panel E, the compression effect of the ELB associated with reduced sensitivity of interest rates dominates the amplification effect associated with increased macroeconomic uncertainty, and the ELB constraint reduces the absolute size of term premiums (this can also be seen based on the volatility of term premiums, by comparing the fourth and final columns of Panel F). In our model, in which term premiums are negative at the ELB, the reduction in the absolute size of term premiums means an increase in term premiums.

The detailed account of how our model matches the averages and volatilities of existing term premium estimates, both above and at the ELB, marks a clear departure from related analysis by Gourio and Ngo, 2020. They only report the model-implied 10-year term premium which is on average positive but shows little variability, and do not discuss the negative term premiums that are observed up to intermediate maturities.

Comparative Statics: Before closing the analysis of this section, we study alternative parameterizations of the model to clarify the role of several key parameters. Specifically, we focus on the parameters that control the stochastic volatility of \(\delta_t\) and \(a_t\), respectively, as well as the inflation and inertia coefficient in the monetary policy rule. To save space, we summarize the takeaways regarding the volatility parameters here in the main text, and delegate the rest of the details, including a table of the results, to Appendix H.

Removing stochastic volatility from \(a_t\) largely affects moments above the ELB, since at the ELB, the impact of \(\delta_t\) is dominant and thus the specification of \(a_t\) matters less. In particular, the removal leads to less macroeconomic volatility and higher average yields above the ELB, compared to the baseline. Meanwhile, both the averages and volatilities of term premiums are significantly smaller, to the point of little time variation. The result that stochastic volatility is an important source of term premium variability is in line with the literature, as discussed in Section 3.3.

Next, consider removing stochastic volatility from \(\delta_t\) instead. In this case, the variation in \(\delta_t\) is never large enough for the model to hit the ELB in the first place due to the small average volatility \(\bar{\sigma}_\delta\). For the model to be occasionally constrained by the ELB, we could increase \(\bar{\sigma}_\delta\). However, this comes at a cost of counterfactual dynamics above the ELB, such as more volatile output, inflation and yields.

Finally, consider removing stochastic volatility from both \(a_t\) and \(\delta_t\). In this case, the ELB constraint binds occasionally, but mostly due to positive shocks to \(a_t\), which seems clearly counterfactual. Similar to the previous parameterization where only \(a_t\) has stochastic volatility, increasing \(\bar{\sigma}_\delta\) leads to average term premiums above the ELB that are too small. In addition, term premium volatility becomes negligible both above and at the ELB.

3.4.2 Dynamics

We now describe the dynamics of our model in response to shocks that send the policy rate to the ELB to gain further insights on how the ELB constraint affects the term structure of interest rates. We are particularly interested in how the term structure behaves when the economy is near the entry to or exit from the ELB state, insights that cannot be gleaned from comparing moments above and at the ELB.

Figure 3 plots the median responses of macro and term structure variables to a negative \(\delta_t\) shock. The size of the shock is set such that the initial declines in output and inflation in the model with the ELB constraint broadly coincide with the decline in their data counterparts between the business cycle peak of 2007:Q4 and trough of 2009:Q2, defined by the NBER. For each variable, the blue solid and dashed lines are for the models with and without the ELB constraint, respectively.

Figure 3: Impulse Responses to δ_t Shocks
* We plot \(median[X_{t+h}\mid \varepsilon_{\delta,t}=-\vartheta_{\delta}, \bar{\Omega}]\) for a given variable \(X_t\). The blue solid lines are the responses from the model with the ELB constraint, and the blue dashed lines are the responses from the model without the ELB constraint. The gray vertical lines indicate the timing of liftoff from the ELB.

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In response to the decrease in \(\delta_t\) (second row, right, blue line), the nominal short rate falls from its average level and reaches the ELB (first row, left). Output (first row, right) and inflation (second row, left) initially drop by about 5 percent and 3 percentage points, respectively. As the initial shock fades, output and inflation gradually return to their steady states, supported by accommodative monetary policy. The liftoff occurs at quarter 7. As explained in Section 2, the declines in output and inflation are notably larger in the presence of the ELB constraint than in the absence of it.

The responses of select nominal yields are shown in the third row. After the negative shock, nominal yields decline for all maturities. The 1-year (or 4-quarter) yield drops to the ELB, and the 5-year (or 20-quarter) yield falls to about 2 percent, before gradually increasing. Note that, while the 5-year yield is comfortably above the ELB throughout the horizon, it is still affected by the presence of the ELB; were it not for the ELB, the 5-year yield would be lower, as shown by the dashed line (third row, right).

