## 2. Calibration Values

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### Introduction and research question
- Can the shape of yield curves be rationalized by consumers’ optimal behavior given empirically observed income and inflation dynamics and central bank policy behavior, including the zero lower bound (ZLB)?
- Model features summarized:
  - Consumers with Epstein-Zin-Weil preferences (inverse IES σ, CRRA α).
  - Nominal interest rate follows a monetary policy rule with smoothing parameter φr and inflation response φπ.
  - Income and inflation decomposed into stationary (gaps) and non-stationary (long-run trend) components.
  - Term premiums defined as:
    - Nominal term premium ψNn,t ≡ Rn,t − R̂ n,t.
    - Real term premium ψRn,t ≡ rn,t − r̂ n,t.
    - Inflation term premium ψIn,t ≡ ψNn,t − ψRn,t.

### Model structure and key calibration values
- Endowment economy decomposition:
  - log(Yt) = log(y∗t) + log(yt); gt ≡ y∗t / y∗t−1 with E(gt) = g∗.
  - log(Πt) = log(π∗t) + log(πt); ξt ≡ π∗t / π∗t−1.
- Monetary policy rule (in levels):
  - Rt = Rφr t−1 [ R∗t (Πt / π∗t) φπ ]1−φr.
- Detrending applied by Pt, π∗t and/or y∗t for tractability; supply of n-period bonds for n>1 set to zero in equilibrium so one-period nominal bonds are the only savings instrument.
- Key calibration values (Table 2 referenced):
  - β = 0.9985
  - σ = 2.0
  - α = 40 for US and 20 for UK
  - φb = 0.001
  - b∗ = 4.8
  - φπ = 2.0 for US and 1.0 for UK
  - φr = 0.8
  - Habit formation parameter used in robustness check: h = 0.25

### Empirical estimation of income and inflation processes
- Two specifications estimated:
  - Trend-stationary VAR(1) for Xt ≡ [log(yt), log(πt)]′: Xt+1 = A1 Xt + εX,t+1, εX,t+1 ∼ N(0, ΣX).
  - Stochastic-trend VAR(1) for Xt ≡ [log(yt), log(πt), log(gt)]′: Xt+1 = A0 + A1 Xt + εX,t+1, εX,t+1 ∼ N(0, ΣX); plus AR(1) for log(ξt).
- Priors and identification highlights:
  - Trend inflation AR(1) prior Beta[0.95, 0.03]; σξ / σπ = 0.01.
  - In stochastic trend model tight prior Beta[0.80, 0.05] for ρy.
  - Improper Uniform[-1,1] priors for ρyπ, ρπy, σπg, σπy (estimated by data).
- Data samples:
  - US: “PCE deflator (chain price index)” and “Personal Income”; sample 1959Q2 - 2017Q4.
  - UK: “Long Term Price Indicator of Consumer Goods and Services” and “Real Household Disposable Income”; sample 1959Q2 - 2017Q4.

### Estimation results — stylized empirical findings
- Stochastic trend case:
  - Positive association over the business cycle (Phillips curve): ρyπ > 0 and ρπy > 0; σπy > 0.
  - Inflation negatively correlated with long-run income growth: σπg < 0.
  - Implication: inflation positively correlated with the stationary income gap and negatively correlated with growth of long-run income.
- Trend-stationary case:
  - Lagged income gap positively affects inflation (ρpy > 0); lagged inflation negatively affects income gap (ρyp < 0); contemporaneous shock correlation σπy < 0.
  - Scatter plots show a positive association between πt and yt−1 despite σπy < 0.

### Quantitative solution methods and discretization
- VAR(1) approximated by first-order Markov chain over discretized grids Gy, Gπ, and Gg using Terry and Knotek II (2011) method.
- AR(1) ξt approximated by Tauchen’s method.
- Preference and market friction calibrations reiterated:
  - β = 0.9985; σ = 2.0; φb = 0.001; b∗ = 4.8; φr = 0.8; φπ = 2.0 (US), φπ = 1.0 (UK); α = 40 (US), α = 20 (UK).

