## wpiea2019240-print-pdf

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---

### Model purpose and core mechanism
- Examines how oil price movements affect sovereign spreads in an oil-dependent emerging market economy using a stochastic general equilibrium model with sovereign debt and endogenous default.
- Government chooses consumption and whether to default; default carries costs and may lead to temporary autarky with exogenous re-entry probability 휃.
- International investors are risk-neutral and price one-period bonds based on expected default probability, yielding bond price schedule 푞푡(·) = (1 − E{훿(·)})/(1 + 푟푡).
- Core mechanism: higher oil prices raise government revenues and output co-moving with oil, reducing default likelihood and sovereign spreads; lower oil prices do the reverse.

### Empirical motivation and stylized facts
- Sample: six resource-rich economies with quarterly data 1995q1–2016q3: Angola, Colombia, Indonesia, Mexico, Nigeria, Russia.
- Observed correlation between oil prices and mean interest rate spreads across the six economies: correlation coefficient = - 0.36 (statistically significant at better than the 1 percent confidence level).
- Individual country correlations (spreads vs oil price):
  - Angola = -0.6
  - Colombia = -0.63
  - Indonesia = -0.03
  - Mexico = -0.09
  - Nigeria = -0.28
  - Russia = -0.56
- Real crude oil price in sample: low ≈ $US 25 in 2001q4; peak = $US 124 in 2008q2.
- Mean correlation between real GDP and oil price across the six economies = 0.2.
- Individual country correlations (GDP vs oil price):
  - Angola = 0.76
  - Colombia = 0.41
  - Indonesia = -0.06
  - Mexico = 0.14
  - Nigeria = 0.26
  - Russia = 0.29
- Data construction: nominal GDP deflated by national GDP deflator; oil price deflated by US CPI; HP filter smoothing parameter = 1600 for cyclical components.

### Stochastic processes and calibrated parameters
- Non-oil output process:
  - 푦푡 = 휇y + zt, zt = 휌z zt−1 + εt z, εt z ~ N(0, 휎z2).
  - Quarterly output persistence = 0.841.
  - Output shock standard deviation = 0.015.
  - Output shock half-life = 4.1 quarters.
- Oil price process:
  - 푝푡 = 휇p + zht, zht = 휌h zht−1 + εt h, εt h ~ N(0, 휎h2).
  - Quarterly oil price persistence = 0.725.
  - Oil price shock standard deviation = 0.136.
  - Oil price shock half-life = 2.2 quarters.
- Transition probabilities: Tauchen (1986) discretization for joint (output, oil price) shocks; estimated mean cross-correlation between output and oil price across countries = 휌yp = 0.05.
- Key calibrated numeric parameters:
  - 휌 = .025 (Discount rate)
  - 훽 = .976 (Discount factor)
  - 휎 = 70 (Coefficient of relative risk aversion)
  - 휉 = .50 (Inter-temporal elasticity of substitution)
  - 휃 = .25 (Probability of re-entry into capital market)
  - 휓 = .95 (1 − output cost of default)
  - 푠̄ = .132 (Oil revenue share)
  - 푟∗ = .004 (Risk free interest rate)
  - r̂ = .016 (Risk premium (exogenous)), with r = r∗ + r̂

### Default mechanism, costs, and re-entry
- Default truncates output and oil revenues by factor governed by 휓 ∈ (0,1):
  - Under default, yD t = { y t if y t < ȳ ; 휓 ȳ if y t > ȳ } and similarly for oil revenues with p̄ and s̄.
  - 휓 captures proportion retained (1 – output cost of default) and is set to .95 in calibration.
- Probability of re-entry following default per quarter: 휃 = .25.
  - Monte Carlo simulation over 1000 runs yields average exclusion duration ≈ just over 2 years given 휃 = .25.
  - Literature note: referenced findings report longer typical exclusion (example: 5.7 years in Richmond and Dias (2009)).

### Oil dependence calibration (data-based measures and country shares)
- Two sample averages (1995q1–2016q4):
  - Average share of oil and gas rents in GDP across the sample = 6% (World Bank WDI).
  - Average share of petroleum in total exports across the sample = 35% (UN Comtrade).
- Country-specific oil shares (%) presented:
  - Angola: GDP share 39.9; Export share 59.5
  - Colombia: GDP share 4.1; Export share 24.5
  - Indonesia: GDP share 2.9; Export share 9.7
  - Mexico: GDP share 3.7; Export share 13.0
  - Nigeria: GDP share 18.6; Export share 57.3
  - Russia: GDP share 10.1; Export share 32.2
  - All (sample averages): GDP share 13.2; Export share 32.7
- Calibrated oil revenue share in the model: 푠̄ = .132.

