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.
The figure consists of \(2\times 3\) panels of equilibrium functions. The horizontal axis in every panel is the three-state demand shock \(\delta_t\), with states \(\delta_L=-\delta\), \(\delta_M=0\), and \(\delta_H=\delta\). Blue dotted lines with circle markers indicate the functions from the model with the ELB constraint, and light-blue dashed lines with plus markers indicate the functions from the model without the ELB constraint.
The top row plots the one-period nominal policy rate \(r_t^{(1)}\), output \(\hat y_t\), and inflation \(\hat\pi_t\), respectively. In both models, all three variables increase as the demand shock rises. Without the ELB, the relationships are linear. With the ELB, the policy rate is lower in all states compared to the policy rate without the ELB, and in particular, zero in the low and medium states. Output and inflation are also lower in all states compared to output and inflation without the ELB, but they are particularly lower in state \(L\). In other words, the equilibrium functions are nonlinear.
The bottom row plots the two-period nominal yield \(r_t^{(2)}\), the two-period yield spread \(r_t^{(2)}-r_t^{(1)}\), and the two-period nominal term premium \(tp_t^{(2)}\). Without the ELB, the two-period yield increases linearly with the demand shock, the yield spread declines linearly, and the term premium is constant and negative. With the ELB, the two-period yield is close to zero in the low and medium states and significantly positive in the high state. The yield spread is nonlinear, zero in the low state, positive in the medium state, and negative in the high state. The term premium is zero in the low state, negative in the medium state, and more negative in the high state. In particular, the ELB compresses the term premium in the low state, while amplifying the term premium in the high state.
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.
The figure consists of \(2\times 2\) panels of U.S. nominal yields and term premiums from 1999:Q1 through 2019:Q2. The shaded gray areas indicate NBER recessions. A vertical line at 2008:Q4 indicates the beginning of the first ELB period.
The top-left panel plots nominal Treasury yields at 3-month, 1-year, 2-year, and 5-year maturities. The yields are relatively high around 2000 and again before the 2007–2009 recession, fall sharply during recessions, and remain near zero at short maturities for several years after 2009 before rising again. The yield curve is generally upward sloping with the 5-year yield staying above the shorter-maturity yields most of the time.
The remaining three panels plot term premium estimates for 1-year, 2-year, and 5-year maturities. In each panel, four series are shown: the dark-blue solid line is the average of three term premium measures, the blue dash-dotted line is the survey-based measure, the blue dashed line is the Kim–Priebsch measure, and the light-blue dash-dotted line is the Andreasen–Jorgensen–Meldrum measure. The estimates generally comove but differ in level and volatility, especially at longer maturities. Average term premiums are mostly positive in the early part of the sample, decline over time, and become close to zero or negative for much of the ELB period. The 5-year term premium is more volatile than the shorter-maturity term premiums, both above and at the ELB.
Figure 3: Impulse Responses to δt
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.
The figure consists of \(4\times 2\) panels of impulse responses to a negative risk-preference shock \(\delta_t\). The horizontal axis shows quarters from 0 to 15. Blue solid lines are responses from the model with the ELB constraint, and blue dashed lines are responses from the model without the ELB constraint. Gray vertical lines indicate the timing of liftoff from the ELB in the model with the ELB constraint.
The panels are ordered from left to right in each row as follows: (first row) the nominal 1-quarter (or short) rate \(R_t^{(1)}\) and output \(Y_t\); (second row) inflation \(\Pi_t\) and the risk-preference shock \(\delta_t\) together with its conditional volatility \(\sigma_{\delta,t}\); (third row) the 1-year nominal yield \(R_t^{(4)}\) and the 5-year nominal yield \(R_t^{(20)}\); and (fourth row) the 1-year nominal term premium \(tp_t^{(4)}\) and the 5-year nominal term premium \(tp_t^{(20)}\).