### Main quantitative findings — equilibrium yield curve and drivers
- Stylized facts (data samples 1959Q1-2017Q1 for US and 1957Q1-2017Q1 for UK):
  - 10-year spread (relative to short rate) is 1.07% in US and 0.84% in UK.
  - Volatility of long-term rates slightly smaller but close to short-term rates.
- Model performance:
  - Stochastic trend baseline model accounts for upward-sloping yield curves and positive sizable term premiums in US and UK when calibrated to estimated income and inflation processes.
  - Trend-stationary model fails to generate sizable term premiums; yield curve too flat.
- Mechanism reconciling Phillips curve with positive term premiums:
  - Income decomposition: inflation positively correlated with cyclical income yt but negatively correlated with long-run income growth gt.
  - Under permanent income logic, consumption driven mainly by long-run income → consumption growth negatively correlated with inflation → positive term premiums while preserving Phillips curve for cyclical income.
- Composition of nominal term premiums:
  - A substantial part of nominal term premiums is accounted for by real term premiums, particularly in US.
  - Inflation term premiums do not play a dominant role in explaining the upward-sloping yield curve in baseline results.
- Monetary policy effects:
  - Real term premiums driven by cov(1/rt+1, mt,t+1) < 0: aggressive monetary response to inflation raises real rates when inflation hikes occur; combined with negative correlation of consumption growth and inflation, makes real bonds risky.
  - Comparative statics: increasing φπ steepens the equilibrium yield curve; φπ values illustrated include 1.0, 2.0, and 3.0.
  - Baseline φπ = 2.0 for US vs φπ = 1.0 for UK helps explain cross-country differences in term premiums and in volatility of interest rates relative to inflation.

### Term structure of interest rate volatility
- Empirical puzzle: observed volatility of long-term interest rates is relatively large compared to theoretical predictions.
- Model result:
  - Time-varying trend inflation π∗t is a key driver of relatively large volatility of long-term interest rates.
  - When trend inflation is fixed, relative volatility of long-term rates would be much smaller.
  - Detrending series (HP filter) shows relative volatility of detrended long-term rates is much smaller and closer to the model without time-varying trend inflation.
- Implication: cointegration between interest rates and slow-moving trend inflation explains a large share of long-term rate volatility; term premium volatility need not be large to match observed long-term volatility.

### Macroeconomic moments for consumption growth
- Model-generated macro moments for quarterly, yearly, and 2-year consumption growth broadly consistent with data, particularly for longer-term growth measures that matter for 5- and 10-year term premiums.

### Low-for-long counterfactual (permanently low growth and low inflation) and policy implications
- Counterfactual specification:
  - Set log(g∗) = 0 and log(π∗t) = 0.
  - Monetary policy with ZLB: Rt = max { 1.0, Rφr t−1 [ π∗t g∗ (Πt/π∗t) φπ ]1−φr } with g∗ = 1 and π∗t = 1 for all t.
- Simulation results:
  - Low-for-long economy associated with lower level of interest rates and significantly flatter yield curves: term spreads for 5-year and 10-year bonds are almost half in the low-for-long economy relative to baseline.
  - Decline in real term premiums accounts for almost all of the decline in nominal term premiums; inflation risk premiums are almost unchanged or slightly higher.
- Mechanism near ZLB:
  - Above ZLB: real bond returns increase with inflation because of active policy response → real bonds are poor hedges → positive real term premiums.
  - Near ZLB: central bank constrained in raising nominal rates → weakens positive correlation between real SDF and real interest rates → reduces real term premiums → long bonds become more like insurance in the low-for-long economy.
- Policy and financial stability implication:
  - Flatter yield curves in a low-for-long environment reduce banks’ maturity transformation profits, exacerbating already lower profitability from ZLB-era net interest margin compression and posing higher financial stability risk.