### Numerical solution and bond pricing
- Numerical methods: value function iteration; state-space discretization for debt, output, and oil revenues; Markov chain for stochastic processes; solution via guess-and-verify for bond price schedule until convergence.
- Discretization used in numerical solution:
  - 11 output states
  - 11 oil price states
  - 125 debt states
- Steady-state/analytic Ramsey bond price expression (simplified case): q = 1/R = (1−λ)/R(δ′(λ−d) − δ+1) + 1/(d δ′ R) (equation 16) — interpretation emphasized non-linearity: pricing depends on fixed default cost λ, interest rate R, debt d, default probability δ and sensitivity δ′.
- Equilibrium bond pricing schedules (numerical):
  - Bond price increases in foreign assets (i.e., less negative debt → higher q).
  - Pricing schedule for ‘high’ output lies above that for ‘low’ output.
  - Low oil prices shift pricing schedules downward (lower oil revenue → higher default likelihood → lower bond price).
  - Comparison to model without oil: bond pricing schedule exhibits larger steps without oil; turning off oil revenues substantially worsens predictive power for spreads.

### Model performance against data moments
- Simulation details:
  - Simulated over 1000 periods; data period 1995q1 to 2016q4; HP filter smoothing parameter = 1600.
- Key calibrated moments and shock statistics:
  - Persistence of productivity: .841
  - Persistence of oil price: .725
  - Std dev. of output shocks: .015
  - Std dev. of oil price shocks: .136
  - Cross correlation: .045
- Table 3a — AR(1) and Std. Dev. (Data vs Model)
  - Output:
    - Data AR(1): 0.841
    - Data Std. Dev.: 0.032
    - Model AR(1): 0.893
    - Model Std. Dev.: 0.043
  - Oil price:
    - Data AR(1): 0.725
    - Data Std. Dev.: 0.198
    - Model AR(1): 0.657
    - Model Std. Dev.: 0.195
  - Consumption:
    - Data AR(1): 0.898
    - Data Std. Dev.: 0.026
    - Model AR(1): 0.977
    - Model Std. Dev.: 0.107
  - Spread:
    - Data AR(1): 0.645
    - Data Std. Dev.: 0.022
    - Model AR(1): 0.304
    - Model Std. Dev.: 0.010
- Table 3b — Cross-correlations between sovereign spreads and shocks (Data vs Model)
  - Output dev:
    - Data: -0.138
    - Model: -0.148
  - Oil price dev:
    - Data: -0.279
    - Model: -0.160
- Model evaluation summary:
  - Calibration reproduces ordering of autoregressive properties.
  - Model underpredicts persistence and variability of the interest rate spread (by about a half) and overestimates volatility of consumption.
  - Model successfully generates negative correlation between spreads and the oil price and produces a higher correlation between spreads and oil price than between spreads and output, consistent with data for oil-dependent emerging markets.
  - Model predicts default (or zero bond price) at relatively low debt-GDP ratios (sometimes below 10 percent) in adverse oil and output states; this may diverge from observed cases with debt ratios up to 75 percent without default.

### Sensitivity analysis (high-level findings)
- Parameters varied: rate of time preference 휌, oil share s̅, covariance of shocks 휌yp, risk-aversion 휎, inter-temporal substitutability 휉.
- Oil dependence and impatience:
  - Parameter ranges explored:
    - Rate of time preference: 휌 = [0.01,0.05] per quarter.
    - Oil share parameter: s̅ = [0.02,0.6].
  - Effects:
    - Higher impatience reduces persistence of interest rate spread but increases interest rate spread (via higher default probability).
    - Higher oil share increases spread persistence.
    - Both higher impatience and higher oil dependency strengthen negative correlation between interest rate spreads and oil price movements.
  - Counterfactual: turning off oil revenues dramatically reduces spread persistence and the countercyclical correlation with oil price.
- Covarying shocks:
  - Central estimate for covariance parameter: 휌yp = 0.045; explored range 0.01 to 0.10.
  - Findings:
    - Auto-regressive properties of interest rate spread follow an inverse-U with respect to shock correlation: reduced persistence for very low and very high correlations.
    - Mechanisms: higher correlation increases default risk but also induces sovereigns to reduce indebtedness, producing offsetting effects.
- Preferences:
  - Parameter ranges: 휎 ∈ [2,100]; 휉 ∈ [0.1,0.5].
  - Findings:
    - Persistence of interest rate spread relatively unaffected by changes in either parameter.
    - Higher IES (larger 휉) → greater negative correlation between spreads and movements in the oil price.
    - Mechanism: higher IES → consumption more volatile and debt path more stable, affecting default risk dynamics.