When the ELB constraint is present, the short rate falls to the ELB and remains there until quarter 7. Without the ELB constraint, the short rate falls much more to significantly negative values before retracing back to its long-run average. In contrast, output and inflation decline much more when the ELB constraint is present than in the model without the ELB. The negative shock also raises the conditional volatility of the risk-preference shock for an extended period. Nominal yields fall at all maturities, but their declines are smaller in absolute value in the model with the ELB than in the model without the ELB. The 1-year term premiums in both models initially fall, but in the model with the ELB constraint, the term premium declines further in the next quarter before it recovers, producing an inverse hump-shaped response. Meanwhile, the 5-year term premiums fall and begin to recover to their average levels after the next quarter in both models. That said, relative to the unconstrained model, the decline and subsequent increase in term premiums are more compressed while the ELB is binding in the model with the ELB.
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).
The figure shows \(1\times 3\) panels of expected ELB duration, nominal yields, and term premiums from 2007:Q2 through 2015:Q4; around the first U.S. ELB episode following the financial crisis. The dashed gray vertical line indicates 2008:Q4, the quarter in which the policy rate reached the ELB, and the solid gray vertical line indicates 2011:Q3, the quarter in which calendar-based forward guidance was introduced.
The left panel plots the expected duration of the ELB in quarters, based on the Survey of Primary Dealers (SPD) and the Survey of Professional Forecasters (SPF). Expected ELB duration rises sharply around the introduction of calendar-based forward guidance in August 2011. It reaches about 10 quarters, or 2.5 years, by the end of 2012 and then declines as liftoff approaches. The policy rate ultimately lifts off from the ELB in December 2015.
The middle panel plots 1-year, 2-year, and 5-year nominal yields. All three yields decline sharply as the financial crisis unfolds. The 1-year and 2-year yields remain very low for much of the ELB period, while the 5-year yield stays noticeably higher. As expected ELB duration later declines, the 5-year yield rises sharply around mid-2013, the 2-year yield begins to rise gradually around mid-2013, and the 1-year yield starts to increase around mid-2014.
The right panel plots term premiums for 1-year, 2-year, and 5-year maturities, where each measure is the average of the survey-based and two model-based measures used in the paper. The 1-year and 2-year term premiums are generally negative through 2011:Q2. Around the introduction of forward guidance, both compress toward zero as expected ELB duration rises, remain near zero until around mid-2013, and then gradually become negative again as expected ELB duration declines. The 5-year term premium behaves differently: it is more volatile, rises initially through about mid-2009 before turning negative, and does not show the same compression toward zero when expected ELB duration rises sharply around 2011. It then rises sharply again around mid-2013 and declines in 2015.
Figure 5: Impulse Responses to Two Consecutive δt
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.
The figure consists of \(2\times 2\) panels of impulse responses of yields and term premiums. The x-axis runs from quarter 0 through quarter 15. Blue solid lines show responses to two consecutive negative \(\delta_t\) shocks of the same size, arriving in quarters 1 and 2, for the model with the ELB constraint. Blue dashed lines show the baseline responses to a single negative \(\delta_t\) shock in quarter 1 for the same model, which are the same as the responses in Figure 3. The vertical lines indicate the timing of liftoff from the ELB under the corresponding scenarios; the second shock delays liftoff relative to the one-shock case.
The top-left and top-right panels plot the 1-year nominal yield \(R_t^{(4)}\) and the 5-year nominal yield \(R_t^{(20)}\). Relative to the one-shock case, the second negative shock keeps the 1-year yield close to the ELB for several additional quarters. It also lowers the 5-year yield for longer, although the effect is less pronounced at the longer maturity.
The bottom-left and bottom-right panels plot the corresponding 1-year and 5-year nominal term premiums. For the 1-year term premium, the first shock makes the premium negative, while the second shock compresses it back toward zero by extending the expected duration of the ELB. As liftoff approaches and the ELB becomes less binding, the 1-year term premium becomes more negative again before eventually increasing to its average value. In contrast, the 5-year term premium is not compressed by the second shock and remains negative for longer.
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.
