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### Introduction and paper objectives
- Context: Since the global financial crisis, major central banks undertook sizeable purchases of long-term bonds while expected short-term policy rates were constrained by the effective lower bound (ELB).
- Objective: Compare alternative withdrawal strategies of monetary stimulus in large advanced economies and analyze domestic propagation and international spillovers of:
  - (i) conventional interest rate tightening with gradual long-term portfolio runoff, and
  - (ii) quicker portfolio unwind (quantitative tightening, QT) accompanied by slower short-term policy rate increases.
- Empirical strategy: Use US data and high-frequency yield-curve-based proxies for shocks (Lewis (2023) building on Swanson (2021) and Gurkaynak et al. (2005)); exploit heteroscedasticity in intraday yield curve data on announcement days; estimate local projections à la Jorda (2005).
- Key conceptual model features to match empirical facts:
  - Two-country New Keynesian model with asset market segmentation (short- and long-term bonds imperfect substitutes; only long-term bonds internationally tradable).
  - Cognitive discounting (moderate degree per Gabaix (2020)).
  - Kimball quasi-kinked demand aggregators (Kimball, 1995) yielding a state-dependent Phillips curve.
- Main modelling implication: Cognitive discounting plus Kimball preferences imply QE becomes the only effective monetary instrument when the ELB is expected to bind for about three years and longer.
- Nonlinear/asymmetric implication: Asset sales need not mirror purchases; QE has stronger effects on domestic output than same-sized QT while producing broadly similar inflation effects.

### Empirical results and stylized facts
- Estimation framework:
  - Local projections: X_{t+h} − X_{t−1} = c_h + β_h ε_t + sum_{p=1}^P d_p Z_{t−p} + u_{t+h}, with var(u_{t+h}) = Ω_h.
  - Controls Z: lags of ΔX_t, industrial production growth, PCE inflation, stock returns, change in unemployment rate, the 10 year term premium, oil prices, OECD-wide industrial production and prices.
  - Forecast errors u_{t+h} assumed t-distributed with degrees of freedom ν and variance Ω_h.
  - Sample: monthly from 1996M1 to 2019M12; lag length P = 12; report quarterly-frequency results for DSGE comparison.
  - Bayesian estimation via Gibbs sampling; 10,000 iterations with 5,000 burn-in for robustness in Appendix C.
- Monetary policy shock identification:
  - Conventional policy shocks and LSAP shocks via Lewis (2023); similar results with Swanson (2021).
  - Lewis (2023): conventional shocks affect first two Fed Fund future contracts; LSAP shocks shift 5- or 10-year yields and move S&P 500 in opposite direction; forward guidance and information shocks included as controls.
- Key empirical findings:
  - Both one standard deviation contractionary conventional policy shocks and LSAP shocks trigger similar negative effects on PCE inflation and industrial production.
  - Only conventional policy significantly elevates short- and medium-term interest rates; LSAPs propagate via significant increases in ten year term premiums and depreciation of the dollar nominal effective exchange rate (NEER).
  - LSAPs significantly decrease the ten year term premium; conventional policy tightening does not compress the ten year term premium similarly.
  - Expansionary LSAP shocks depreciate the dollar NEER more than conventional interest rate shocks.
  - Normalization by impact on long-term yields:
    - QE (LSAP) has roughly 2.5 times smaller effects on domestic industrial production than conventional policy when normalized by long-rate impact.
    - Example: the conventional shock increases the 10 year rate around FOMC meetings by 0.1 SD units on impact while the unconventional shock raises this variable by 0.24 SD units on impact.
  - Cross-border spillovers (ten-year term premiums across 18 AEs and 14 EMEs):
    - US LSAPs compress 10-year term premiums in both advanced and emerging market economies.
    - Conventional US policy shocks have either insignificant or negative effects on foreign term premiums.
- Empirical stylized facts to match:
  - LSAP shocks propagate mainly through exchange rates and long-dated term premiums.
  - QE has roughly 2.5 times smaller effects on domestic industrial production than conventional policy when normalized by impact on long rates.
  - Central bank asset sales have considerably smaller effects on domestic demand than conventional policy shocks when normalized by impact on long-term nominal yields.

### Model structure, calibration, and mechanics
- Model overview:
  - Two-country DSGE nested on Kolasa and Wesołowski (2020); home economy relative size ω ∈ (0,1); foreign variables denoted with ∗.
  - Agents: two household types (restricted and unrestricted); final and intermediate goods producers; asset market segmentation across maturities and international holdings.
- Households and preferences:
  - Two types j ∈ {r, u}; share of restricted households ω_r ∈ (0,1).
  - Utility U^j_t = E^j_t sum_{s=0}^∞ β_j^s exp{ε^d_{t+s}} [ log(c^j_{t+s}) − (n^j_{t+s})^{1+φ}/(1+φ) ].
  - Cognitive discounting: perceived law X_{t+1} − X = m_j G_X(X^s_t − X^s, ε_{t+1}), with 0 ≤ m_j ≤ 1; m_j = 1 implies rational expectations.
- Asset structure and segmentation:
  - Long-term bonds as perpetuities with decay κ ∈ (0,1]; price P_{L−s,t} = κ^s P_{L,t}; R_{L,t} ≡ P_{L,t}^{−1} + κ.
  - Restricted households trade only long-term bonds; unrestricted households face transaction costs when trading long-term bonds.
- Government and central bank:
  - Fiscal constraint: B^f_{H,t} + P_{L,t} B^f_{H,L,t} + T_t + Φ^c_t = R_{t−1} B^f_{H,t−1} + P_{L,t} R_{L,t} B^f_{H,L,t−1} + P_t g_t.
  - Monetary policy: R_t = max{1, ˜R_t} with ˜R_t = R_{t−1}^{γ_r} [ R (π_t/π)^{γ_π} (y_t/y)^{γ_y} ]^{1−γ_r} exp{ε^r_t}.
  - Central bank LSAP rule: QE_t ≡ P_{L,t} B^c_{H,L,t} = B^c_{H,t} with QE_t ≡ [ 1 + (1−ρ) ( P_{L,t}/P_{L,t−1} R_{L,t} − 1 / P_{L,t−1} − 1 ) ] QE_{t−1} + ε^c_t.
- Asset market and UIP:
  - Long-UIP: E^u_t ˆR^1_{L,t+1} = E_t ˆR^{1,∗}_{L,t+1} + E_t ΔˆS_{t+1}.
  - Linearized arbitrage: E^u_t ˆR^1_{L,t+1} = ˆR_t + ζ_F ˆB^u_{L,t}.
- Calibration (selected exact values):
  - Small economy size ω = 0.01
  - Share of restricted households ω_r = 0.15
  - Inv. Frisch elasticity φ = 2
  - Discount factor, unrestricted β_u = 0.99875
  - Discount factor, restricted β_r = 0.9975
  - Cognitive discounting, unrestricted m_u = 0.95
  - Cognitive discounting, restricted m_r = 1
  - Steady-state inflation π = 1.005
  - Long-term bond duration D = 40
  - Transaction cost on long-term bonds ξ = 0.015
  - Price markup μ = 1.15
  - Kimball parameter ψ = -12
  - Calvo probability for domestic production θ_H, θ^⋆_F = 0.75
  - Calvo probabilities (AE, EME): θ^⋆_H = 0.667, 0.88; θ_F = 0.667, 0.5
  - Price indexation in home economy (AE, EME) ζ = 0, 0.75
  - Elast. of substitution ν = 0.8
  - Home-bias η = 0.75
  - Interest rate smoothing γ = 0.9
  - Interest rate response to inflation γ_π = 2
  - Interest rate response to output gap γ_y = 0.125
  - Reinvestment strategy ρ = 0.5525
  - Targeted steady-state ratios:
    - Government consumption to GDP g/y = 0.2
    - Public debt to GDP b^f/y = 2.5
    - Net foreign assets to GDP = 0
    - Long-term bonds to total bonds θ_L = 0.5
    - Central bank assets (QE) = 0 (steady state)
    - Home long-term bonds held by residents = 0.75 of B^f_{H,L}
- QE mechanics and runoff:
  - Passive reinvestment parameter ρ = 0.5525 governs reinvestment/runoff dynamics consistent with empirical term-premium and output responses.
  - Evolution of nominal QE portfolio: P^{⋆}_{L,t} B^{c,⋆}_{F,L,t} = [1 + (1−ρ) (−1/P^{⋆}_{L,t−1} + Π^{⋆}_{L,t} R^{⋆}_{L,t−1})] P^{⋆}_{L,t−1} B^{c,⋆}_{F,L,t−1} + ε^{c,⋆}_t.
  - Real value eroded by positive inflation even if nominal QE is held constant.