### Policy-relevant implications and recommendations
- The calibrated model explains a large proportion of countercyclical movements in interest-rate spreads in oil-dependent emerging markets via the optimal default channel.
- Policy recommendations implied:
  - Save out of oil revenues to manage volatility in borrowing costs due to sovereign default risk; build up liquid buffers and/or pay down debt to insulate the economy from commodity-driven movements in risk premiums and reduce “stop-go” cycles of investment and growth.
  - Buffers should be larger the stronger the link between oil prices and the domestic economy, the more impatient policy makers are, and the more willing they are to substitute current for future consumption.
  - Continue focus on economic diversification to reduce co-movements of spreads with oil prices and increase resilience to commodity price shocks.
  - Recognize that prolonged periods of substantial interest rate premiums for oil-dependent governments can lower investment and capital stocks and hinder growth.
  - Suggested model extension: incorporate domestic physical capital to analyze trade-offs between precautionary savings and domestic investment under endogenous interest rate premiums.

*Source: IMF Working Paper (section 2, Calibration) — wpiea2019240-print-pdf*

### 2. Output and the oil price _______________________________________________________________________ 11

### 2. Output and the oil price

### Section headings and page references
- 2. Output and the oil price — 11
- 3. Bond pricing schedule — 18
- A1. Dependence and impatience — 28
- A2. Covarying shocks — 29
- A3. Preferences — 30

### Tables
- 1. Oil shares — 14

*https://www.imf.org/-/media/files/publications/wp/2019/wpiea2019240-print-pdf.pdf*

### 2. Calibration  ____________________________________________________________________________________ 16

### 2. Calibration

### Model purpose and key mechanism
- Examines how oil price movements affect sovereign spreads in an oil-dependent emerging market economy using a stochastic general equilibrium model with sovereign debt and endogenous default.
- Government chooses consumption and whether to default; default carries costs and may lead to temporary autarky with exogenous re-entry probability 휃.
- International investors are risk-neutral and price one-period bonds based on expected default probability, yielding bond price schedule 푞푡(·) = (1 − E{훿(·)})/(1 + 푟푡).
- Core mechanism: higher oil prices raise government revenues and output co-moving with oil, reducing default likelihood and sovereign spreads; lower oil prices do the reverse.

### Empirical motivation and stylized facts preserved from the data
- Sample: six resource-rich economies with quarterly data 1995q1–2016q3: Angola, Colombia, Indonesia, Mexico, Nigeria, Russia.
- Observed negative correlation between oil prices and mean interest rate spreads across the six economies: correlation coefficient = - 0.36 (statistically significant at better than the 1 percent confidence level).
- Individual country correlations (spreads vs oil price): Angola = -0.6, Colombia = -0.63, Indonesia = -0.03, Mexico = -0.09, Nigeria = -0.28, Russia = -0.56.
- Real crude oil price ranged between a low of around $US 25 in 2001q4 and a peak of $US 124 in 2008q2 in the sample period.
- Mean correlation between real GDP and oil price across the six economies = 0.2.
- Individual country correlations (GDP vs oil price): Angola = 0.76, Colombia = 0.41, Indonesia = -0.06, Mexico = 0.14, Nigeria = 0.26, Russia = 0.29.
- Sample construction notes: nominal GDP deflated by national GDP deflator; oil price deflated by US CPI; HP filter with smoothing parameter = 1600 applied to obtain cyclical components.

### Stochastic processes and solution features (as calibrated)
- Non-oil output process:
  - 푦푡 = 휇y + zt, zt = 휌z zt−1 + εt z, εt z ~ N(0, 휎z2).
  - Quarterly output persistence = 0.841; output shock standard deviation = 0.015.
  - Output shock half-life = 4.1 quarters.
- Oil price process:
  - 푝푡 = 휇p + zht, zht = 휌h zht−1 + εt h, εt h ~ N(0, 휎h2).
  - Quarterly oil price persistence = 0.725; oil price shock standard deviation = 0.136.
  - Oil price shock half-life = 2.2 quarters.
- Transition probabilities for joint (output, oil price) shocks constructed using Tauchen (1986) discretization, allowing covariance between processes; estimated mean cross-correlation between output and oil price across countries = 휌yp = 0.05.