### Appendices — detrending and robustness checks
- Appendix I — Detrending: reformulated variables and equilibrium conditions (selected expressions preserved as in source)
  - Detrended nominal bond holdings:
    - b_t = B_t/(P_t y*_t π*_t)
    - b_{n,t} = B_{n,t}/(P_t y*_t π*_t^n)
  - Detrended nominal interest rate and n-period bond prices:
    - \tilde{R}_t = R_t/π*_t
    - \tilde{Q}_{n,t} = π*_t^n Q_{n,t}
  - Detrended consumption:
    - c_t = C_t/y*_t
  - Reformulated budget constraint (equation (6)):
    - c_t + b_t \tilde{R}_t + ∑_{n>1} \tilde{Q}_{n,t} b_{n,t} + Φ(b_t \tilde{R}_t) = 1 + b_{t−1} g_t π_t ξ_t + ∑_{n>1} \tilde{Q}_{n−1,t} b_{n,t−1} g_t π_t ξ_{t}^n
  - Reformulated monetary policy rule (equation (7)):
    - \tilde{R}_t = (\tilde{R}_{t−1} ξ_t)^{φ_r} [g^* π^{φ_π}_t]^{1−φ_r}
  - Detrended nominal SDF and Euler equations presented in reformulated form in the appendix.
- Appendix II — Robustness checks (marginal changes from baseline; US equilibrium yield curve)
  1. CRRA sensitivity:
     - α tested at α = 20, α = 40, α = 150.
     - Finding: size of term premiums highly sensitive to α; qualitative result of positive and sizable term premiums preserved across a wide range.
  2. Habit formation (h = 0.25):
     - Value function changed to include V_t = { (c_t − h c_{t−1})^{1−σ} + β E_t[V_{t+1}^{1−α}]^{1/(1−α)} }^{1/(1−σ)}.
     - Finding: habit formation has very small effects on term premiums.
  3. Fixed trend inflation:
     - Finding: time-varying trend inflation barely affects size of term premiums because trend inflation is very slow-moving with very small volatility.
  4. Monetary policy responding to income gap y_t (additional parameter φ_y):
     - Policy rule modified to include (Y_t/y*_t)^{φ_y}.
     - Finding: term premiums barely changed but become slightly higher than baseline.
- Robustness conclusion:
  - Model replicates positive and sizable term premiums under estimated inflation and income co-movement across robustness variations.
  - Highly sensitive: CRRA coefficient.
  - Small/no effect: habit formation (h = 0.25), time-varying vs fixed trend inflation, monetary policy responding to output gap (slightly higher term premiums).

*Source: wp18242 (IMF Working Paper wp18242)*

### 2.  Calibration Values .................................................................................................

### 2.  Calibration Values

### Sections listed in the source
- 2.  Calibration Values ...............................................................................................................37
- 3. Stylized Facts: Shape of Yield Curves in the United States and the United Kingdom ........38
- 4. Equilibrium Yield Curve in the United States and the United Kingdom .............................39
- 5. Volatility of Nominal Interest Rate Relative to Inflation Volatility ....................................40
- 6. Macroeconomic Moments for Consumption Growth ...........................................................41

### Figures listed in the source
- 1. Phillips Curve in the Stochastic Trend Model: log(y_t) and log(π_t) ................................3
- 2. Long-run Income Growth and Inflation: log(g_t) and log(π_t) ........................................3
- 3. Phillips Curve in the Trend Stationary Model: log(y_t) and log(π_t) ................................37
- 4. Equilibrium Yield Curve in the Simulated Model ................................................................38
- 5. Equilibrium Yield Curve with Different Monetary Policy Rules .........................................39
- 6. Relative Volatility for Each Maturity ...................................................................................40
- 7. Equilibrium Yield Curve in a Different Setting  ...................................................................4

### Appendices listed in the source
- I. Detrending.............................................................................................................................3 
- II. Robustness Check ................................................................................................................33

*https://www.imf.org/-/media/files/publications/wp/2018/wp18242.pdf*

### References ................................................................................ ............................