### Quantitative experiments: transmission and state dependence
- Relative potency (calibrated outcomes):
  - Conventional policy rate hikes lower output by about 1 percent, and inflation by 0.3 percent (baseline experiment).
  - Quantitative tightening: a 1.5 percent sell-off of central bank bond holdings affects output by an order of magnitude less than conventional rate hikes, though it raises the long-term nominal rate more persistently.
  - QE in a liquidity trap: large scale purchases totaling almost 20 percent of baseline GDP can close the output gap after three years and raise inflation notably.
- State dependence:
  - QE is substantially more effective when the policy rate is constrained by the ELB and the economy is in a recession.
  - Forward guidance is largely ineffective in a prolonged liquidity trap due to cognitive discounting (m_u = 0.95) and Kimball aggregation (ψ = −12).
- Term structure and exchange rate channels:
  - Transaction costs on long bonds create wedges that drive term premiums; increases in supply of long-term bonds (QT) raise expected returns and term premiums, reducing consumption of restricted agents.
  - Long-UIP calibrated implication: a 100bp hike in foreign long-term rates increases domestic same-maturity rates by about 35bp under a Taylor-type rule.
  - QT produces more persistent appreciation and larger medium-to-long-run effects on term premiums and the real exchange rate than equivalent conventional rate hikes.

### International spillovers: exit strategies and recipient responses
- Exit scenario setup:
  - After QE and forward guidance, a source economy faces unanticipated positive demand and adverse supply shocks, causing an inflation surge; central bank tightens via QT and/or short-term rate hikes.
- Two alternative exit strategies sized to generate the same source-country inflation and output deviations:
  - Strategy A: more rapid asset sell-off (QT) combined with a smaller short-term interest rate hike.
  - Strategy B: conventional tightening via short-term rate hikes while allowing QE portfolio to run down.
- Spillover patterns (calibrated):
  - Both strategies lower recipient-country output; short-term rate tightening is associated with smaller adverse output effects compared to QT, especially in emerging markets.
  - QT induces larger exchange rate depreciation and larger increases in term premiums in recipient economies than equivalent short-rate tightening, leading to larger output declines abroad.
  - Under EM calibration (θ^⋆_H = 0.88, θ_F = 0.5, ζ = 0.75), larger pass-through and sticky export prices amplify adverse spillovers from QT.
  - If recipient EMs peg NEER and defend the peg via interest-rate policy, QT in the source causes especially large contractions in the recipient because the recipient central bank must tighten aggressively to maintain the peg.
- Mechanism magnifiers:
  - 85 percent of consumers assumed financially unconstrained and respond more strongly to short-term policy rate changes than to term-premium changes.
  - QT leads to larger capital flows and larger depreciation pressures on recipient currencies, widening term-premium increases abroad.

### Policy implications and recommendations
- Domestic stabilization at the ELB:
  - QE is an effective tool to compress term-premiums and stimulate output when policy rates are at the ELB and inflation is well below target; it can be more potent than forward guidance.
- Monetary exit and global spillovers:
  - Prefer conventional short-term policy rate hikes over large asset sell-offs (QT) to achieve domestic disinflation with smaller international spillovers.
  - Short-rate tightening achieves similar domestic disinflation with smaller adverse effects on foreign output and term-premiums.
  - QT tends to cause larger capital flows, larger exchange rate depreciation in recipient countries, and stronger increases in foreign term-premiums.
- For recipient economies:
  - Flexible exchange rate and inflation-targeting frameworks reduce adverse spillovers from source-country QT relative to fixed or heavily managed exchange rate regimes.
  - Fixed NEER regimes (pegs) force recipient central banks into much larger interest-rate hikes to defend the peg, producing sharper contractions and persistent core inflation declines.

### Robustness, appendices, and additional experiments
- Robustness checks:
  - Alternative LSAP identifications (Swanson (2021), Jarociński (2024)) produce comparable IRFs (Appendix B figures).
  - Local projections with time-varying t errors and Bayesian priors detailed in Appendix C; Gibbs sampler settings and priors specified (e.g., ν_0 = 20, θ_0 = 0.0001, T_0 = 1).
- Behavioral and aggregation experiments (Appendix I):
  - Without cognitive discounting (rational agents) the economy is more resilient in output, requiring over 50% larger demand shocks to generate a 6% on-impact output contraction; forward guidance becomes almost as potent as QE in that rational case.
  - Kimball aggregation has a larger role than cognitive discounting in muting inflation declines relative to output declines; without Kimball aggregation, inflation falls markedly more for identical shocks.
- Technical appendices:
  - Appendix D: asset market segmentation, household adjustment-cost specifications, firm aggregators, Kimball aggregator form, government and central bank balance sheets.
  - Appendix E: closed-form term premium and duration derivations (D = R / (R − κ)).
  - Appendix F–H: QE mechanics, runoff/reinvestment, linear run-off AR(1) approximation.