Turning to term premium dynamics, we first note that, due to countercyclical volatility in \(\delta_t\), the negative shock to \(\delta_t\) increases its own conditional volatility in the initial period, which stays constant at a higher level for an extended period of time (second row, right, dash-dotted line). The heightened volatility of \(\delta_t\) leads to negative term premiums as nominal bonds become more attractive as hedging instruments when volatility in demand increases. This effect can be seen most clearly from the responses of term premiums in the economy without the ELB (bottom row, blue dashed lines).

However, the ELB constraint introduces additional dynamics. As shown in the bottom left panel, the 1-year term premium shows an (inverse) hump-shaped response, in which it declines for another quarter after the initial decline, before rising back to its average (positive) level. The response is in contrast to the 5-year term premium (bottom right), which declines to its trough immediately after the initial shock. The reason is, since the ELB constraint is expected to bind in the near term, the 1-year term premium is compressed towards zero more strongly than its 5-year counterpart in the initial quarter. In the second quarter, the 1-year term premium falls further because the expected ELB duration is now shorter, and thus less binding, which implies future policy rates can react to macroeconomic risk more easily and hence bonds become a better hedging instrument. Later on, as \(\delta_t\) returns closer to its average and the economy recovers, the volatility of \(\delta_t\) (\(\sigma_{\delta,t}\)) declines (second row, right) and the effect of productivity shocks starts to dominate. As a result, nominal bonds become riskier and the term premium increases. The 5-year term premium (bottom right) incorporates the expected decline in \(\sigma_{\delta,t}\) quickly, and turns from positive to negative right after the initial shock. However, even at the 5-year horizon, term premiums are compressed by the ELB. We revisit these results in the context of the recent U.S. experience in Section 4 and show that a sequence of shocks can generate further variation in the short-maturity term premium in ways that help us understand the data better.

Overall, the ELB constraint exacerbates the recession induced by the negative \(\delta_t\) shocks and generally dampens the negative response of yields and term premiums that would have occurred in the absence of the ELB constraint. At the same time, the compression effect of the ELB introduces interesting nonlinear term premium dynamics at the ELB.

4 Model-based Account of Two ELB Episodes in the U.S.

To further understand term structure dynamics around the ELB, we zoom in on the first ELB episode in the U.S. following the financial crisis. In particular, we investigate how yields and term premiums evolved in conjunction with the expected ELB duration throughout the episode and how our model can account for it.

Figure 4: Nominal Yields and Term Premiums at the ELB
—The First ELB Episode—
* The expected ELB duration is computed as the time to the first quarter when the median federal funds rate forecast exceeds 37.5 basis points. Data are at the FOMC frequency for the SPD and otherwise quarterly. The term premiums are the average of the survey- and model-based term premiums. The gray vertical lines indicate 2008:Q4 (dashed) and 2011:Q3 (solid).

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Figure 4 shows survey-based expectations for the time until liftoff (left panel), nominal yields (middle panel), and term premiums (right panel) of 1-, 2-, and 5-year maturities since 2007:Q2, just before the onset of the financial crisis.⁴⁴ Nominal yields were well above the ELB, and term premiums were largely positive across maturities at the time. However, as the financial crisis unfolded, yields declined sharply. Shorter-maturity term premiums declined, while the 5-year term premium fluctuated and rose initially until about mid-2009, before turning negative. Around the introduction of “calendar-based” forward guidance in August 2011 (the solid vertical lines), the expected ELB duration began to increase and reached about 2.5 years (10 quarters) by the end of 2012. Thereafter, the expected ELB duration fell.⁴⁵ The policy rate lifted off from the ELB in December 2015. Consistent with this evolution of the expected ELB duration, yields declined and remained low for a period of time as the expected duration was elevated around its peak. The 1-year yield started to increase around mid-2014, while the 2-year yield gradually rose beginning as early as mid-2013. Meanwhile, the 5-year yield increased sharply circa mid-2013.