The figure shows \(1\times 3\) panels of expected ELB duration, nominal yields, and term premiums from 2019:Q3 through 2022:Q1, covering the period immediately before and during the second U.S. ELB episode following the Covid-19 crisis. The gray vertical line indicates 2020:Q2.
The left panel plots expected ELB duration in quarters from the SPD and SPF. Expected duration rises sharply after the policy rate reaches the ELB in March 2020, reaching 14 to 16 quarters depending on the survey. It then declines much faster than during the first ELB episode and is close to zero by 2022:Q1.
The middle panel plots 1-year, 2-year, and 5-year nominal yields. All three yields fall discretely in early 2020 as the policy rate reaches the ELB. The 1-year yield remains near zero for about a year and a half before beginning to retrace. The 2-year and especially the 5-year yields begin rising earlier, reflecting expectations of future liftoff. By 2022:Q1, all three yields have risen substantially from their ELB-period lows.
The right panel plots survey-based term premiums for 1-year, 2-year, and 5-year maturities. The 2-year and 5-year term premiums are generally negative near the end of 2019 and drop further by 2020:Q1 after the policy rate reaches the ELB. Shorter-maturity term premiums then move back toward zero and remain near zero until around mid-2021. The subsequent dynamics contrast sharply with the first ELB episode: during 2021, term premiums rise as liftoff approaches rather than becoming more negative. In particular, the 5-year term premium, which remained negative during the first ELB episode, rises steadily and becomes positive during the second episode.
Figure 7: Impulse Responses to δt 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).
The figure consists of \(2\times 2\) panels of impulse responses of yields and term premiums for different model specifications. The x-axis runs from quarter 0 through quarter 15. The top row plots the responses of the 1-year nominal yield \(R_t^{(4)}\) and the 5-year nominal yield \(R_t^{(20)}\), while the bottom row plots the responses of 1-year and 5-year nominal term premiums. Blue dashed lines show the responses from the baseline model with the ELB constraint and two consecutive negative \(\delta_t\) shocks. Light-blue dash-dotted lines add negative TFP shocks in the same two periods. Blue solid lines further allow a permanent increase in the relative volatility of TFP beginning after the initial shocks. Gray vertical lines indicate liftoff from the ELB under the different scenarios.
Adding negative TFP shocks brings liftoff forward and raises both the 1-year and 5-year yields relative to the baseline with only \(\delta_t\) shocks. For term premiums, however, the effect differs by maturity. Negative TFP shocks make the 1-year term premium more negative, while they raise the 5-year term premium.
When the permanent increase in relative TFP volatility is also introduced, both term premiums are significantly higher than in the baseline after the initial decline. The 1-year term premium is compressed towards zero and stays there for a few quarters before increasing to positive values as liftoff approaches, and the 5-year term premium rises sharply after the increase in volatility and becomes positive.
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.
The figure consists of \(4\times 2\) panels of impulse responses to a negative risk-preference shock \(\delta_t\). The horizontal axis shows quarters from 0 to 15. Blue solid lines are responses under the Reifschneider–Williams (RW) rule, and blue dashed lines are responses under the baseline Taylor rule with the ELB constraint. Gray vertical lines indicate the timing of liftoff from the ELB under the corresponding rules.
The panels are ordered from left to right in each row as follows: (first row) the nominal 1-quarter (or short) rate \(R_t^{(1)}\) and output \(Y_t\); (second row) inflation \(\Pi_t\) and the risk-preference shock \(\delta_t\) together with its conditional volatility \(\sigma_{\delta,t}\); (third row) the 1-year nominal yield \(R_t^{(4)}\) and the 5-year nominal yield \(R_t^{(20)}\); and (fourth row) the 1-year nominal term premium \(tp_t^{(4)}\) and the 5-year nominal term premium \(tp_t^{(20)}\).