*Source: wpiea2024073-print-pdf - Appendix I.*

### 1.    Introduction1

### 1. Introduction1

### Major themes and structure
- Overview chapter: "Introduction1"
- Sets stage for empirical analysis, model development, and policy analysis across subsequent sections

### 2. Empirical Results5

### Subsections and focus areas
- 2.1. Monetary Policy Shocks5
- 2.2. Estimation and Specification6
- 2.3. Transmission of Conventional Monetary Policy and LSAPs7

### Implied content scope
- Empirical identification of monetary policy shocks
- Estimation strategies and specification choices
- Empirical assessment of transmission channels for conventional monetary policy and large-scale asset purchases (LSAPs)

### 3. Model and Calibration9

### Subsections and focus areas
- 3.1. Households10
- 3.2. Firms12
- 3.3. Government14
- 3.4. Market Clearing Conditions15
- 3.5. Exogenous shocks15
- 3.6. Calibration15
- 3.7. Solution18

### Implied content scope
- Structural model components: household and firm behavior, government sector
- Market clearing and shock processes
- Calibration details and numerical solution methods

### 4. Transmission of Asset Purchases and Interest Rate Policy in the Model19

### Subsections and focus areas
- 4.1. Comparing the Effects of QT and Conventional Tightening19
- 4.2. Effects of Forward Guidance and QE in a Liquidity Trap22
- 4.3. Comparing the Effects of QE and QT24

### Implied content scope
- Comparative analysis of quantitative tightening (QT) versus conventional policy tightening
- Interaction of forward guidance and quantitative easing (QE) during liquidity trap episodes
- Direct comparison of QE and QT effects within the model

### 5. International Spillovers from Central Bank Exit Strategies26

### Subsections and focus areas
- 5.1. The Inflation Surge Scenario26
- 5.2. Spillovers from Alternative Exit Strategies28
- 5.3. The Role of Policy Conduct in Recipient Economies30

### Implied content scope
- Scenarios analyzing inflationary episodes and central bank exit strategies
- Cross-border spillovers under alternative exit paths
- Importance of recipient-economy policy responses

### 6. Conclusions32

### Implied content scope
- Synthesis of empirical findings and model results
- Policy implications regarding exit strategies, QE/QT, and international spillovers

### Supplementary Material

### Appendices and robustness material
- References34
- Appendix A. Additional Empirical Results38
- Appendix B. Robustness Checks39
- Appendix C. Estimation Details41
- Appendix D. The Role of Asset Market Segmentation42
- Appendix E. Term Premium and Duration43
- Appendix F. The Government and Central Bank Balance Sheets44
- Appendix G. Extended Model Equations49
- Appendix H. The Linear Run-Off Specification50

*Source: wpiea2024073-print-pdf - 1.    Introduction1*

### Appendix I.    The Impact of Cognitive Discounting and Kimball Preferences51

### Appendix I.    The Impact of Cognitive Discounting and Kimball Preferences51

### Introduction
- Context: Since the global financial crisis, central banks in major economies (US Federal Reserve, European Central Bank, Bank of Japan, Bank of England) undertook sizeable purchases of long-term bonds to provide stimulus and anchor inflation expectations; expected short-term policy rates were expected to be constrained by the effective lower bound (ELB) for an extended period.
- Objective: Compare alternative ways of withdrawing monetary stimulus in large advanced economies and analyze domestic propagation and international spillovers of (i) conventional interest rate tightening with gradual long-term portfolio runoff and (ii) quicker portfolio unwind (quantitative tightening, QT) accompanied by slower short-term policy rate increases.
- Empirical strategy: Use US data and high-frequency yield curve-based proxies for shocks constructed by Lewis (2023), building on Swanson (2021) and Gurkaynak et al. (2005); exploit heteroscedasticity in intraday yield curve data on monetary policy announcement days; estimate local projections à la Jorda (2005).
- Key empirical summary: One standard deviation contractionary shocks of both types (conventional policy rate and LSAPs) trigger similar negative effects on PCE inflation and industrial production; only conventional policy significantly elevates short and medium term interest rates, while LSAPs propagate via significant increases in ten year term premiums and depreciation of the dollar nominal effective exchange rate.
- Cross-border empirical pattern: US quantitative tightening tends to increase term premiums abroad (both in advanced and emerging market economies), while conventional policy rate shocks have either insignificant or negative effects on foreign term premiums.
- Model features to capture these facts:
  - Two-country New Keynesian model with asset market segmentation (Chen et al., 2012 style): short- and long-term bonds are imperfect substitutes due to portfolio transaction costs that depend on agent positions and central bank asset purchases; only long-term bonds are internationally tradable (ensures realistic exchange rate dynamics and international term premium comovement per Kolasa and Wesołowski (2020)).
  - Cognitive discounting: moderate degree per Gabaix (2020), reducing household sensitivity to future policy paths and mitigating the forward guidance puzzle.
  - Kimball quasi-kinked demand aggregators (Kimball, 1995) replacing standard log-linear Dixit-Stiglitz: induces state-dependent Phillips curve (flatter in slumps, steeper when inflation is elevated), attenuating the forward guidance puzzle in long-lived ELB episodes.
- Main modelling implication: Together, cognitive discounting and Kimball preferences imply QE becomes the only effective monetary instrument when the ELB is expected to constrain short-term rates for a prolonged period (e.g., for about three years and longer).
- Nonlinear and asymmetric implications:
  - Effects of large scale asset sales do not necessarily mirror those of asset purchases because they can occur along different segments of the Phillips curve and imply different short-term rate adjustments due to ELB asymmetry.
  - QE has stronger effects on domestic output than same-sized QT while producing broadly similar inflation effects; this is attributed to QE being deployed at the ELB when the Phillips curve is flat, and inflation responses being similar because the Phillips curve steepens during recoveries.
- International policy conclusion: Exit via conventional interest rate hikes is generally favored over asset sales because conventional tightening can achieve similar domestic objectives with smaller negative spillovers on other countries, especially emerging market economies; this aligns with the IMF’s Integrated Surveillance Decision (ISD) inclination toward alternatives with fewer negative spillovers.
- Paper roadmap: Section 2 (empirical framework and key results), Section 3 (model and calibration), Section 4 (transmission mechanisms), Section 5 (spillovers and robustness), Section 6 (conclusion).