1- and 2-year term premiums were negative until the second quarter of 2011. However, around the introduction of forward guidance, both 1- and 2-year term premiums compressed towards zero, as the expected duration of the ELB jumped. Both term premiums hovered around zero until mid-2013, after which they gradually turned negative as the expected duration of the ELB decreased. Meanwhile, the 5-year term premium was volatile, but broadly declined from its peak in 2009, and did not compress when the expected ELB duration increased sharply in the second quarter of 2011.⁴⁶

Figure 5: Impulse Responses to Two Consecutive δ_t Shocks
* The blue solid lines are the responses to two consecutive shocks of the same size (adjusted by standard deviation); \(median[X_{t+h}\mid \varepsilon_{\delta,t}=\varepsilon_{\delta,t+1}=-\vartheta_{\delta}, \bar{\Omega}]\) for a given variable \(X_t\), and the blue dashed lines are the baseline responses to a one-time shock; \(median[X_{t+h}\mid \varepsilon_{\delta,t}=-\vartheta_{\delta}, \bar{\Omega}]\) for a given variable \(X_t\). The vertical lines indicate the timing of liftoff from the ELB for each response.

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From the impulse responses of our model in the previous section, we do not see short-maturity term premiums compressed from negative values towards zero as expected ELB duration rises, and term premiums later becoming negative again, as expected ELB duration falls. However, by simply assuming two consecutive negative shocks that hit \(\delta_t\) unexpectedly instead of one shock in the initial period, we can replicate a similar pattern in short-maturity term premium dynamics. Indeed, as shown in the bottom-left panel of Figure 5, after the initial shock makes the 1-year term premium negative, the second shock compresses the term premium toward zero (blue solid line), as the expected ELB duration increases (implied by the difference between the location of the solid and dashed vertical lines). Afterwards, the term premium drops again (before rising) as the economy approaches liftoff and the compression effect of the ELB constraint fades. In contrast, the 5-year term premium is not as compressed, since longer-maturity yields are less sensitive to the current ELB episode, consistent with the data (right panel, blue solid line). One could loosely interpret the first shock as representing a shock that led to the financial crisis, and the second one as the shock that extended the expected ELB duration further around the timing of the date-based forward guidance.⁴⁷

The U.S. experienced another ELB episode following the Covid-19 crisis in early 2020. Figure 6 summarizes expected ELB duration, yields and term premiums during this episode, analogous to Figure 4. Similar to 2011, the expected ELB duration increased sharply after the policy rate hit the ELB, but the pace of decline afterwards was significantly faster (left panel). Reflecting these expectations, yields fell discretely in March 2020 as the policy rate reached the ELB, but the 1-year yield began to retrace after about a year and a half, much earlier than the first ELB episode. The 2- and 5-year yields started to increase earlier accounting for the future liftoff, similar to developments before the liftoff in 2015 (middle panel).

2- and 5-year term premiums were generally negative towards the end of 2019 and dropped notably at the end of the first quarter of 2020 after the policy rate hit the ELB (right panel). However, term premiums of shorter maturities quickly reverted towards zero and stayed there until around mid-2021. These developments appear to be broadly similar to the first ELB episode. However, in sharp contrast to the first ELB episode, term premiums in 2021 increased as the policy rate approached liftoff. In addition, while the 5-year term premium remained negative in the first ELB episode, it increased steadily to a positive level in the second episode.

Figure 6: Nominal Yields and Term Premiums at the ELB
—Covid-19 Crisis Episode—
* The expected ELB duration is computed as the time to the first quarter when the median federal funds rate forecast exceeds 37.5 basis points. Data are at the FOMC frequency for the SPD and otherwise quarterly. The term premiums are computed based on the Blue Chip survey. The gray vertical lines indicate 2020:Q2.

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Though our model is calibrated to the economy before the pandemic, we can conduct some simple counterfactual model simulations that help us interpret such term structure developments since the onset of the pandemic crisis. To this end, we hit the baseline model with both a negative TFP shock and a negative \(\delta_t\) shock in the initial period. In addition, we hit the model with the same shocks again in the next period, similar to the experiment above.⁴⁸ The combination of negative demand and supply shocks to describe the initial impact of the pandemic is consistent with Chen et al., 2021 and Eichenbaum et al., 2022. Our contribution is to analyze its impact on the term structure. The impulse responses for selected yields and term premiums are shown in Figure 7.⁴⁹ The blue dashed lines are the responses from the baseline model with the \(\delta_t\) shock only, identical to the solid lines in Figure 5. The light blue dash-dotted lines are the responses that include the negative TFP shock. When the model is hit with negative TFP shocks which raise inflation, the policy rate is raised earlier than when the shocks are absent. Thus, the shorter-term yield—represented by the 1-year yield in the figure—rises faster compared to the baseline (top left panel). The longer-term yield—represented by the 5-year yield in the figure—is also higher compared to the baseline (top right panel). Such model dynamics are broadly consistent with the shorter duration of the second ELB episode and the associated faster increase in yields.

