## _wp10100

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### Introduction and research question
- Business cycles in small emerging economies differ from those in developed economies: higher, more volatile and countercyclical interest rates; higher output volatility; more countercyclical net exports; and higher consumption volatility relative to income volatility.
- State-dependent interest rate schedules are commonly used in emerging-economy models; sovereign-default models provide microfoundations for such interest-rate schedules.
- The paper studies the numerical solution of the sovereign-default model used in Aguiar and Gopinath (2006) and Arellano (2008) and evaluates the effects of numerical approximation errors on simulated interest-rate behavior.
- Main methodological comparison: discrete state space technique (DSS) versus interpolation methods (Chebyshev collocation and cubic splines).

### Major findings (simulation and approximation)
- DSS with grid specifications used in Aguiar and Gopinath (2006) and Arellano (2008) can produce significant approximation errors that materially affect simulated interest-rate behavior.
- Example quantitative comparisons:
  - The standard deviation of the interest rate spread in the authors’ simulations is less than half of the values reported in Aguiar and Gopinath (2006) when using more accurate methods.
  - Correlations between spread and income:
    - Using more accurate methods for Aguiar and Gopinath (2006): correlation ≈ -0.6 in the parameterization with shocks to the income level.
    - Using more accurate methods for Aguiar and Gopinath (2006): correlation ≈ 0.1 in the parameterization with shocks to the growth rate of income.
    - Aguiar and Gopinath (2006) report correlations of 0.5 (first parameterization) and -0.03 (second parameterization).
- Implication: Results cast doubt on the conclusion that shocks to income growth make spreads countercyclical in these models; approximation error can reverse or materially change the sign and magnitude of key moments.

### Numerical methods performance and computation time
- Interpolation methods (cubic splines and Chebyshev collocation) produce robust results and allow the sovereign to choose borrowing from a continuous set and endowment realizations off the grid.
- DSS convergence toward interpolation-method results requires a significantly larger number of grid points than commonly used in the literature.
- Relative computation times reported for a representative calibration:
  - Solving the model with cubic splines to obtain a solution "not affected by spurious spread volatility": less than 20 minutes.
  - Solving the model with DSS with evenly spaced grid points to obtain a similarly reliable solution: over 45 hours.
- Algorithmic improvement for DSS:
  - A one-loop algorithm that iterates simultaneously on the value and bond price functions reduces computation time relative to a two-loop algorithm.
  - Example: DSS one-loop algorithm takes 31 seconds to solve for the baseline model in Arellano (2008) versus 182 seconds for the two-loop algorithm, using Arellano’s convergence criteria and grid specifications.

### Model summary (structure and primitives)
- Economy: small open economy receiving a stochastic endowment stream of a single tradable good.
- Endowment form: y_t = A e^{z_t} Γ_t, where A is constant, z_t is transitory component, Γ_t is permanent component.
- Transitory shock process: z_t = (1−ρ_z) μ_z + ρ_z z_{t−1} + ε^z_t, with ε^z_t ∼ N(0,σ^2_z), and |ρ_z|<1.
- Permanent component: Γ_t = g_t Γ_{t−1}, with ln(g_t) = (1−ρ_g)(ln(μ_g)−m) + ρ_g ln(g_{t−1}) + ε_t, |ρ_g|<1, ε_t ∼ N(0,σ^2_g), and m = 1/2 σ^2_g / (1−ρ^2_g).
- Preferences: representative agent with CRRA utility u(c) = (c^{1−γ} − 1)/(1−γ), γ is coefficient of relative risk aversion.
- Decisions each period: whether to default on previously issued debt; choice of next-period borrowing/saving.
- Default costs:
  - Exclusion from capital markets; after default, access is regained each period with probability ψ ∈ [0,1].
  - An output loss φ(y) in periods of exclusion.
- Bonds: one-period non-contingent bonds promising one unit of the good next period; lenders are risk neutral, can borrow/lend at risk-free rate r, and have perfect information.
- Bond price satisfying zero-profit condition when government issues b′ bonds:
  q(b′,z,Γ,g) = 1/(1+r) [ 1 − ∫ ∫ d(b′,z′,g′,Γ,g′) F_Z(dz′ | z) F_G(dg′ | g) ],
  where d(b,z,Γ,g) is indicator of optimal default (1 if default optimal, 0 otherwise).
- Recursive equilibrium elements (Definition 1):
  - Set of value functions V(b,z,Γ,g), V^1(z,Γ,g), V^0(b,z,Γ,g).
  - Set of policies for asset holdings b′(b,z,Γ,g) and default decisions d(b,z,Γ,g).
- Bellman equations:
  - V(b,z,Γ,g) = max_{d∈{0,1}} { (1−d) V^0(b,z,Γ,g) + d V^1(z,Γ,g) }.
  - Excluded sovereign value:
    V^1(z,Γ,g) = u(y − φ(y)) + β ∫ ∫ [ ψ V(0,z′,g′,Γ,g′) + (1−ψ) V^1(z′,g′,Γ,g′) ] F_Z(dz′ | z) F_G(dg′ | g).
  - Paying sovereign Bellman:
    V^0(b,z,Γ,g) = max_{b′} { u(y + b − q(b′,z,Γ,g) b′) + β ∫ ∫ V(b′,z′,g′,Γ,g′) F_Z(dz′ | z) F_G(dg′ | g) }.