### Empirical Results
- Estimation framework:
  - Local projections: X_{t+h} − X_{t−1} = c_h + β_h ε_t + sum_{p=1}^P d_p Z_{t−p} + u_{t+h}, with var(u_{t+h}) = Ω_h.
  - Controls Z include lags of the dependent variable ΔX_t, industrial production growth, PCE inflation, stock returns, change in the unemployment rate, the 10 year term premium, oil prices, and OECD-wide industrial production and prices.
  - Forecast errors u_{t+h} assumed t-distributed with degrees of freedom ν and variance Ω_h (equivalent to allowing heteroscedasticity).
  - Sample: monthly from 1996M1 to 2019M12; lag length P = 12; report quarterly-frequency results for comparison with DSGE model.
  - Bayesian estimation: Gibbs sampling to approximate posterior of impulse responses β_h; lag augmentation to handle LP residual autocorrelation; references to Montiel Olea and Plagborg-Møller (2021) and Cloyne et al. (2022) for coverage properties.
- Monetary policy shocks:
  - Focus on conventional policy rate shocks and unconventional LSAP shocks using Lewis (2023) proxies; similar results obtained with Swanson (2021) shocks.
  - Swanson (2021) and Gurkaynak et al. (2005) construct intraday yield-curve factors: conventional policy shock loads on the 3-month yield; LSAP shock restricted by pre-2009 small variance criterion.
  - Lewis (2023) relaxes time-constant loadings by exploiting heteroscedasticity on announcement days; labels shocks via historical decomposition: conventional shocks affect first two Fed Fund future contracts; LSAP shocks shift 5- or 10-year yields and move S&P 500 in opposite direction; Lewis also identifies forward guidance and an information shock (included as controls).
- Key empirical findings (Figure 2 and Figure 3 summaries):
  - One standard deviation expansionary shocks considered; comparisons of IRFs for conventional monetary policy shocks vs LSAP shocks.
  - Interest rates: Looser conventional monetary policy is estimated to slowly decrease the one-month interest rate; LSAPs show somewhat positive then insignificant short-rate responses.
  - Output and inflation: Both shocks are inflationary and increase industrial production (industrial production used as a monthly proxy for total output).
  - Term premiums and exchange rate:
    - LSAPs significantly decrease the ten year term premium; conventional policy tightening does not compress the ten year term premium in the same way.
    - Expansionary LSAP shocks depreciate the dollar nominal effective exchange rate (NEER), while conventional interest rate shocks have a considerably more modest effect on the exchange rate.
  - Normalization and relative potency:
    - If shocks are normalized by impact on long-term yields, LSAPs would be associated with a notably smaller impact on industrial production but a relatively larger propagation to the nominal exchange rate.
    - Empirical implication: quantitative easing in the United States has roughly 2.5 times smaller effects on domestic industrial production when normalized by the shocks’ impact on long rates.
    - Example statistic: the conventional shock increases the 10 year rate around FOMC meetings by 0.1 SD units on impact while the unconventional shock raises this variable by 0.24 SD units on impact.
  - International spillovers to 10-year term premiums (Figure 3):
    - Full sample: ten year term-premiums for 18 AEs and 14 EMEs (Adrian et al. (2013) methodology).
    - Subsample: inflation targeters with de facto freely floating exchange rates per IMF AREAERS (2022); AE average: Australia, Canada, Norway, Sweden, UK; EME subset: Chile, Mexico, Poland, Russia.
    - US LSAPs compress 10-year term premiums in both advanced and emerging market economies; conventional policy shows either insignificant or negative spillover effects on foreign term premiums.
- Empirical stylized facts to replicate in the DSGE model:
  - Large scale asset purchase shocks mainly propagate through exchange rates and long-dated term premiums.
  - QE has roughly 2.5 times smaller effects on domestic industrial production than conventional policy when normalized by impact on long rates.
  - Central bank asset sales have considerably smaller effects on domestic demand than conventional policy shocks when normalized by their impact on long-term nominal yields.

### Model and Calibration
- Model overview:
  - Two-country DSGE that nests Kolasa and Wesołowski (2020); focus on home economy with symmetric foreign counterpart (foreign variables denoted with an asterisk).
  - Global population normalized to unity; domestic economy relative size ω ∈ (0,1).
  - Agents: two household types (restricted and unrestricted); final and intermediate goods producers supplying domestic and foreign markets.
- Households:
  - Types indexed j ∈ {r, u}; share of restricted households ω_r ∈ (0,1).
  - Utility (type j): U^j_t = E^j_t sum_{s=0}^∞ β_j^s exp{ε^d_{t+s}} [ log(c^j_{t+s}) − (n^j_{t+s})^{1+φ}/(1+φ) ], where ε^d_t is preference shock, β_j ∈ (0,1) subjective discount factor, φ > 0 Frisch elasticity inverse, and E^j_t is subjective expectation operator of type j.
  - Cognitive discounting (Gabaix, 2020): households shrink expectations toward steady state. Perceived law of motion for any variable X_t taken as given by type j: X_{t+1} − X = m_j G_X(X^s_t − X^s, ε_{t+1}), where 0 ≤ m_j ≤ 1 is the cognitive discounting parameter; m_j = 1 implies rational expectations.
- Asset structure and segmentation:
  - Long-term bonds modeled as perpetuities with exponentially decaying coupon 1, κ, κ^2,... with κ ∈ (0,1]; price of a bond issued s periods ago P_{L−s,t} = κ^s P_{L,t}, so only contemporaneous long bond price P_{L,t} needs tracking.
  - Asset access and transaction costs summarized in Table 1 (restricted households trade only in long-term bonds; unrestricted households face transaction costs when trading long-term bonds; restricted investors face negligible costs) — table denotes units of short- or long-term bonds held and indicates transaction-cost-associated holdings via gray background.
  - Notation: R_{L,t} ≡ P_{L,t}^{−1} + κ denotes gross yield to maturity on domestic long-term bonds.
- Restricted households’ budget constraint (in nominal terms with prices P_t and nominal exchange S_t):
  - P_t c^r_t + [P_{L,t} B^r_{H,L,t}] + [1 + Γ^r_t] [S_t P^⋆_{L,t} B^r_{F,L,t}] + T^r_t
    = [P_{L,t} R_{L,t} B^r_{H,L,t−1}] + [S_t P^⋆_{L,t} R^⋆_{L,t} B^r_{F,L,t−1}] + W_t n^r_t + D^r_t + Ξ^r_t.
- Model calibration and additional details are introduced (model nests prior framework and introduces cognitive discounting and Kimball preferences to match empirical regularities); further calibration specifics and extensions are described in later sections of the paper.