Interestingly, shorter-maturity term premiums decline in response to the addition of negative TFP shocks (bottom left panel). This decline is, in fact, consistent with a faster pace of liftoff from the ELB due to the TFP shock (top left panel); the weakening of the compression effect of the ELB leads to more negative term premiums. Meanwhile, longer-maturity term premiums increase slightly (bottom right panel). This increase is because the negative TFP shocks raise the volatility of TFP and increase term premiums in our model. While this effect is also present for shorter maturities, it is outweighed by the effect of the ELB constraint.

Figure 7: Impulse Responses to δ_t Shocks
—Adding TFP Shocks and Increased Volatility of TFP—
* The blue dashed lines are the median impulse responses from the baseline model with the ELB constraint, the light blue dash-dotted lines are the responses where negative TFP shocks are added on impact, and the blue solid lines are the responses where a further increase in \(\bar{\sigma}_a/\bar{\sigma}_\delta\) occurs. The gray vertical lines indicate the timing of liftoff from the ELB corresponding to each scenario (the dashed and dash-dotted lines coincidentally overlap).

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The term premium dynamics from the simulations are somewhat at odds with what we observed during the second ELB episode in which (1) shorter-maturity term premiums did not decrease approaching liftoff, and (2) longer-maturity term premiums showed a notable increase to a positive level during the ELB period. One simple way to justify these observations is to further assume that the relative volatility of TFP with respect to \(\delta_t\) (\(\bar{\sigma}_a/\bar{\sigma}_\delta\)) shifted up (unexpectedly) following the initial TFP shock. This assumption is motivated by the empirical evidence of shifts in economic volatility following the pandemic (e.g., Carriero et al., 2024).⁵⁰ The responses under this scenario are shown in blue solid lines. An increase in TFP volatility pulls forward the timing of liftoff (solid vertical line) and generally increases yields, similar to the negative TFP shock (top panels). More importantly, the effects on term premiums are very different. First, the shorter-maturity term premium is \(higher\) compared to the baseline response after the initial drop (bottom-left panel). In particular, the term premium increases from zero to positive as the liftoff approaches. This is because the permanent increase in TFP volatility exerts a strong force in raising the term premium which more than offsets the downward force from the weakening of the ELB constraint mentioned earlier. Second, by a similar mechanism, the longer-maturity term premium increases sharply and becomes positive (bottom-right panel).⁵¹

So far, modeling time-varying volatility seems to have received little attention in equilibrium models that focus on post-pandemic macroeconomic developments. However, second moments are a key driver of risk premiums. Our analysis shows that the changes in volatility may indeed be crucial in understanding term structure dynamics since the pandemic crisis.

5 Monetary Policy and Equilibrium Yield Curves at the ELB

The structural feature of our model makes it a natural framework to study how monetary policy affects the dynamics of yields and term premiums in a coherent manner. In this section, we focus on how a commitment to keep the policy rate at the ELB for longer than what would be warranted above the ELB (or, accommodative “forward guidance”) affects the term structure of interest rates as well as the macroeconomy in our quantitative model.

We model this alternative policy based on the work of Reifschneider and Williams, 2000—the “RW rule”—, where we augment the baseline monetary policy rule as follows: \[\begin{align} \begin{split}R_t^{(1)}=\max \left[ R_{ELB},R_t^{*}-\phi_{RW}J_t \right], \\ J_t= J_{t-1}+(R_{t-1}^{(1)}-R_{t-1}^{*}). \end{split}\tag{31} \end{align}\] The shadow rate \(R_{t}^{*}\) is the same as in (28) and \(\phi_{RW}\geq 0\) controls the degree of extra accommodation when past policy rates have been constrained by the ELB. \(J_t\) is the cumulative past deviation of the policy rate from the shadow rate. This rule implies that the timing of liftoff depends more strongly on the cumulative shortfalls in inflation and output than our baseline rule. Note that in the model without the ELB constraint, this rule is identical to our baseline rule, and when \(\phi_{RW}=0\), the RW rule collapses to the baseline rule with the ELB constraint (27).⁵² We set \(\phi_{RW}=0.5\), a value taken from Reifschneider and Roberts, 2006, and conduct a similar experiment where we compare the impulse responses of the economy to a negative demand shock under the baseline rule and the RW rule, respectively.