### wp18242 - References ................................................................................ ............................

### Introduction and research question
- Can the shape of yield curves be rationalized by consumers’ optimal behavior given empirically observed income and inflation dynamics and central bank policy behavior, including the zero lower bound (ZLB)?
- The paper analyzes a model with optimal savings as a buffer stock where consumers have Epstein-Zin-Weil preferences and nominal interest rates follow a monetary policy rule with smoothing parameter φr and inflation response φπ.

### Model structure and key mechanisms
- Endowment economy with real income Yt decomposed as log(Yt) = log(y∗t) + log(yt), where yt is the stationary (income gap) and y∗t is the non-stationary (long-run income) component; growth of long-run income gt ≡ y∗t / y∗t−1 with E(gt) = g∗.
- Inflation Πt ≡ Pt/Pt−1 decomposed as log(Πt) = log(π∗t) + log(πt), with ξt ≡ π∗t / π∗t−1 and πt the inflation gap.
- Monetary policy rule: Rt = Rφr t−1 [ R∗t (Πt / π∗t) φπ ]1−φr (Rt depends on last period’s interest rate; φr and φπ denote smoothing and inflation response).
- Consumers maximize Epstein-Zin-Weil utility Vt with inverse IES σ and CRRA α; nominal SDF Mt,t+1 and real SDF mt,t+1 defined as in text.
- Term premiums:
  - Nominal term premium ψNn,t ≡ Rn,t − R̂ n,t.
  - Real term premium ψRn,t ≡ rn,t − r̂ n,t.
  - Inflation term premium ψIn,t ≡ ψNn,t − ψRn,t.
- To ensure tractability, supply of n-period bonds for n>1 set to zero in equilibrium so one-period nominal bonds are the only savings instrument; detrending by Pt, π∗t and/or y∗t is applied.

### Empirical estimation of income and inflation processes
- Two specifications: stochastic trend model and trend-stationary model.
- Trend-stationary VAR(1) for Xt ≡ [log(yt), log(πt)]′: Xt+1 = A1 Xt + εX,t+1, εX,t+1 ∼ N(0, ΣX).
- Stochastic-trend VAR(1) for Xt ≡ [log(yt), log(πt), log(gt)]′: Xt+1 = A0 + A1 Xt + εX,t+1, εX,t+1 ∼ N(0, ΣX), with AR(1) for log(ξt): log(ξt) = ρξ log(ξt−1) + εξ,t, εξ,t ∼ iid N(0, σξ).
- Priors and identification:
  - Trend inflation AR(1) prior Beta[0.95, 0.03]; σξ / σπ = 0.01.
  - In stochastic trend model tight prior Beta[0.80, 0.05] for ρy.
  - Improper Uniform[-1,1] priors for ρyπ, ρπy, σπg, σπy (estimated by data).
- Data: US “PCE deflator (chain price index)” and “Personal Income”; UK “Long Term Price Indicator of Consumer Goods and Services” and “Real Household Disposable Income”; sample 1959Q2 - 2017Q4 for US and 1959Q2 - 2017Q4 for UK.

### Estimation results — core empirical findings
- Stochastic trend case (posterior results summarized in text):
  - Positive association over business cycle (Phillips curve): ρyπ > 0 and ρπy > 0; σπy > 0.
  - Inflation is negatively correlated with long-run income growth: σπg < 0.
  - Implication: inflation is positively correlated with the stationary income gap but negatively correlated with growth of long-run income.
- Trend-stationary case:
  - Lagged income gap positively affects inflation (ρpy > 0) while lagged inflation negatively affects income gap (ρyp < 0); contemporaneous shock correlation σπy < 0.
  - Despite σπy < 0, scatter plots still show a positive association between πt and yt−1 (a form of the Phillips curve).