*Appendix I. The Impact of Cognitive Discounting and Kimball Preferences51 — Source: IMF working paper (content unit: wpiea2024073-print-pdf - Appendix I.)*

### Appendix D.

### Appendix D (wpiea2024073-print-pdf - Appendix D)

### Model structure and key equations
- Households: two types — "restricted" and "unrestricted" — with home nominal exchange rate S_t and adjustment cost Γ^r_t specified by
  - 1 + Γ^r_t = exp( ξ^r [ S_t P^⋆_{L,t} B^r_{F,L,t}/(P_{L,t} B^r_{H,L,t}) − s P^⋆_{L} b^r_{F,L}/(P_{L} b^r_{H,L}) ] ), with ξ^r set positive but very small.
- Unrestricted agents face transaction costs ζ_{H,t} and ζ_{F,t} on short-term bond trading; bond holding costs are rebated lump-sum through Ξ^u_t.
- Firms:
  - Final goods aggregator: ˜y_t = [ η^{1/ν} y_{H,t}^{(ν−1)/ν} + (1−η)^{1/ν} y_{F,t}^{(ν−1)/ν} ]^{ν/(ν−1)} with η∈(0,1), ν>0.
  - Kimball aggregator G(x) = φ/(1+ψ) [ (1+ψ)x^{−ψ} ]^{1/φ} − φ/(1+ψ) + 1, with ψ≤0; steady-state markup μ = φ / [ (1−φ)(1+ψ) + φ ].
  - Intermediate firms: y_{H,t}(i)+y^⋆_{H,t}(i) = exp{ε^z_t} n_t(i) − f; Calvo stickiness with probabilities θ_H and θ^⋆_H and indexation parameter ζ.
- Government and central bank:
  - Fiscal constraint: B^f_{H,t} + P_{L,t} B^f_{H,L,t} + T_t + Φ^c_t = R_{t−1} B^f_{H,t−1} + P_{L,t} R_{L,t} B^f_{H,L,t−1} + P_t g_t.
  - Monetary policy: R_t = max{1, ˜R_t} with ˜R_t = R_{t−1}^{γ_r} [ R (π_t/π)^{γ_π} (y_t/y)^{γ_y} ]^{1−γ_r} exp{ε^r_t}.
  - Central bank LSAP (QE) size: QE_t ≡ P_{L,t} B^c_{H,L,t} = B^c_{H,t} with rule QE_t ≡ [ 1 + (1−ρ) ( P_{L,t}/P_{L,t−1} R_{L,t} − 1 / P_{L,t−1} − 1 ) ] QE_{t−1} + ε^c_t, with 0≤ρ<1.
- Market clearing and portfolio conditions:
  - Goods: ˜y_t = ω_r c^r_t + (1−ω_r) c^u_t + g_t.
  - Long-UIP (international arbitrage for long bonds): E^u_t ˆR^1_{L,t+1} = E_t ˆR^{1,∗}_{L,t+1} + E_t ΔˆS_{t+1}.
  - Linearized arbitrage linking expected returns: E^u_t ˆR^1_{L,t+1} = ˆR_t + ζ_F ˆB^u_{L,t}.

### Calibration and steady-state targets (selected exact values)
- Size of the small economy ω = 0.01
- Share of restricted households ω_r = 0.15
- Inv. Frisch elasticity of labor supply φ = 2
- Discount factor, unrestricted households β_u = 0.99875
- Discount factor, restricted households β_r = 0.9975
- Cognitive discounting, unrestricted households m_u = 0.95
- Cognitive discounting, restricted households m_r = 1
- Steady-state inflation π = 1.005
- Long-term bond duration D = 40
- Transaction cost on long-term bonds ξ = 0.015
- Price markup μ = 1.15
- Kimball parameter ψ = -12
- Calvo probability for domestic production θ_H, θ^⋆_F = 0.75
- Calvo probabilities (AE, EME): θ^⋆_H = 0.667, 0.88; θ_F = 0.667, 0.5
- Price indexation in home economy (AE, EME) ζ = 0, 0.75
- Elast. of substitution ν = 0.8
- Home-bias η = 0.75
- Interest rate smoothing γ = 0.9
- Interest rate response to inflation γ_π = 2
- Interest rate response to output gap γ_y = 0.125
- Reinvestment strategy ρ = 0.5525
- Targeted steady-state ratios:
  - Government consumption to GDP g/y = 0.2
  - Public debt to GDP b^f/y = 2.5
  - Net foreign assets to GDP = 0
  - Long-term bonds to total bonds θ_L = 0.5
  - Central bank assets (QE) = 0 (steady state)
  - Home long-term bonds held by residents = 0.75 of B^f_{H,L}

### Transmission: QE vs QT vs conventional rate policy (key quantitative findings)
- Relative potency:
  - Conventional policy rate hikes lower output by about 1 percent, and inflation by 0.3 percent (calibrated outcome under baseline experiments).
  - Quantitative tightening: a 1.5 percent sell-off of central bank bond holdings affects output by an order of magnitude less than conventional rate hikes, though it raises the long-term nominal rate more persistently.
  - QE during a liquidity trap (ELB bound): large scale purchases totaling almost 20 percent of baseline GDP can close the output gap after three years and raise inflation notably over the forecast horizon.
- State dependence:
  - QE is substantially more effective in stimulating output when the policy rate is constrained by the effective lower bound and the economy is in a recession.
  - Forward guidance is largely ineffective in a prolonged liquidity trap due to cognitive discounting (m_u = 0.95) and Kimball aggregation (ψ = −12) — the forward guidance puzzle is mitigated.
- Term structure and exchange rate channels:
  - Transaction costs on long bonds (ζ terms) create wedges that drive term premiums; increases in supply of long-term bonds (QT) raise expected returns and term premiums, reducing consumption of restricted agents.
  - Long-UIP (Equation (23)) implies that an increase in a country’s expected long-term bond return tends to appreciate its currency on impact; calibrated implication: a 100bp hike in foreign long-term rates increases domestic same-maturity rates by about 35bp under a Taylor-type rule.
  - QT produces a more persistent appreciation and larger medium-to-long-run effects on term premiums and the real exchange rate than equivalent conventional rate hikes.