Figure 8: Effect of Accommodative Forward Guidance at the ELB
* We plot \(median[X_{t+h}\mid \varepsilon_{\delta,t}=-\vartheta_{\delta}, \bar{\Omega}]\) for a given variable \(X_t\). The blue solid lines are the responses from the model under the RW rule, and the blue dashed lines are the responses from the model under the Taylor rule with the ELB constraint. The gray vertical lines indicate the timing of liftoff from the ELB corresponding to each rule.

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The results are shown in Figure 8. As shown in the top-left panel, the policy rate is kept at the ELB for longer—by about 4 quarters—under the RW rule (blue solid line) than under the baseline rule (blue dashed line). With the nominal rate staying at the ELB for longer, the path of real rates is lower (not shown), which stimulates consumption of forward-looking households. An increase in consumption is associated with an increase in output (top right), which raises marginal costs of production and thus inflation (second row, left). The mechanism through which the RW rule stimulates economic activities at the ELB is the same as the mechanism through which the central bank stimulates economic activities at the ELB under optimal commitment policy and is well known in the macro literature.⁵³

Consistent with the lower path of short-term interest rates, the paths of other nominal yields are lower under the RW rule than under the baseline rule (third row). Note that, while the difference in the short-term rate between these two policy rules emerges well after the initial shock, the difference in longer-term yields shows up at the onset of the crisis, reflecting the forward-looking nature of longer-term yields.

The panels in the last row plot the effects of adopting the RW rule on nominal term premiums. In contrast to the effects on yields and the expected path of future short rates, term premiums at the ELB are higher under the RW rule than under the baseline rule for both maturities. The intuition can be understood through our discussion in Section 2. The RW rule extends the period in which the policy rate is at the ELB and thus the interest rate sensitivity becomes even lower, compressing term premiums. Also, the more accommodative policy achieved by the RW rule leads to smaller macroeconomic volatility, further compressing term premiums. Accordingly, the absolute size of term premiums is lower, or, in this case, term premiums are less negative, under the RW rule than under the baseline rule.

Note that this result implies that the policy accommodation achieved through the decline in the expected short-rate path is partially offset by an increase in term premiums. However, since the effect of forward guidance is to compress the absolute size of term premiums, this result depends crucially on the risk exposure of bonds. For example, contrary to our baseline model in which demand shocks play a dominant role at the ELB, suppose TFP shocks play a dominant role, making average term premiums positive even at the ELB. Then, forward guidance could lower term premiums, amplifying the decline in the expected short-rate path. We present this case in Appendix I.

We used the forward guidance policy to motivate our analysis of the effects of adopting the RW rule. However, our analysis may be useful in understanding the effects of the central bank’s other unconventional policies, such as QE. Some have argued that QE stimulates economic activities partly by signaling the central bank’s commitment to an extended period of more accommodative policies in the future (e.g., Woodford, 2012, Bauer and Rudebusch, 2014, and Bhattarai et al., 2023). If that is the case, then the RW rule can be seen as capturing this signaling effect of QE policy.

6 Conclusion

We have studied the term structure of interest rates in a New Keynesian model with an occasionally binding ELB constraint. Using a stylized model, we showed that the ELB constraint induced state dependency in term premiums by affecting macroeconomic uncertainty and the sensitivity of interest rates to economic activity. Further extending the model to match key features of output, inflation, yield curves, and term premium estimates in the U.S., we found that the compression effect of the ELB constraint was strong, reducing the absolute level of term premiums on average, while generating distinct dynamics around liftoff that would not arise without the constraint. We also used the model to provide a structural interpretation of term structure dynamics during the two recent ELB episodes.

We additionally investigated how the central bank’s forward guidance to keep the policy rate at the ELB longer than previously expected affects the term structure of interest rates. We demonstrated that the adoption of such a policy reduced the expected short-rate path and the absolute size of term premiums. That is, if demand shocks are dominant and bonds are a hedge against economic downturns, term premiums are negative on average, and forward guidance increases term premiums. By contrast, if supply shocks are dominant and bonds are risky, term premiums are positive on average, and forward guidance decreases term premiums.

At a time when central banks in advanced economies have been providing various unconventional policies at or near the ELB, we believe that studying how the ELB constraint affects the dynamics of the term structure of interest rates is an essential step towards a deeper understanding of these policies. Extending our model to explicitly include the central bank’s balance sheet is an interesting avenue for future research.

Footnotes