### Robustness, recommendations for numerical implementation
- DSS inefficiencies can be mitigated by:
  - Increasing the number of asset-grid points substantially relative to common practice.
  - Concentrating DSS grid points at asset levels where the bond price is most sensitive to borrowing and at asset levels visited more frequently in simulations.
  - Using a one-loop algorithm to iterate on value and price functions simultaneously.
- Interpolation methods (cubic splines, Chebyshev collocation) are computationally efficient and produce robust results with fewer grid points.
- DSS inefficiencies are less severe when:
  - The bond price function is less sensitive to borrowing level, which occurs in parameterizations with shocks to income growth and in parameterizations with output-cost specifications similar to Arellano (2008).

### Parameterization
- Three parameterizations solved:
  - Model I and Model II: Aguiar and Gopinath (2006) with φ(y) = λy.
  - Model III: Arellano (2008) with φ(y) = { y − λ if y > λ; 0 if y ≤ λ } (equation (5)).
- Period length: one quarter.
- Parameter values (Table 1 excerpts):
  - Risk aversion σ: 2 2 2
  - Interest rate r: 1% 1% 1.7%
  - Discount factor β: 0.8 0.8 0.953
  - Probability of redemption ψ: 10% 10% 28.2%
  - Loss of output λ: 2% 2% 0.969 E(y)
  - Output scale A: 1 1 10
  - Mean growth rate μ_g: 1.006 1.006 1
  - Mean (log) transitory productivity μ_z: (-1/2)σ_z^2 (-1/2)σ_z^2 0
  - Transitory shock standard deviation σ_z: 3.4% 0 2.5%
  - Transitory shock autocorrelation ρ_z: 0.9 NA 0.945
  - Growth shock standard deviation σ_g: 0 3% 0
  - Growth shock autocorrelation ρ_g: NA 0.17 NA

### Computation
- Numerical method: value function iteration; find V0 and V1.
- Models I and II solved in de-trended form (normalize by μ_g y_{t−1}).
- When objective may not be globally concave:
  - Use global search for candidate borrowing level, then non-linear optimization routine.
  - First-order Taylor approximation to evaluate value functions outside grids when using interpolation.
  - No extrapolation when using DSS.
- Implementation:
  - Fortran 90, serial mode on Unix with Intel Xeon 5160 processors at 3.0 GHz.
- Grid strategies (Table 2):
  - Various DSS grids (original, coarse, finer, finest), one Chebyshev collocation grid, one cubic spline interpolation grid.
  - For Model III with interpolation: two endowment grids (below λ and above λ) because V1 displays a kink at y = λ.
  - Original DSS grids inferred from available codes of Aguiar and Gopinath and Arellano.

### Results — algorithmic comparison (one-loop vs two-loop)
- Standard practice: DSS with two loops (outside loop on bond price, inside loop on value functions).
- Finding: one-loop algorithm (iterate simultaneously on value and bond price functions) reduces computation time substantially.
  - Example (Model III, original DSS grids): one-loop takes 31 seconds; two-loop takes 182 seconds.
  - Computation time per value function iteration: one-loop 0.11 seconds; two-loop 0.13 seconds (bond price not updated every iteration in two-loop).
- Decision: remainder of paper uses one-loop algorithms; initial guesses from finite-horizon last-period solutions, except when using finest grids (there use initial guess from finer grids with linear interpolation).