### QE in a liquidity trap: mechanisms and calibrated responses
- Experiment design and magnitudes:
  - Adverse demand shocks drive the economy well below potential and keep inflation below target so the policy rate binds at the ELB for an extended period (about three years in the reported scenario).
  - Forward guidance to keep rates at ELB for almost 5 years has very small stimulative effects.
  - Adding QE of almost 20 percent of baseline GDP produces a notable term-premium compression and output stimulus; QE acts as a "policy lever of last resort."
- Role of Kimball aggregation:
  - State-dependent Phillips curve slope: low slope in downturns (so QE raises output more than inflation in a recession) and higher slope during recoveries (so similar policy moves affect inflation more outside recessions).

### International spillovers of exit strategies (quantified comparisons)
- Exit scenario: after QE and forward guidance, the large source economy experiences unanticipated positive demand and adverse supply shocks, causing an inflation surge; central bank tightens via combinations of QT and/or short-term rate hikes.
- Main calibrated comparisons and outcomes:
  - Two alternative exit strategies are sized to generate the same deviations of source-country inflation and output:
    - Strategy A: more rapid asset sell-off (QT) combined with a smaller short-term interest rate hike.
    - Strategy B: conventional tightening via short-term rate hikes while allowing QE portfolio to run down.
  - Spillover patterns:
    - Both exit strategies lower recipient-country output, but tightening via short-term policy rates is associated with smaller adverse output effects compared to QT, especially in emerging markets (EMs).
    - QT induces larger exchange rate depreciation and larger increases in term premiums in recipient economies than equivalent short-rate tightening, leading to larger output declines abroad.
    - Under EM calibration, larger pass-through and sticky export prices amplify adverse spillovers from QT (θ^⋆_H = 0.88, θ_F = 0.5, ζ = 0.75).
  - Policy-regime sensitivity:
    - If recipient EMs peg the nominal effective exchange rate (NEER) and defend the peg via interest-rate policy, QT in the source economy causes especially large contractions in the recipient, because the recipient central bank must tighten aggressively to maintain the peg.
- Quantified policy parameters reinforcing results:
  - 85 percent of consumers are assumed financially unconstrained and hence respond more strongly to short-term policy rate changes than to term-premium changes.
  - Reinvestment/runoff parameter ρ = 0.5525 implies central bank maturing asset reinvestment consistent with empirical response of term premiums and output to LSAP.

### Policy implications and recommendations (from model results)
- For domestic stabilization when the policy rate is at the ELB and inflation is well below target:
  - QE (large scale asset purchases) is an effective tool to compress term-premiums and stimulate output; it can be more potent than forward guidance.
- For monetary exit and global spillovers:
  - Prefer conventional short-term policy rate hikes over large asset sell-offs (QT) when the goal is to tighten policy with smaller international spillovers, because:
    - Short-rate tightening achieves similar domestic disinflation with smaller adverse effects on foreign output and term-premiums.
    - QT tends to cause larger capital flows, larger exchange rate depreciation in recipient countries, and stronger increases in foreign term-premiums.
- For recipient economies:
  - Flexible exchange rate and inflation-targeting frameworks reduce the adverse spillovers from source-country QT relative to fixed or heavily managed exchange rate regimes.
  - Fixed NEER regimes (pegs) force recipient central banks into much larger interest-rate hikes to defend the peg, producing sharper contractions and persistent core inflation declines.

*Source: Appendix D of wpiea2024073-print-pdf (https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024073-print-pdf.pdf).*

### References

### wpiea2024073-print-pdf - References

### Key cited works
- Lists foundational and recent contributions on monetary policy transmission, quantitative easing, spillovers, and empirical methods, including:
  - Adrian, T., R. K. Crump, and E. Moench (2013)
  - Adrian, T., C. Erceg, S. Gray, and R. Sahay (2021a); Adrian et al. (2021b)
  - Chen, H., V. Curdia, and A. Ferrero (2012)
  - Georgiadis, G. and M. Jarociński (2023)
  - Jarociński, M. (2024)
  - Swanson, E. T. (2021, 2023)
  - Kolasa, M., S. Ravgotra, and P. Zabczyk (2022)
  - Kolasa, M. and G. Wesołowski (2020, 2024)
  - Geweke (1993); Koop (2003)

### Appendix A — Empirical results (figures)
- Figure 10: Compares IRFs to a one standard deviation expansionary US conventional monetary policy shock and a large scale asset purchase (LSAP) shock across variables including S&P500, Unemployment Rate, 1 Year Rate. Panels show quarterly responses up to horizon "04812".
- Figure 11: Compares IRFs from a local projection with quadratic terms for asymmetric transmission to 1 SD expansionary (QE) and contractionary (QT) shocks for IP, PCE inflation, SP500, Unemployment rate, 10 yr TP. Panels use horizons denoted "04812".
- Note: Figures report median and 68% error bands for shocks.

### Appendix B — Robustness checks (identifications)
- Figures 12–15 reproduce IRFs from main figures using alternative identifications:
  - Swanson (2021) identification for LSAP shock (Figures 12 and 14).
  - Jarociński (2024) identification for LSAP shock (Figures 13 and 15).
- Panels include variables: 1 month rate, PCE inflation, IP, 10 yr TP, NEER, and separate panels for Advanced Economies and Emerging Market Economies. Horizontal axes labeled "04812"; vertical scales indicated explicitly (e.g., -0.35, -0.175, 0, 0.175, 0.35).