### Default, costs, and re-entry
- Default truncates output and oil revenues by factor governed by parameter 휓 ∈ (0,1), creating asymmetric default costs:
  - Under default, yD t = { y t if y t < ȳ ; 휓 ȳ if y t > ȳ } and similarly for oil revenues with p̄ and s̄.
  - 휓 captures proportion retained (1 – output cost of default) and is set to .95 in calibration.
- Probability of re-entry following default per quarter: 휃 = .25.
  - Monte Carlo simulation over 1000 runs yields average exclusion duration ≈ just over 2 years given 휃 = .25.
  - Note: referenced literature finds longer typical exclusion (e.g., 5.7 years in Richmond and Dias (2009)).

### Preference specification
- Epstein-Zin recursive preferences (Caldara et al., 2012) used so risk-aversion and IES are separable.
- Parameter choices:
  - Inter-temporal elasticity of substitution (IES) 휉 = .50.
  - Coefficient of relative risk aversion 휎 = 70.
  - Discount rate parameter set as 휌 = .025 (impatience); implied discount factor 훽 = 1/(1 + 휌) = .976.

### Oil dependence calibration
- Two data-based measures reported (1995q1–2016q4):
  - Average share of oil and gas rents in GDP across the sample = 6% (World Bank WDI).
  - Average share of petroleum in total exports across the sample = 35% (UN Comtrade).
- Country-specific oil shares (%) presented:
  - Angola: GDP share 39.9; Export share 59.5
  - Colombia: GDP share 4.1; Export share 24.5
  - Indonesia: GDP share 2.9; Export share 9.7
  - Mexico: GDP share 3.7; Export share 13.0
  - Nigeria: GDP share 18.6; Export share 57.3
  - Russia: GDP share 10.1; Export share 32.2
  - All (sample averages): GDP share 13.2; Export share 32.7
- Calibrated oil revenue share in the model: 푠̄ = .132.

### Other financial parameters and numerical solution
- Risk-free interest rate r* = .004.
- Exogenous risk premium r̂ = .016; model sets r = r* + r̂.
- Numerical methods: model solved using value function iteration / numerical methods appropriate for non-linear optimal default problem (details in appendix C).
- Sensitivity analysis on key parameters (oil dependence, impatience, shock co-movement, preference structure) reported in appendix D.

### Key quantitative calibration table (preserved values)
- 휌 = .025 (Discount rate)
- 훽 = .976 (Discount factor)
- 휎 = 70 (Coefficient of relative risk aversion)
- 휉 = .50 (Inter-temporal elasticity of substitution)
- 휃 = .25 (Probability of re-entry into capital market)
- 휓 = .95 (1 − output cost of default)
- 푠̄ = .132 (Oil revenue share)
- 푟∗ = .004 (Risk free interest rate)
- r̂ = .016 (Risk premium (exogenous)), with r = r∗ + r̂

### Policy-relevant implications (from the calibrated model)
- Model can explain a large proportion of countercyclical movements in interest-rate spreads in oil-dependent emerging markets via the optimal default channel.
- Policy recommendation implied by the model:
  - It may be optimal to smooth volatility risks associated with oil price shocks by building up liquid buffers and/or paying down debt to insulate the economy from commodity-driven movements in risk premiums and reduce “stop-go” cycles of investment and growth.

*Source: IMF Working Paper (section 2, Calibration) from the provided PDF content.*

### 0.02 Real interest rate

### wpiea2019240-print-pdf - 0.02 Real interest rate

### Key calibrated moments and shock statistics
- Persistence of productivity: .841
- Persistence of oil price: .725
- Std dev. of output shocks: .015
- Std dev. of oil price shocks: .136
- Cross correlation: .045