### Results — simulations (selected exact figures)
- Simulation methodology:
  - Models I and II: 500 simulation samples × 1,500 periods; use last 500 periods to compute moments; HP filter smoothing parameter 1,600; report average across 500 samples.
  - Model III: follow Arellano (2008): extract samples satisfying criteria (default immediately after end of sample; sample length 74 periods; last exclusion at least two periods before sample start); statistics are average across 2,000 samples of 74 periods.
- Notation:
  - y = log income, c = log consumption.
  - TB = trade balance (output minus consumption) as fraction of income Y.
  - Rs = interest rate spread (annual).
  - σ denotes standard deviation in percentage points; ρ denotes correlations.
- Key comparative computational observations (selected exact figures from Table 3):
  - Model I (Original DSS vs Cheb coll. vs Spline):
    - σ(y): 4.33 (Original) … 4.35 (Spline)
    - σ(Rs): 0.05 (Original) … 0.01 (Spline)
    - ρ(Rs,y): 0.56 (Original) … -0.59 (Spline)
    - Defaults per 10,000 quarters: 378 (Original) … 88 (Cheb/Spline)
    - Mean debt output ratio (%): 27 (Original) … 25 (Cheb/Spline)
    - Time to converge: 2’:8” (Original) … 13’:11” (Spline)
  - Model II:
    - σ(y): 4.46 (Original) … 4.43 (Cheb/Spline)
    - σ(Rs): 0.33 (Original) … 0.07 (Cheb/Spline)
    - ρ(Rs,y): 0.02 (Original) … 0.09 (Cheb/Spline)
    - Defaults per 10,000 quarters: 23 (Original) … 22 (Cheb/Spline)
    - Mean debt output ratio (%): 19 (all)
    - Time to converge: 1’:13” (Original) … 19’:42” (Spline)
  - Model III:
    - σ(y): 5.81 (Original) … 5.63 (Spline)
    - σ(Rs): 6.20 (Original) … 2.70 (Spline)
    - E(Rs): 3.78 (Original) … 3.34 (Spline)
    - Defaults per 10,000 quarters: 77 (Original) … 74 (Cheb/Spline)
    - Mean debt output ratio (%): 54 (Original) … 44 (Cheb/Spline)
    - Time to converge: 31” (Original) … 29’:58” (Spline)
- Consistency: Chebyshev collocation results are consistent with cubic spline interpolation results.
- Convergence: DSS results converge toward interpolation results as:
  - endowment grid width increases,
  - number of endowment grid points increases,
  - number of asset grid points increases.

### Results — grid width effects
- Narrow endowment grids can bias default probabilities downward:
  - If the lowest endowment grid point is above default thresholds, DSS may impute zero default probability for certain borrowing levels.
  - Downward bias in default probability increases value of market access and sustains higher borrowing levels.
- Figure 1 (Model I): shows optimal savings b'(y,b) as function of y for endowment grids of increasing width; wider grids reduce upward bias in borrowing.

### Results — number of endowment grid points
- Discretization of income shocks introduces discontinuities in bond price schedule; coarser endowment grids amplify these discontinuities.
- Table 4 (Model III comparisons):
  - DSS (200/21) σ(Rs) = 6.20; DSS (200/211) σ(Rs) = 3.84; DSS (2000/21) σ(Rs) = 4.91
  - Interpretation: tenfold increase in endowment grid points (21 → 211) reduces σ(Rs) from 6.20 to 3.84, holding asset grid constant.
  - Tenfold increase in asset grid points (200 → 2000), holding endowment grid fixed, reduces σ(Rs) from 6.20 to 4.91.
- Figure 2:
  - Left panel: zero-profit bond price q(b', y) vs b' for y at unconditional mean; coarser endowment grids produce more pronounced discontinuities (steps).
  - Right panel: government objective as function of borrowing level shows spurious convexities where bond price is flat, leading to distorted optimal saving choices.