### Appendix C — Estimation details (local projections with time-varying t errors)
- Empirical model specification:
  - Equation A.1: X_{t+h} − X_{t−1} = c_h + β_h ε_t + Σ_{p=1}^P d_p Z_{t−p} + u_{t+h}
  - Var(u_{t+h}) = Ω_h; Ω_h = σ^2 λ_t
  - 1/λ_t denotes time-varying volatility; residuals follow Student’s t with degrees of freedom ν under Gamma prior for λ_t (Geweke (1993)).
- Priors:
  - Hierarchical prior for λ_t: P(λ_t) = Γ(1,ν) (A.4)
  - Prior for ν: P(ν) = Γ(ν_0,2) with ν_0 = 20 (A.5)
  - Prior for regression coefficients B_h: P(B) ∼ N(B_0, Σ_{B0}) with B_0 = 0 and Σ_{B0} diagonal elements = 10
  - Prior for σ^2: IG(θ_0, T_0) with θ_0 = 0.0001 and T_0 = 1
- Gibbs sampler:
  - Posterior approximated using 10,000 iterations with a burn-in of 5,000.
  - Conditional draws:
    - G(λ_t|Ξ): Gamma with mean (ν+1)/(1/σ u_t^2 + ν) and degrees of freedom ν+1.
    - G(ν|Ξ): Non-standard; sampled via Random Walk Metropolis Hastings with proposal ν_new = ν_old + g^{1/2} ε, ε ∼ N(0,1); acceptance probability G(ν_new|Ξ)/G(ν_old|Ξ); g chosen to keep acceptance rate around 40%.
    - G(σ^2|Ξ): Inverse Gamma with scale parameter e′ e + θ_0 and degrees of freedom T + T_0 after transforming residuals e_t = u_t × λ_t^{0.5}.
    - G(B_h|Ξ): Normal after GLS transformation; variance V = (Σ_{B0}^{−1} + x′ x)^{−1}; mean = V (Σ_{B0}^{−1} B_0 + x′ y).

### Appendix D — Asset market segmentation
- Two dimensions of segmentation:
  - Across maturities: Restricted households cannot trade short-term bonds; unrestricted ones face portfolio adjustment costs. Limits arbitrage between short- and long-term bonds, producing fluctuations in term premiums that affect real activity (references: Chen et al. (2012), Kolasa and Wesołowski (2020)).
  - International: Unrestricted households cannot trade short-term bonds issued abroad; motivated by IMF Coordinated Portfolio Investment Survey evidence that short-term debt securities have a fairly stable market share typically oscillating below 5%, with foreign debt assets held by small open economies even less quantitatively relevant. International segmentation helps the model avoid counterfactual surges into small economies and generate realistic capital flow magnitudes.

### Appendix E — Term premium and duration (closed-form derivation)
- Term premium definition:
  - TP_t = R_{L,t} − R^EH_{L,t}, where R^EH_{L,t} is the counterfactual yield to maturity in absence of transaction costs.
- Duration definition and derivation:
  - D ≡ Σ_{s=1}^{+∞} κ^{s−1} s / R_s P (where for a consol R − κ = 1/P), manipulation yields:
  - Let x ≡ κ / R, then D = R / (R − κ).
  - Final formula used in paper: D ≡ R / (R − κ).

### Appendix F — Government and central bank balance sheets; QE mechanics
- Notation distinguishes fiscal issuance (superscript f), central bank issuance (superscript c), consolidated government (superscript g).
- Central bank (CB) balance sheet operational assumptions for large source economy:
  - CB holds short-term positions ̃B^{c,⋆}_{F,t} and long-term B^{c,⋆}_{F,L,t}, financed by issuing central bank commercial paper ̃B^{cp,⋆}_{F,t}, which is 100% of CB liabilities and indistinguishable from short-term government bonds.
  - Define net short-term CB position B^{c,⋆}_{F,t} ≡ ̃B^{c,⋆}_{F,t} − ̃B^{cp,⋆}_{F,t}.
  - Consolidated simplified CB balance sheet assets P^{⋆}_{L,t} B^{c,⋆}_{F,L,t} and liabilities B^{c,⋆}_{F,t}.
  - Simplified government debt definitions: B^{g,⋆}_{F,t} ≡ B^{f,⋆}_{F,t} − B^{c,⋆}_{F,t}; B^{g,⋆}_{F,L,t} ≡ B^{f,⋆}_{F,L,t} − B^{c,⋆}_{F,L,t}.
  - Size of QE is P^{⋆}_{L,t} B^{c,⋆}_{F,L,t}.
- Holding period profits of QE:
  - Φ^{c,⋆}_t ≡ R^{⋆}_{t−1} B^{c,⋆}_{F,t−1} + P^{⋆}_{L,t} R^{⋆}_{L,t} B^{c,⋆}_{F,L,t−1}.
- CB budget constraint with full rebating of carry costs and losses:
  - P^{⋆}_{L,t} B^{c,⋆}_{F,L,t} + B^{c,⋆}_{F,t} = R^{⋆}_{t−1} B^{c,⋆}_{F,t−1} + P^{⋆}_{L,t} R^{⋆}_{L,t} B^{c,⋆}_{F,L,t−1} − Φ^{c,⋆}_t
  - Alternatively simplified to P^{⋆}_{L,t} B^{c,⋆}_{F,L,t} + B^{c,⋆}_{F,t} = 0 under the “clean” balance sheet assumption each period.

### Runoff, revaluation, and reinvestment mechanics
- Mechanical change in long-term asset value when CB purchased P^{⋆}_{L,t−1} B^{c,⋆}_{F,L,t−1}:
  - Coupon worth B^{c,⋆}_{F,L,t−1} and market value κ P^{⋆}_{L,t} B^{c,⋆}_{F,L,t−1}.
  - Change Ψ_t ≡ κ^{⋆} P^{⋆}_{L,t} B^{c,⋆}_{F,L,t−1} − P^{⋆}_{L,t−1} B^{c,⋆}_{F,L,t−1}.
  - Decomposed into runoff (negative) and revaluation: Ψ_t = −B^{c,⋆}_{F,L,t−1} (runoff) + [κ^{⋆} P^{⋆}_{L,t} B^{c,⋆}_{F,L,t−1} − P^{⋆}_{L,t−1} B^{c,⋆}_{F,L,t−1} + B^{c,⋆}_{F,L,t−1}] (revaluation).
  - Using R^{⋆}_{L,t} ≡ (P^{⋆}_{L,t})^{−1} + κ^{⋆}, revaluation simplifies to (P^{⋆}_{L,t} R^{⋆}_{L,t} − P^{⋆}_{L,t−1}) B^{c,⋆}_{F,L,t−1} = (Π^{⋆}_{L,t} R^{⋆}_{L,t} − 1) P^{⋆}_{L,t−1} B^{c,⋆}_{F,L,t−1}.
- Passive reinvestment parameter ρ and total reinvestment Θ_t:
  - Θ_t ≡ ρ [1/P^{⋆}_{L,t−1} − Π^{⋆}_{L,t} R^{⋆}_{L,t} + 1] P^{⋆}_{L,t−1} B^{c,⋆}_{F,L,t−1} + ε^{c,⋆}_t (ε^{c,⋆}_t denotes active reinvestment).
- Evolution of nominal QE portfolio:
  - P^{⋆}_{L,t} B^{c,⋆}_{F,L,t} = [1 + (1−ρ) (−1/P^{⋆}_{L,t−1} + Π^{⋆}_{L,t} R^{⋆}_{L,t−1})] P^{⋆}_{L,t−1} B^{c,⋆}_{F,L,t−1} + ε^{c,⋆}_t.
  - If ρ = 1 and ε^{c,⋆}_t = 0, nominal value stays constant.
- Real equivalent after dividing by P^{⋆}_t:
  - P^{⋆}_{L,t} b^{c,⋆}_{F,L,t} = [1 + (1−ρ)(−1/P^{⋆}_{L,t−1} + Π^{⋆}_{L,t} R^{⋆}_{L,t−1})] Π^{⋆}_t P^{⋆}_{L,t−1} b^{c,⋆}_{F,L,t−1} + ε^{c,⋆}_t / P^{⋆}_t.
  - Positive inflation erodes real value of CB portfolio even if nominal value is held fixed.