### Quantitative simulation and calibration
- Markov approximation: VAR(1) approximated by first-order Markov chain over discretized grids Gy, Gπ, and Gg using Terry and Knotek II (2011) method; AR(1) ξt approximated by Tauchen’s method.
- Calibration highlights:
  - Preference parameters: β = 0.9985; σ = 2.0.
  - Steady-state bond holdings b∗ = 4.8.
  - Bond holding cost quadratic: Φ( b̃t Rt ) ≡ φb / 2 ( b̃t Rt − b∗ R∗ )2 ̃Rt with φb = 0.001.
  - Monetary policy: φr = 0.8; φπ = 2.0 for US and φπ = 1.0 for UK.
  - CRRA α chosen to fit average term premiums: α = 40 for US and α = 20 for UK in baseline fit.

### Main quantitative findings — equilibrium yield curve and drivers
- Stylized facts from data (1959Q1-2017Q1 for US and 1957Q1-2017Q1 for UK):
  - 10-year spread (relative to short rate) is 1.07% in US and 0.84% in UK.
  - Volatility of long-term rates slightly smaller but close to short-term rates.
- Model performance:
  - Stochastic trend baseline model successfully accounts for upward sloping yield curves and positive sizable term premiums in US and UK when calibrated to estimated income and inflation processes.
  - Trend-stationary model fails to generate sizable term premiums; yield curve too flat.
- Mechanism reconciling Phillips curve with positive term premiums:
  - Income decomposition is key: inflation positively correlated with cyclical income (yt) but negatively correlated with growth of long-run income (gt).
  - Under the permanent income hypothesis, consumption is mainly driven by long-run income, so consumption growth is negatively correlated with inflation, producing positive term premiums while preserving the Phillips curve for cyclical income.
- Composition of nominal term premiums:
  - A substantial part of nominal term premiums is accounted for by real term premiums, particularly in US.
  - Inflation term premiums do not play a dominant role for explaining the upward sloping yield curve in the baseline results.
- Monetary policy effects:
  - Real term premiums are driven by cov(1/rt+1, mt,t+1) < 0: aggressive monetary response to inflation raises real rates when inflation hikes occur and, combined with negative correlation of consumption growth and inflation, makes real bonds risky.
  - Comparative statics: increasing φπ steepens the equilibrium yield curve; φπ values illustrated include 1.0, 2.0, and 3.0.
  - Baseline φπ = 2.0 for US vs φπ = 1.0 for UK helps explain cross-country differences in term premiums and in volatility of interest rates relative to inflation.

### Term structure of interest rate volatility
- Empirical puzzle: observed volatility of long-term interest rates is relatively large compared to theoretical predictions.
- Model result: time-varying trend inflation π∗t is a key driver of the relatively large volatility of long-term interest rates.
  - When trend inflation is fixed, relative volatility of long-term rates would be much smaller.
  - Detrending series (HP filter) shows relative volatility of detrended long-term rates is much smaller and closer to the model without time-varying trend inflation.
- Implication: cointegration between interest rates and slow-moving trend inflation explains a large share of long-term rate volatility; term premium volatility need not be large to match observed long-term volatility.

### Macroeconomic moments for consumption growth
- Model-generated macro moments for quarterly, yearly, and 2-year consumption growth are broadly consistent with data, particularly for longer-term growth measures that matter for 5- and 10-year term premiums.