### Results — number of asset grid points
- Coarse asset grids produce steps in optimal borrowing and exaggerated spread movements.
- Figure 3 (Model II):
  - Using DSS with 400 vs 5000 asset points (endowment grid with 1500 points): steps in b' and q(b', g) shrink as asset grid density increases.
  - Discrete borrowing availability can preclude adjustments, exacerbating spread volatility and contaminating ρ(Rs, y).

### Computation efficiency and DSS grid design
- DSS can be improved by concentrating asset grid points in ranges where bond price is most sensitive and where states are frequently visited.
- Table 5 (Model III): comparing different DSS asset point allocations (200 evenly, 500 evenly, 500 non-evenly concentrated, Chebyshev, Spline):
  - σ(Rs): 6.20 (DSS 200) → 3.38 (DSS 500 non-evenly) → 2.71 (Chebyshev) → 2.70 (Spline)
  - Time to converge: 31” (DSS 200) → 67’:51” (DSS 500 evenly) → 68’:5” (DSS 500 non-evenly) → 2 hours (Chebyshev) → 29’:58” (Spline)
- Observation: spline interpolation time is lower than Chebyshev collocation for the three parameterizations; in some cases Chebyshev is slower than coarse DSS despite coarse DSS producing small inaccuracies.
- Trade-off: uneven DSS grids require tailoring to parameterization, complicating calibration and comparative statics.

### Robustness of interpolation methods
- Table 6: interpolation results (Chebyshev and Spline) are robust to increasing grid resolution.
  - Model I, Model II, Model III statistics remain stable across alternative (asset, endowment) grid sizes reported.
  - Example: Model I σ(y) ≈ 4.34 across Cheb (15,10), Cheb (30,20), Spline (30,15), Spline (50,30).

### den Haan and Marcet (1994) test of numerical accuracy
- Test implementation: evaluate Euler-equation residuals using 5,000 samples × 1,500 periods; remove first 10 periods and periods of exclusion except periods with default and first 10 periods after exclusion.
- Table 7: fraction of samples where test statistic is below (left 5%) or above (right 5%) χ^2 critical values:
  - Interpolation methods (Spline, Chebyshev) produce values close to 5% in lower and upper tails for many cases (e.g., Spline lower 5%: 5.54% Model I; Chebyshev lower 5%: 5.56% Model I).
  - DSS coarse and DSS finer often show large departures (e.g., DSS coarse upper 5% for Model I and Model II = 100.0%).
- Caveat: poor DSS performance in the test may be driven by approximation errors in computing q1 (derivative of the zero-profit bond price with respect to borrowing level):
  - DSS approximation q1(b_i, y_j) = [q(b_{i+Δ}, y_j) − q(b_{i−Δ}, y_j)] / (b_{i+Δ} − b_{i−Δ}) with Δ = 1.
  - Sample distribution of the test statistic is sensitive to Δ; optimal Δ depends on parameterization and grid; bond price steps make q1 highly sensitive.
  - Conclusion: cannot definitively attribute DSS poor test performance to lack of equilibrium approximation accuracy.

### Robustness of findings in Aguiar and Gopinath (2006) and Arellano (2008)
- Comparison of reported results with interpolation-based recomputations (Table 8 and Table 9):
  - Aguiar and Gopinath (2006) (Model I vs Model II):
    - Reported ρ(Rs, y): 0.51 (Model I), -0.03 (Model II).
    - Interpolation recomputation: ρ(Rs, y) = -0.59 (Model I), 0.09 (Model II).
    - Implication: their claim that Model II (growth shocks) improves fit by producing countercyclical interest rates is called into question.
  - Arellano (2008) (Table 9):
    - Arellano reported σ(Interest rate spread) = 24.3; recomputed with spline: σ(Rs) = 8.84; Chebyshev: 9.04.
    - More than half of the spread volatility in Arellano (2008) attributable to DSS grid inaccuracies.
- Counterfactual exercise (Figure 4): feeding Model III with Argentine GDP 1993–2001:
  - Original DSS grids produce a higher predicted spread prior to 2001 default than in 1995 crisis, consistent with Arellano (2008).
  - Interpolation methods and finest DSS grids produce spread behavior indistinguishable from each other and different from original DSS results.
- Optimal policies (Figure 5):
  - Optimal saving rules and implied interest rates from interpolation or dense DSS grids differ significantly from original DSS-grid-derived policies used in the cited papers.