### Pricing sovereign debt — analytic Ramsey expression
- Steady-state bond price expression (from the simplified Ramsey version with fixed cost of default and unitary re-entry probability):
  - 푞푞 =
    1
    푅푅
    =
    1−휆휆
    푅푅
    (
    훿훿
    ′
    (
    휆휆−푑푑
    )
    −훿훿+1
    )
    +
    1
    푑푑훿훿
    ′
    푅푅
    (equation 16)
- Interpretation and comparative statics:
  - The steady-state price of sovereign debt is a non-linear combination of the fixed cost of default, 휆휆, the interest rate charged to the sovereign, 푅푅, the level of debt, 푑푑, the probability of default, 훿훿, and the sensitivity of this probability to the level of debt, 훿훿
    ′
    .
  - If default is not possible (훿훿=훿훿
    ′
    =0 and 휆휆=0) the expression collapses to 푞푞=1푅푅
    ⁄
    , i.e., bond price inversely related to 푅푅.
  - When default is possible, a larger 훿훿
    ′
    (default decision more sensitive to debt) lowers bond price and raises the effective interest rate.
  - 푅푅 itself is endogenous: it depends on the level of debt and the output and oil price states.

### Nonlinear bond pricing schedules and role of output and oil prices
- Numerical solution (full model) produces an equilibrium bond pricing schedule:
  - Bond price (vertical axis) is plotted against level of indebtedness (horizontal axis) for different output and oil price states (Figure 3).
  - Solid lines: pricing schedules for ‘high’, ‘medium’ and ‘low’ output states (chosen as the 3rd, 6th and 9th positions respectively) given a ‘high’ oil price.
  - Properties of the schedule:
    - Bond price is increasing in foreign assets (negative assets = debt); more negative asset position → higher default probability → lower bond price.
    - Pricing schedule for the ‘high’ output state lies above that for the ‘low’ output state because higher output reduces default probability.
  - Dashed lines: same schedules for the ‘low’ oil price state.
    - Low oil prices shift pricing schedules down because oil revenue enters additively into the budget constraint; lower oil price → reduced revenues → higher default likelihood → lower bond price.
  - Comparison to model without oil:
    - Bond price remains a downward-sloping, non-linear function of output.
    - Without oil the bond pricing schedule exhibits significantly larger steps.
- Model prediction note:
  - The model predicts default (or a zero bond price) at relatively low debt-GDP ratios (sometimes below 10 percent) in adverse oil price and output states; this may not match observed cases where debt ratios can be as high as 75 percent without default.

### Model performance: ability to match data moments
- Simulation details:
  - Model calibrated and solved via value-function iteration.
  - Simulated over 1000 periods.
  - Data period: 1995q1 to2016q4.
  - Series detrended with Hodrick-Prescott (HP) filter using smoothing parameter of 1600.
- Table 3a — AR(1) and Std. Dev. (Data vs Model)
  - Output:
    - Data AR(1): 0.841
    - Data Std. Dev.: 0.032
    - Model AR(1): 0.893
    - Model Std. Dev.: 0.043
  - Oil price:
    - Data AR(1): 0.725
    - Data Std. Dev.: 0.198
    - Model AR(1): 0.657
    - Model Std. Dev.: 0.195
  - Consumption:
    - Data AR(1): 0.898
    - Data Std. Dev.: 0.026
    - Model AR(1): 0.977
    - Model Std. Dev.: 0.107
  - Spread:
    - Data AR(1): 0.645
    - Data Std. Dev.: 0.022
    - Model AR(1): 0.304
    - Model Std. Dev.: 0.010
- Table 3b — Cross-correlations between sovereign spreads and shocks (Data vs Model)
  - Output dev:
    - Data: -0.138
    - Model: -0.148
  - Oil price dev:
    - Data: -0.279
    - Model: -0.160
- Model evaluation summary:
  - The calibration fits the data relatively well and reproduces the ordering of autoregressive properties.
  - The model underpredicts the persistence and variability of the interest rate spread (by about a half) and overestimates volatility of consumption.
  - The model successfully generates the negative correlation between spreads and the oil price (countercyclicality) and produces a higher correlation between spreads and the oil price than between spreads and output, consistent with the data for oil-dependent emerging markets.
  - Sensitivity analysis (appendix D) highlights the importance of: the role of oil, covarying shocks, and deterministic preferences.