### Low-for-long counterfactual (permanently low growth and low inflation) and policy implications
- Counterfactual specification: set log(g∗) = 0 and log(π∗t) = 0; monetary policy with ZLB: Rt = max { 1.0, Rφr t−1 [ π∗t g∗ (Πt/π∗t) φπ ]1−φr } with g∗ = 1 and π∗t = 1 for all t.
- Simulation results:
  - Low-for-long economy associated with lower level of interest rates and significantly flatter yield curves: term spreads for 5-year and 10-year bonds are almost half in the low-for-long economy relative to baseline.
  - Decline in real term premiums accounts for almost all of the decline in nominal term premiums; inflation risk premiums are almost unchanged or slightly higher.
- Mechanism near ZLB:
  - Above ZLB, real bond returns increase with inflation due to active policy response, making real bonds poor hedges and producing positive real term premiums.
  - Near ZLB, central bank cannot raise nominal rates much in response to inflation, weakening positive correlation between real SDF and real interest rates and reducing real term premiums — long bonds become more like insurance in the low-for-long economy.
- Policy and financial stability implication:
  - Flatter yield curves in a low-for-long environment reduce banks’ maturity transformation profits, exacerbating already lower profitability from ZLB-era net interest margin compression and posing higher financial stability risk.

### Conclusion and avenues for future research
- Contributions:
  - Demonstrates that consumer optimization with income decomposed into cyclical and long-run components (estimated from US and UK data) can rationalize upward-sloping yield curves and positive term premiums while preserving the Phillips curve.
  - Shows that monetary policy response to inflation is critical in shaping yield curves and that a permanently low growth and low inflation environment flattens the yield curve via changes in policy behavior around the ZLB.
- Suggested future work:
  - Embed the estimated income and inflation relationships endogenously within a general equilibrium model to test robustness to structural change.
  - Apply the framework to dynamics of term premiums over time, predictability of excess bond returns, and responses around exits from low interest rate policies.

*Source: wp18242 - References ................................................................................ ................................................ 28; PDF: wp18242 - References ................................................................................ ............................; canonical URL: https://www.imf.org/-/media/files/publications/wp/2018/wp18242.pdf*

### References

### wp18242 - References (Appendices I–II: Detrending and Robustness)

### Appendix I — Detrending: method and reformulated equations
- Purpose: make the model stationary by detrending all non-stationary variables by P_t, π*_t, and/or y*_t and reformulating the budget constraint, monetary policy rule, and Euler equations using detrended variables.
- Detrended nominal bond holdings:
  - b_t = B_t/(P_t y*_t π*_t)
  - b_{n,t} = B_{n,t}/(P_t y*_t π*_t^n)
  - Note: nominal bond holdings are detrended by the price level, long-run income, and trend inflation because bond returns are cointegrated with those variables on the balanced growth path.
- Detrended nominal interest rate and n-period bond prices:
  - \tilde{R}_t = R_t/π*_t
  - \tilde{Q}_{n,t} = π*_t^n Q_{n,t}
  - Note: n-period bond holdings and prices are detrended by π*_t raised to maturity n.
- Detrended consumption:
  - c_t = C_t/y*_t
- Reformulated budget constraint (after dividing original budget constraint by P_t and y*_t):
  - c_t + b_t \tilde{R}_t + ∑_{n>1} \tilde{Q}_{n,t} b_{n,t} + Φ(b_t \tilde{R}_t) = 1 + b_{t−1} g_t π_t ξ_t + ∑_{n>1} \tilde{Q}_{n−1,t} b_{n,t−1} g_t π_t ξ_{t}^n
  - Labeled as equation (6) in the source.
- Reformulated monetary policy rule (dividing both sides by π*_t):
  - \tilde{R}_t = (\tilde{R}_{t−1} ξ_t)^{φ_r} [g^* π^{φ_π}_t]^{1−φ_r}
  - Presented as equation (7) in the source.
- Detrended nominal stochastic discount factor (SDF):
  - \tilde{M}_{t,t+1} ≡ π*_{t+1} M_{t,t+1}
  - = β Π_{t+1}/π*_{t+1} (C_{t+1}/C_t)^{−σ} [V_{t+1} / E_t(V_{t+1}^{1−α})^{1/(1−α)}]^{σ−α}
  - = β π_{t+1} (c_{t+1}/c_t)^{−σ} [v_{t+1} / E_t[(v_{t+1} g_{t+1})^{1−α}]^{1/(1−α)}]^{σ−α} g_{t+1}^{−α}
  - where c_t = C_t/y*_t and v_t = V_t/y*_t.
- Reformulated Euler equations using \tilde{M}_{t,t+1}:
  - \tilde{R}_t E_t[\tilde{M}_{t,t+1} ξ_{t+1}] = 1
  - E_t[\tilde{Q}_{n−1,t+1} \tilde{M}_{t,t+1} ξ^n_{t+1}] = \tilde{Q}_{n,t}, ∀ n > 1.
- Reformulated n-period bond prices for risk-neutral agents:
  - 1/\tilde{R}_t E_t[\tilde{\hat{Q}}_{n−1,t+1} ξ^{n−1}_{t+1}] = \tilde{\hat{Q}}_{n,t}, ∀ n > 1.