### Conclusions (section VII)
- Using DSS with inappropriate grid specifications introduces approximation errors that materially affect results in Aguiar and Gopinath (2006) and Arellano (2008):
  - Misleading inference that shock-to-growth specification (Model II) improves model fit for interest rate-current account co-movement.
  - More than half of reported spread volatility in those studies arises from approximation errors.
- Interpolation methods (Chebyshev collocation, cubic spline) can be significantly more efficient and more accurate than evenly spaced DSS for these default models.
- DSS efficiency can be improved by:
  - Using a one-loop algorithm.
  - Concentrating asset grid points where bond price is most sensitive and where states are observed frequently in simulations.
- Practical recommendation: prefer interpolation methods or carefully designed DSS grids (wider endowment range, more endowment grid points, denser asset grids or concentrated grids) to mitigate approximation errors and obtain reliable policy and moment implications.

### Appendix. Computational strategy
- Model assumptions and scope
  - Discussion assumes that shocks affect only the endowment level and not the growth rate of the endowment.
  - Value functions V0 and V1 are approximated across (b,y)∈[b̄ b]×[ȳ y].
  - When b or y takes values outside [b̄ b]×[ȳ y], value functions are approximated using a first-order Taylor approximation evaluated at the closest point in [b̄ b]×[ȳ y].
- Chebyshev collocation implementation
  - V0 and V1 are approximated as a weighted sum of Chebyshev polynomials for all (b,y)∈[b̄ b]×[ȳ y].
  - Outside the approximation domain, first-order Taylor approximation at the closest point is used.
- Cubic spline interpolation implementation
  - Evenly distributed grids of asset positions and endowment shocks are defined.
  - Grid vectors and matrices of values for V0 and V1 are used to compute breakpoints and coefficients for the piecewise cubic representation using routine CSDEC from the IMSL library (based on de Boor (1977), chapter 4).
  - Evaluation procedure for V0 at a point (b,y) in [b̄ b]×[ȳ y]:
    - Interpolate over asset positions to compute the vector (V0(b,y1),...V0(b,yNy)), where Ny denotes the number of grid points for endowment shocks.
    - Interpolate over endowment positions to compute V0(b,y).
  - When asset position or endowment shock falls outside minimum or maximum grid values, evaluate value functions using a first-order Taylor approximation.
  - The not-a-knot condition is used to determine the value of the derivatives at the end points.
- Algorithm to solve for equilibrium with interpolation methods
  1. Initial guesses
     - Use continuation values at the last period of the finite-horizon version of the model as initial guesses:
       - For (bi,yj) on the grid: V(0)0(bi,yj) = u(yj+bi)
       - V(0)1(yj) = u(yj−φ(yj))
  2. Optimization on grid
     - For each grid point of asset levels and endowment shocks, solve the optimization problem defined in equations V-V0.
     - Find a candidate for the optimal borrowing level using a global search procedure.
     - Use the candidate as an initial guess in the optimization routine UVMIF from the IMSL library (quasi-Newton method) to find the maximum.
     - Each evaluation of the borrower’s objective computes the expectation E(V(b′,y′|y)) using Gauss-Legendre quadrature points and weights, and using V(0)0 and V(0)1 to approximate next-period continuation values.
     - The bond price function q(b,y) is evaluated using the optimal default decision derived from V(0)0 and V(0)1.
     - Solutions at each grid point produce new continuation values V(1)0 and V(1)1.
  3. Convergence criterion
     - Evaluate whether the maximum absolute deviation between the new and previous continuation values is below 10^{-6}.
     - If below 10^{-6}, a solution has been found.
     - If not, repeat the optimization using V(1)0 and V(1)1 to compute expected value functions at each grid point and to evaluate q(b,y).
     - Repeat until the maximum absolute deviation between new and previous continuation values is below 10^{-6}.
- Additional notes on solution properties
  - The algorithm only imposes differentiability on V0 and V1.
  - The algorithm may capture discontinuities in the optimal saving rule (as illustrated in Figure 5) or kinks in the bond price function (as illustrated in Figure 2).
  - Note 10: For that reason, when solving Model III, we partition the grid for endowment shocks. The output cost assumed in Arellano (2008) displays a kink at y=λ, which generates a kink in the function V1.