### Policy implications and recommendations (from Conclusion)
- Empirical and model findings:
  - There is a counter-cyclical relationship between oil prices and interest-rate spreads in oil-dependent emerging market economies.
  - The model delivers an endogenous link between oil prices and interest rate risk premiums.
- Policy recommendations and implications:
  - Save out of oil revenues to manage volatility in borrowing costs due to sovereign default risk.
    - Buffers should be larger the stronger the link between oil prices and the domestic economy, the more impatient policy makers are, and the more willing they are to substitute current for future consumption.
  - Continue focus on economic diversification to:
    - Reduce co-movements of spreads with oil prices.
    - Increase resilience to commodity price shocks.
  - Recognize that prolonged periods of substantial interest rate premiums for oil-dependent governments can lower investment and capital stocks and hinder growth.
  - Future model extensions suggested:
    - Incorporate domestic physical capital to analyze trade-offs between precautionary savings and domestic investment under endogenous interest rate premiums.

### Appendix highlights
- Ramsey representation (re-entry probability set to unity):
  - Government chooses consumption 푐푐
    푡푡
    , risky government debt 푑푑
    푡푡
    , and safe foreign assets 푓푓
    푡푡
    .
  - Wealth definition: 휔휔
    푡푡
    =푦푦
    푡푡
    +푝푝
    푡푡
    푠푠̅+푓푓
    푡푡−1
    −푑푑
    푡푡−1
    �1−훿훿
    푡푡
    (
    푦푦
    푡푡
    ,푝푝
    푡푡
    ,푑푑
    푡푡
    )
    �−휆휆
    (
    푦푦
    푡푡
    ,푝푝
    푡푡
    )
    훿훿
    푡푡−1
    (
    푦푦
    푡푡−1
    ,푝푝
    푡푡−1
    ,푑푑
    푡푡−1
    )
  - Default indicator: 훿훿
    푡푡
    (
    푦푦
    푡푡
    ,푝푝
    푡푡
    ,푑푑
    푡푡−1
    )
    ∈{0,1}, with 훿훿=1 indicating default. 훿훿=0 if 푑푑
    푡푡−1
    ≤0 (no default when sovereign has positive assets).
  - First-order conditions lead to consumption Euler equation, optimal debt equation, and default Euler equation; solving in steady state yields the bond pricing schedule reproduced above.
- Bivariate Markov shocks (Tauchen-style discretization)
  - Bivariate AR(1) form for shocks (풛풛
    푡푡
    =흆흆풛풛
    푡푡−1
    +휺휺
    푡푡
    ) with 휺휺
    푡푡
    ~푁푁(흁흁,횺횺), 흁흁=�
    0
    0
    �.
  - Variance-covariance of innovations: 횺횺=�
    휎휎
    푥푥
    휎휎
    푥푥푦푦
    휎휎
    푦푦푥푥
    휎휎
    푦푦
    �.
  - State-space grid and transition matrix constructed using bivariate normal CDF F(.). State-space factor r set to 3.
  - Matlab code developed to estimate the transition matrix and simulate the Markov chain.

### Numerical solution method
- Value-function iteration used to solve model numerically.
- Discretize state-space for debt, output and oil revenues; use Markov chain for stochastic processes.
- Solve for policy functions by guess-and-verify: guess bond price schedule and value functions, compute optimal policies and default probabilities, update bond pricing schedule, iterate until convergence.
- Analytical simplifications:
  - Euler equation collapses to standard RBC Euler when default not an option.
  - A single parameterization of fixed default cost and risk-free rate can make optimal debt choice collapse to that of a simple RBC model with perfect capital markets.
  - Without default, optimal debt is decreasing in the elasticity of the sovereign's interest rate with respect to indebtedness.

*Source: wpiea2019240-print-pdf - 0.02 Real interest rate*

### 1. Choose parameters values for:

### 1. Choose parameters values for:

### Numerical solution procedure
- Parameters to choose: �휌휌,휃휃,휎휎,휉휉,휓휓,푠푠̅,푟푟,휌휌
  푦푦
  ,휌휌
  푝푝
  ,휌휌
  푦푦푝푝
  ,휎휎
  푦푦
  ,휎휎
  푝푝
- Discretization:
  - 11 output states for 푦푦
  - 11 oil price states for 푝푝
  - 125 debt states
  - (Repeated statement: "I use 11 output states, 11 oil price states and 125 debt states.")
- Iteration algorithm:
  - Start with a guess of the bond price schedule 푞푞0(푑푑,푦푦,푝푝) =11 +푟푟⁄
  - Solve the sovereign's problem by value function iteration to obtain optimal policy functions for consumption, asset holdings and default choice.
  - Given policy functions, compute the probability of default, 훿훿(푑푑′,푦푦,푝푝).
  - Update bond pricing schedule using this probability of default and repeat solving until convergence:
    - Convergence criterion: 푞푞0(푑푑,푦푦,푝푝) − 푞푞1(푑푑,푦푦,푝푝) < ∆, where ∆ is a small number.