Under the baseline Taylor rule, the short rate remains at the ELB until quarter 7. Under the RW rule, the short rate remains at the ELB until quarter 11. Output and inflation decline under both rules, but their declines are smaller under the RW rule. The risk-preference shock and its conditional volatility follow the same paths under the two rules. The 1-year and 5-year nominal yields decline under both rules and are lower under the RW rule. The 1-year and 5-year nominal term premiums also decline after the shock, but are generally higher, or less negative, under the RW rule than under the baseline Taylor rule. The differences across the two rules diminish as the variables return toward their long-run values.
Appendix
Figure C.1: Equilibrium Functions of the Continuous-State
Model
* Blue solid lines indicate variables of the model with the ELB
constraint, and blue dashed lines indicate variables of the model
without the ELB constraint. The gray vertical lines indicate the
threshold state where the ELB binds.
The figure consists of \(2\times 3\) panels of equilibrium functions for various variables. The horizontal axis in every panel is the \(\delta_t\) shock, a continuous variable ranging approximately from \(-0.02\) to \(0.02\). Blue solid lines indicate functions from the model with the ELB constraint, and blue dashed lines indicate functions from the model without the ELB constraint. Gray vertical lines indicate the threshold state where the ELB binds.
The top row plots the one-period nominal policy rate \(r_t^{(1)}\), output \(\hat y_t\), and inflation \(\hat\pi_t\), respectively. Without the ELB, all three variables increase smoothly and approximately linearly with \(\delta_t\). With the ELB, the policy rate is zero at sufficiently low values of \(\delta_t\), while output and inflation are below their values in the model without the ELB and decline more sharply as \(\delta_t\) falls. As \(\delta_t\) rises sufficiently above the ELB threshold, the equilibrium functions in the two models become nearly identical.
The bottom row plots nominal yields \(r_t^{(n)}\), yield spreads \(r_t^{(n)}-r_t^{(1)}\), and nominal term premiums \(tp_t^{(n)}\) for maturities \(n=2\), \(4\), and \(20\). Without the ELB, yields increase approximately linearly with \(\delta_t\), yield spreads decline approximately linearly, and term premiums are nearly constant and negative. With the ELB, all three sets of variables are nonlinear in \(\delta_t\). The differences in yields between the two models are larger at shorter maturities. However, the differences in yield spreads between the two models are larger at longer maturities. For term premiums, the 2-period and 4-period premiums with the ELB constraint approach zero at sufficiently low values of \(\delta_t\), become more negative compared to the term premiums without the ELB over a range near the ELB threshold, and approach their values without the ELB as \(\delta_t\) rises. The 20-period term premium is more negative with the ELB constraint over the range shown and also approaches its value without the ELB as \(\delta_t\) rises.
Figure C.2: Conditional Volatilities and Correlations
* Blue solid lines indicate variables of the model with the ELB
constraint, and blue dashed lines indicate variables of the model
without the ELB constraint. The gray vertical lines indicate the
threshold state where the ELB binds.
The figure contains five panels of conditional volatilities and correlations arranged in two rows. The horizontal axis in every panel is the \(\delta_t\) shock over the same range as in Figure C.1. Blue solid lines indicate results from the model with the ELB constraint, and blue dashed lines indicate results from the model without the ELB constraint. Gray vertical lines indicate the threshold state where the ELB binds.
The top row plots the one-period-ahead conditional volatility of output, inflation, and the one-period nominal policy rate, respectively. Without the ELB, each volatility is approximately constant across \(\delta_t\). With the ELB, output and inflation volatility are substantially higher at low values of \(\delta_t\) and decline toward their values without the ELB as \(\delta_t\) rises. In contrast, policy-rate volatility is close to zero at very low values of \(\delta_t\), rises sharply around the ELB threshold, and approaches its approximately constant value without the ELB as \(\delta_t\) rises.
The bottom row plots the conditional correlation of output with the policy rate and the conditional correlation of inflation with the policy rate. Without the ELB, both correlations are approximately one throughout the state space. With the ELB, both correlations are close to zero at very low values of \(\delta_t\), rise as \(\delta_t\) increases, and approach one above the ELB threshold.