*Source: _wp10100 - 3. A bond price functionq(b / _wp10100 - References*

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

### _wp10100 - References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

### Introduction and research question
- Business cycles in small emerging economies differ from those in developed economies: higher, more volatile and countercyclical interest rates; higher output volatility; more countercyclical net exports; and higher consumption volatility relative to income volatility.
- State-dependent interest rate schedules are commonly used in emerging-economy models; sovereign-default models provide microfoundations for such interest-rate schedules.
- The paper studies the numerical solution of the sovereign-default model used in Aguiar and Gopinath (2006) and Arellano (2008) and evaluates the effects of numerical approximation errors on simulated interest-rate behavior.
- Main methodological comparison: discrete state space technique (DSS) versus interpolation methods (Chebyshev collocation and cubic splines).

### Major findings (simulation and approximation)
- DSS with grid specifications used in Aguiar and Gopinath (2006) and Arellano (2008) can produce significant approximation errors that materially affect simulated interest-rate behavior.
- Example quantitative comparisons:
  - The standard deviation of the interest rate spread in the authors’ simulations is less than half of the values reported in Aguiar and Gopinath (2006) when using more accurate methods.
  - Correlations between spread and income:
    - Using more accurate methods for Aguiar and Gopinath (2006): correlation ≈ -0.6 in the parameterization with shocks to the income level.
    - Using more accurate methods for Aguiar and Gopinath (2006): correlation ≈ 0.1 in the parameterization with shocks to the growth rate of income.
    - Aguiar and Gopinath (2006) report correlations of 0.5 (first parameterization) and -0.03 (second parameterization).
- Implication: Results cast doubt on the conclusion that shocks to income growth make spreads countercyclical in these models; approximation error can reverse or materially change the sign and magnitude of key moments.

### Numerical methods performance and computation time
- Interpolation methods (cubic splines and Chebyshev collocation) produce robust results and allow the sovereign to choose borrowing from a continuous set and endowment realizations off the grid.
- DSS convergence toward interpolation-method results requires a significantly larger number of grid points than commonly used in the literature.
- Relative computation times reported for a representative calibration:
  - Solving the model with cubic splines to obtain a solution "not affected by spurious spread volatility": less than 20 minutes.
  - Solving the model with DSS with evenly spaced grid points to obtain a similarly reliable solution: over 45 hours.
- Algorithmic improvement for DSS:
  - A one-loop algorithm that iterates simultaneously on the value and bond price functions reduces computation time relative to a two-loop algorithm.
  - Example: DSS one-loop algorithm takes 31 seconds to solve for the baseline model in Arellano (2008) versus 182 seconds for the two-loop algorithm, using Arellano’s convergence criteria and grid specifications.

### Model summary (structure and primitives)
- Economy: small open economy receiving a stochastic endowment stream of a single tradable good.
- Endowment form: y_t = A e^{z_t} Γ_t, where A is constant, z_t is transitory component, Γ_t is permanent component.
- Transitory shock process: z_t = (1−ρ_z) μ_z + ρ_z z_{t−1} + ε^z_t, with ε^z_t ∼ N(0,σ^2_z), and |ρ_z|<1.
- Permanent component: Γ_t = g_t Γ_{t−1}, with ln(g_t) = (1−ρ_g)(ln(μ_g)−m) + ρ_g ln(g_{t−1}) + ε_t, |ρ_g|<1, ε_t ∼ N(0,σ^2_g), and m = 1/2 σ^2_g / (1−ρ^2_g).
- Preferences: representative agent with CRRA utility u(c) = (c^{1−γ} − 1)/(1−γ), γ is coefficient of relative risk aversion.
- Decisions each period: whether to default on previously issued debt; choice of next-period borrowing/saving.
- Default costs:
  - Exclusion from capital markets; after default, access is regained each period with probability ψ ∈ [0,1].
  - An output loss φ(y) in periods of exclusion.
- Bonds: one-period non-contingent bonds promising one unit of the good next period; lenders are risk neutral, can borrow/lend at risk-free rate r, and have perfect information.
- Bond price satisfying zero-profit condition when government issues b′ bonds:
  q(b′,z,Γ,g) = 1/(1+r) [ 1 − ∫ ∫ d(b′,z′,g′,Γ,g′) F_Z(dz′ | z) F_G(dg′ | g) ],
  where d(b,z,Γ,g) is indicator of optimal default (1 if default optimal, 0 otherwise).
- Recursive equilibrium elements (Definition 1):
  - Set of value functions V(b,z,Γ,g), V^1(z,Γ,g), V^0(b,z,Γ,g).
  - Set of policies for asset holdings b′(b,z,Γ,g) and default decisions d(b,z,Γ,g).
- Bellman equations:
  - V(b,z,Γ,g) = max_{d∈{0,1}} { (1−d) V^0(b,z,Γ,g) + d V^1(z,Γ,g) }.
  - Excluded sovereign value:
    V^1(z,Γ,g) = u(y − φ(y)) + β ∫ ∫ [ ψ V(0,z′,g′,Γ,g′) + (1−ψ) V^1(z′,g′,Γ,g′) ] F_Z(dz′ | z) F_G(dg′ | g).
  - Paying sovereign Bellman:
    V^0(b,z,Γ,g) = max_{b′} { u(y + b − q(b′,z,Γ,g) b′) + β ∫ ∫ V(b′,z′,g′,Γ,g′) F_Z(dz′ | z) F_G(dg′ | g) }.