### Fiscal policy constraint and steady-state assumptions
- Fiscal authority constraint:
  - B^{f}_{F,t} + P^{⋆}_{L,t} B^{f}_{F,L,t} + T^{⋆}_t + Φ^{c,⋆}_t = R^{⋆}_{t−1} B^{f}_{F,t−1} + P^{⋆}_{L,t} R^{⋆}_{L,t} B^{f}_{F,L,t−1} + P^{⋆}_t g^{⋆}_t
  - Market value of outstanding government debt B^{f}_t ≡ B^{f}_{F,t} + P^{⋆}_{L,t} B^{f}_{F,L,t}
- Baseline fiscal assumption: large-country fiscal authority keeps real market value of debt b^{f}_{F,t} ≡ B^{f}_{F,t} / P^{⋆}_t and composition θ_L constant.
- Government expenditures assumed g_t ≡ g exp{ε^{g}_t}, taxes uniform per capita; transfers and CB profits Φ^{c,⋆}_t rebated to treasury.

### Appendix G — Extended model equations (real variables and balances)
- Introduces real short- and long-term fiscal and central bank positions b^{f,⋆}_{F,t}, b^{f,⋆}_{F,L,t}, b^{c,⋆}_{F,t}, b^{c,⋆}_{F,L,t}.
- Fiscal-side steady relations:
  - b^{f,⋆}_{F,L,t} R^{⋆}_{L,t} − κ^{⋆} b^{f,⋆}_{F,t} + b^{f,⋆}_{F,L,t} R^{⋆}_{L,t} − κ^{⋆} = b^{f,⋆}_{F,L} R^{⋆}_{L} − κ^{⋆} b^{f,⋆}_{F} + b^{f,⋆}_{F,L} R^{⋆}_{L} − κ^{⋆}
  - b^{f,⋆}_{F,t} + b^{f,⋆}_{F,L,t} R^{⋆}_{L,t} − κ^{⋆} = b^{f,⋆}_{F} + b^{f,⋆}_{F,L} R^{⋆}_{L} − κ^{⋆}
- Central bank real dynamics:
  - Self-financing constraint: P^{⋆}_{L,t} B^{c,⋆}_{F,L,t} + B^{c,⋆}_{F,t} = 0 ⇐⇒ b^{c,⋆}_{F,L,t} R^{⋆}_{L,t} − κ^{⋆} = − b^{c,⋆}_{F,t}
  - Real nonlinear QT specification:
    - P^{⋆}_{L,t} b^{c,⋆}_{F,L,t} = [1 + (1−ρ) (−1/P^{⋆}_{L,t−1} + Π^{⋆}_{L,t} R^{⋆}_{L,t−1})] Π^{⋆}_t P^{⋆}_{L,t−1} b^{c,⋆}_{F,L,t−1} + ε^{c,⋆}_t / P^{⋆}_t
    - Equivalently b^{c,⋆}_{F,L,t} R^{⋆}_{L,t} − κ^{⋆} = [1 + (1−ρ) (κ^{⋆} − R^{⋆}_{L,t−1} + (R^{⋆}_{L,t−1} − κ^{⋆})/(R^{⋆}_{L,t} − κ^{⋆}) ) R^{⋆}_{L,t−1}] Π^{⋆}_t b^{c,⋆}_{F,L,t−1} R^{⋆}_{L,t−1} − κ^{⋆} + ε^{c,⋆}_t.

### Appendix H — Linear run-off specification (AR(1) approximation)
- Define CB^{⋆}_t as size of long-term bond position:
  - CB^{⋆}_t = (1 − (1−ρ) (φ_t + φ_t)) Π^{⋆}_t CB^{⋆}_{t−1} + ε^{c,⋆}_t, where φ_t = runoff, φ_t = revaluation.
- Linearizing around steady state with no QE yields AR(1):
  - CB^{⋆}_t ≈ (1 − (1−ρ) (φ + φ)) Π^{⋆} CB^{⋆}_{t−1} + ε^{c,⋆}_t.
- Runoff and revaluation expressed in structural parameters:
  - φ ≡ −1/P^{⋆}_{L} − κ^{⋆} + κ^{⋆} = − R^{⋆}_{L} / DUR^{⋆}_{L}
  - φ ≡ R^{⋆}_{L} / Π^{⋆}_{L}
  - DUR^{⋆}_L denotes consol duration; runoff is pinned down by long-bond duration.

### Appendix I — Impact of cognitive discounting and Kimball preferences (model experiments)
- Figure 16: Simulations with no cognitive discounting (rational agents) comparing effects of Forward Guidance (FG) and QE:
  - Economy without cognitive discounting is more resilient in output: over 50% larger demand shocks required to generate a 6% on-impact output contraction, but accompanied by considerably larger disinflation.
  - Forward guidance in this rational simulation is almost as potent as large scale asset purchases, even when the central bank promises to keep rates lower for six rather than seven periods.
  - Forward guidance has much smaller impact on term premiums, the long-term rate, and exchange rate compared to QE.
- Figure 17: Role of Kimball aggregation:
  - Kimball aggregation plays a larger role than cognitive discounting in muting inflation declines relative to output declines.
  - Without Kimball aggregation, the fall in inflation would be markedly larger for identical underlying shocks.
- Notes:
  - Forward guidance implemented endogenously by targeting the shadow interest rate; becomes strictly positive slightly quicker under rational expectations than under cognitive discounting.
  - Reference to Kolasa et al. (2022) for comprehensive discussion of behavioral discounting in open economy.

*Central Bank Exit Strategies: Domestic Transmission and International Spillovers — Working Paper No. WP/2024/073*

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_Source: https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024073-print-pdf.pdf_