Figure C.3: Conditional Volatilities and Correlations
* Solid lines indicate variables of the model with the ELB
constraint and dashed lines indicate variables of the model without the
ELB constraint. The gray vertical lines indicate the threshold state
where the ELB binds.
The figure consists of \(1\times 3\) panels of conditional volatilities and correlations. The horizontal axis in every panel is the \(\delta_t\) shock over the same range as in Figures C.1 and C.2. Solid lines indicate results from the model with the ELB constraint, and dashed lines indicate results from the model without the ELB constraint. For each model, three yield maturities are shown: \(n=2\), \(4\), and \(20\). Gray vertical lines indicate the threshold state where the ELB binds.
The left panel plots the one-period-ahead conditional volatility of nominal yields \(r_t^{(n)}\). Without the ELB, yield volatility is approximately constant for each maturity and lower at longer maturities. With the ELB, yield volatility is substantially lower at low values of \(\delta_t\), particularly for the 2-period and 4-period yields, and rises toward its value without the ELB as \(\delta_t\) increases. The 20-period yield volatility varies less across \(\delta_t\) and differs less between the two models.
The middle and right panels plot the conditional correlations of output and inflation, respectively, with nominal yields for the three maturities. Without the ELB, the correlations are approximately constant and close to one. With the ELB, the correlations are lower at low values of \(\delta_t\), with the largest differences at the 2-period maturity and the smallest differences at the 20-period maturity. The correlations approach their values without the ELB of 1 as \(\delta_t\) rises above the ELB threshold.
Figure C.4: Term Premium Volatility
* Solid lines indicate variables of the model with the ELB
constraint and dashed lines indicate variables of the model without the
ELB constraint. The gray vertical lines indicate the threshold state
where the ELB binds.
The figure plots the conditional volatility of nominal term premiums \(tp_t^{(n)}\) for maturities \(n=2\), \(4\), and \(20\) as functions of the \(\delta_t\) shock. The horizontal axis is the \(\delta_t\) shock over the same range as in Figures C.1–C.3. Solid lines indicate the volatilities from the model with the ELB constraint, and dashed lines indicate the volatilities from the model without the ELB constraint. The gray vertical line indicates the threshold state where the ELB binds.
Without the ELB, term premium volatility is close to zero at all three maturities throughout the state space. With the ELB, term premium volatility is positive and varies with \(\delta_t\) in a highly nonlinear fashion. For all maturities, volatility is low at high values of \(\delta_t\), and rises as \(\delta_t\) declines toward the ELB threshold and further into the region where the ELB binds. However, volatilities do not increase monotonically as \(\delta_t\) declines; the peak volatility is reached where the ELB is binding, but the timing is different across maturities. The 4-period term premium volatility varies the most, with the highest volatility achieved at the smallest value of \(\delta_t\) shown. Meanwhile the 20-period volatility varies more smoothly, reaching its peak near the ELB threshold.
Figure I.1: Effect of Accommodative Forward Guidance at the ELB
—The Case of Positive Term Premiums 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.
The figure consists of \(2\times 2\) panels of impulse responses under an alternative parameterization of the quantitative model in which productivity shocks play a dominant role at the ELB. The horizontal axis shows quarters from 0 to 15. Blue solid lines are responses under the RW rule, and blue dashed lines are responses under the baseline Taylor rule with the ELB constraint. Gray vertical lines indicate the timing of liftoff from the ELB under the corresponding rules.
The top-left and top-right panels plot the 1-year nominal yield \(R_t^{(4)}\) and the 5-year nominal yield \(R_t^{(20)}\). Both yields fall sharply after the shock and subsequently rise toward their long-run levels. Under the RW rule, both yields are lower than under the baseline Taylor rule during the ELB period. The difference between the two rules becomes larger for the 1-year yield than for the 5-year yield as the economy approaches liftoff under the RW rule.
The bottom-left and bottom-right panels plot the corresponding 1-year and 5-year nominal term premiums. Both term premiums remain non-negative over the horizon shown. Under the RW rule, both term premiums are lower than under the baseline Taylor rule.