### Appendix II — Robustness checks: design and findings
- Four robustness checks conducted for the US equilibrium yield curve (marginal changes from baseline):
  1. Different values for the CRRA coefficient (tests sensitivity of equilibrium term premiums to risk aversion).
     - Figure 7 compares α = 20, α = 40, α = 150 (US case shown).
     - Finding: The size of term premiums is highly sensitive to the CRRA coefficient, but the qualitative result of positive and sizable term premiums is preserved for a relatively wide range of CRRA coefficients.
  2. Habit formation in household consumption.
     - Household value function changed to:
       - V_t = { (c_t − h c_{t−1})^{1−σ} + β E_t[V_{t+1}^{1−α}]^{1/(1−α)} }^{1/(1−σ)}
     - h is set to 0.25 based on previous literature.
     - Finding: Habit formation has very small effects on term premiums. Mechanism: households can smooth consumption in response to negative income shocks by adjusting labor supply and savings, leading to small habit effects (consistent with Rudebusch and Swanson (2008)).
  3. Fixed trend inflation (case without time-varying trend inflation).
     - Finding: Time-varying trend inflation barely affects the size of term premiums because trend inflation is very slow-moving with very small volatility, thus generating little risk for consumers over the business cycle.
  4. Monetary policy responds to income gap y_t in addition to inflation gap π_t.
     - Modified policy rule:
       - R_t = R_t^{φ_r} [π*_t g^* (Π_t/π*_t)^{φ_π} (Y_t/y*_t)^{φ_y}]^{1−φ_r}
     - Finding: When the central bank responds to the income gap, the size of term premiums is barely changed but becomes slightly higher than the baseline. Rationale: higher interest rates (lower bond prices) suppress consumption growth, reinforcing the negative correlation between bond prices and consumption growth.
- Overall robustness conclusion:
  - The model can replicate positive and sizable term premiums under the estimated inflation and income co-movement across robustness variations.
  - Specific qualitative and quantitative sensitivities:
    - Highly sensitive: CRRA coefficient (magnitude of term premiums).
    - Small/no effect: habit formation (h = 0.25 yields very small effect), time-varying vs fixed trend inflation (little effect), monetary policy responding to output gap (slightly higher term premiums).

### Key calibration and parameter values referenced in appendices and tables
- Table 2 calibration values:
  - Discount rate, β = 0.9985
  - Inverse of IES, σ = 2.0
  - Risk averseness, α = 40 for US and 20 for UK
  - Cost for bond holdings, φ_b = 0.001
  - Steady-state savings, b^* = 4.8
  - Response to inflation, φ_π = 2.0 for US and 1.0 for UK
  - Interest rate smoothing, φ_r = 0.8
- Habit formation parameter used in robustness check:
  - h = 0.25

*Source: wp18242 - References (Appendices I and II)*

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_Source: https://www.imf.org/-/media/files/publications/wp/2018/wp18242.pdf_