### Sensitivity
- Purpose: test sensitivity of the model to key parameter calibrations.

#### Oil dependence and impatience
- Parameter ranges explored:
  - Rate of time preference: 휌휌 = [0.01,0.05] per quarter (stated as "which works out at a discount rate roughly in the range of 4-20% per annum.")
  - Oil share parameter: s̅ = [0.02,0.6]
- Computational approach:
  - Given computational weight, choose 5 evenly spaced points covering each parameter range.
- Key findings:
  - Figure A1a: presents AR(1) parameter for the interest rate spread across combinations of impatience and oil dependence.
  - Figure A1b: presents spread's correlation with the oil price for same combinations.
  - Non-linear relationship between deep parameters and data moments.
  - Effects:
    - Higher impatience reduces persistence of interest rate spread but increases interest rate spread (via higher default probability).
    - Higher oil share increases spread persistence (oil price shocks more likely to affect ability to repay).
    - Both higher impatience and higher oil dependency strengthen the negative correlation between interest rate spreads and oil price movements (i.e., more negative correlation).
  - Note on estimation noise: "Given the discretised state-space and the numerical solution method, the data moments estimated from the simulations display significant estimation noise. I therefore display a smoothed version of these estimates in the figures."
- Counterfactual:
  - Turning off oil revenues completely: substantial worsening in predictive power; spread persistence and the countercyclical correlation of the spread and the oil price are dramatically reduced.
- Model vs data moments (Table A1a. Model moments):
  - Data / Model AR(1) and Std. Dev.:
    - Output: 0.841 0.032 / 0.829 0.043
    - Oil price: 0.725 0.198 / 0.656 0.195
    - Consumption: 0.898 0.026 / 0.923 0.035
    - Spread: 0.645 0.022 / 0.001 0.005
- Model correlations (Table A1b. Model correlations):
  - Output dev: -0.138 (Data) / -0.125 (Model)
  - Oil price dev: -0.279 (Data) / -0.090 (Model)

#### Covarying shocks
- Focus: vary strength of correlation between oil price and output shocks.
- Parameter choices:
  - Central estimate: 휌휌
    푦푦푝푝 = 0.045
  - Lower and upper values for covariance parameter: 0.01 and 0.10
  - Solve model for 50 points between these bounds.
  - For each parameter, simulate model over 1000 periods and retrieve data moments.
  - Cross-correlation in the four countries in sample ranges from -0.007 to +.077.
- Key findings:
  - Figures A2a and A2b present spread persistence and spread's correlation with oil prices for combinations of risk aversion and inter-temporal substitutability.
  - Auto-regressive properties of interest rate spread are reduced for very low and very high correlations of output and oil price shocks — an inverse-U relationship.
  - Mechanisms:
    - Effect 1: Increased correlation raises default risk (low output + low oil price more frequent) → raises spreads.
    - Effect 2: As default becomes more frequent, sovereign optimally reduces indebtedness → reduces likelihood of default and spread persistence.
  - Net effect: For moderate correlations, spread persistence is relatively high; for low or high correlations, persistence is low.

#### Preferences
- Recursive preferences allow IES independent from risk-aversion.
- Parameter ranges explored:
  - Risk-aversion: 휎휎 ∈ [2,100]
  - Inter-temporal substitutability (IES): 휉휉 ∈ [0.1,0.5]
- Key findings:
  - Figures A3a and A3b present spread persistence and spread's correlation with oil prices for combinations of risk aversion and inter-temporal substitutability.
  - Persistence of interest rate spread: relatively unaffected by changes in either parameter.
  - Correlation of spread with oil price:
    - Higher IES (easier substitution) → greater negative correlation between spreads and movements in the oil price.
    - Mechanism: Higher IES → consumption more volatile and debt path more stable → when oil price rises again, outstanding debt is reduced relatively slowly → higher default risk when oil prices fall initially → creditors raise risk premium when oil prices fall.
    - A similar, albeit less pronounced, relationship exists with risk aversion.

*Source: wpiea2019240-print-pdf - 1. Choose parameters values for:*

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_Source: https://www.imf.org/-/media/files/publications/wp/2019/wpiea2019240-print-pdf.pdf_