### Robustness, recommendations for numerical implementation
- DSS inefficiencies can be mitigated by:
  - Increasing the number of asset-grid points substantially relative to common practice.
  - Concentrating DSS grid points at asset levels where the bond price is most sensitive to borrowing and at asset levels visited more frequently in simulations.
  - Using a one-loop algorithm to iterate on value and price functions simultaneously.
- Interpolation methods (cubic splines, Chebyshev collocation) are computationally efficient and produce robust results with fewer grid points.
- DSS inefficiencies are less severe when:
  - The bond price function is less sensitive to borrowing level, which occurs in parameterizations with shocks to income growth and in parameterizations with output-cost specifications similar to Arellano (2008).

*Source: _wp10100 - References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .*

### 3. A bond price functionq(b

### _wp10100 - 3. A bond price functionq(b

### Parameterization
- Three parameterizations solved:
  - Model I and Model II: Aguiar and Gopinath (2006) with φ(y) = λy.
  - Model III: Arellano (2008) with φ(y) = { y − λ if y > λ; 0 if y ≤ λ } (equation (5)).
- Period length: one quarter.
- Parameter values (Table 1 excerpts):
  - Risk aversion σ: 2 2 2
  - Interest rate r: 1% 1% 1.7%
  - Discount factor β: 0.8 0.8 0.953
  - Probability of redemption ψ: 10% 10% 28.2%
  - Loss of output λ: 2% 2% 0.969 E(y)
  - Output scale A: 1 1 10
  - Mean growth rate μ_g: 1.006 1.006 1
  - Mean (log) transitory productivity μ_z: (-1/2)σ_z^2 (-1/2)σ_z^2 0
  - Transitory shock standard deviation σ_z: 3.4% 0 2.5%
  - Transitory shock autocorrelation ρ_z: 0.9 NA 0.945
  - Growth shock standard deviation σ_g: 0 3% 0
  - Growth shock autocorrelation ρ_g: NA 0.17 NA

### Computation
- Numerical method: value function iteration; find V0 and V1.
- Models I and II solved in de-trended form (normalize by μ_g y_{t−1}).
- When objective may not be globally concave:
  - Use global search for candidate borrowing level, then non-linear optimization routine.
  - First-order Taylor approximation to evaluate value functions outside grids when using interpolation.
  - No extrapolation when using DSS.
- Implementation:
  - Fortran 90, serial mode on Unix with Intel Xeon 5160 processors at 3.0 GHz.
- Grid strategies (Table 2):
  - Various DSS grids (original, coarse, finer, finest), one Chebyshev collocation grid, one cubic spline interpolation grid.
  - For Model III with interpolation: two endowment grids (below λ and above λ) because V1 displays a kink at y = λ.
  - Original DSS grids inferred from available codes of Aguiar and Gopinath and Arellano.

### Results — algorithmic comparison (one-loop vs two-loop)
- Standard practice: DSS with two loops (outside loop on bond price, inside loop on value functions).
- Finding: one-loop algorithm (iterate simultaneously on value and bond price functions) reduces computation time substantially.
  - Example (Model III, original DSS grids): one-loop takes 31 seconds; two-loop takes 182 seconds.
  - Computation time per value function iteration: one-loop 0.11 seconds; two-loop 0.13 seconds (bond price not updated every iteration in two-loop).
- Decision: remainder of paper uses one-loop algorithms; initial guesses from finite-horizon last-period solutions, except when using finest grids (there use initial guess from finer grids with linear interpolation).

### Results — simulations (Table 3 summary and computation times)
- Simulation methodology:
  - Models I and II: 500 simulation samples × 1,500 periods; use last 500 periods to compute moments; HP filter smoothing parameter 1,600; report average across 500 samples.
  - Model III: follow Arellano (2008): extract samples satisfying criteria (default immediately after end of sample; sample length 74 periods; last exclusion at least two periods before sample start); statistics are average across 2,000 samples of 74 periods.
- Notation:
  - y = log income, c = log consumption.
  - TB = trade balance (output minus consumption) as fraction of income Y.
  - Rs = interest rate spread (annual).
  - σ denotes standard deviation in percentage points; ρ denotes correlations.
- Key comparative computational observations (selected exact figures from Table 3):
  - Model I (Original DSS vs Cheb coll. vs Spline):
    - σ(y): 4.33 (Original) … 4.35 (Spline)
    - σ(Rs): 0.05 (Original) … 0.01 (Spline)
    - ρ(Rs,y): 0.56 (Original) … -0.59 (Spline)
    - Defaults per 10,000 quarters: 378 (Original) … 88 (Cheb/Spline)
    - Mean debt output ratio (%): 27 (Original) … 25 (Cheb/Spline)
    - Time to converge: 2’:8” (Original) … 13’:11” (Spline)
  - Model II:
    - σ(y): 4.46 (Original) … 4.43 (Cheb/Spline)
    - σ(Rs): 0.33 (Original) … 0.07 (Cheb/Spline)
    - ρ(Rs,y): 0.02 (Original) … 0.09 (Cheb/Spline)
    - Defaults per 10,000 quarters: 23 (Original) … 22 (Cheb/Spline)
    - Mean debt output ratio (%): 19 (all)
    - Time to converge: 1’:13” (Original) … 19’:42” (Spline)
  - Model III:
    - σ(y): 5.81 (Original) … 5.63 (Spline)
    - σ(Rs): 6.20 (Original) … 2.70 (Spline)
    - E(Rs): 3.78 (Original) … 3.34 (Spline)
    - Defaults per 10,000 quarters: 77 (Original) … 74 (Cheb/Spline)
    - Mean debt output ratio (%): 54 (Original) … 44 (Cheb/Spline)
    - Time to converge: 31” (Original) … 29’:58” (Spline)
- Consistency: Chebyshev collocation results are consistent with cubic spline interpolation results.
- Convergence: DSS results converge toward interpolation results as:
  - endowment grid width increases,
  - number of endowment grid points increases,
  - number of asset grid points increases.

### Results — grid width effects
- Narrow endowment grids can bias default probabilities downward:
  - If the lowest endowment grid point is above default thresholds, DSS may impute zero default probability for certain borrowing levels.
  - Downward bias in default probability increases value of market access and sustains higher borrowing levels.
- Figure 1 (Model I): shows optimal savings b'(y,b) as function of y for endowment grids of increasing width; wider grids reduce upward bias in borrowing.

### Results — number of endowment grid points
- Discretization of income shocks introduces discontinuities in bond price schedule; coarser endowment grids amplify these discontinuities.
- Table 4 (Model III comparisons):
  - DSS (200/21) σ(Rs) = 6.20; DSS (200/211) σ(Rs) = 3.84; DSS (2000/21) σ(Rs) = 4.91
  - Interpretation: tenfold increase in endowment grid points (21 → 211) reduces σ(Rs) from 6.20 to 3.84, holding asset grid constant.
  - Tenfold increase in asset grid points (200 → 2000), holding endowment grid fixed, reduces σ(Rs) from 6.20 to 4.91.
- Figure 2:
  - Left panel: zero-profit bond price q(b', y) vs b' for y at unconditional mean; coarser endowment grids produce more pronounced discontinuities (steps).
  - Right panel: government objective as function of borrowing level shows spurious convexities where bond price is flat, leading to distorted optimal saving choices.

### Results — number of asset grid points
- Coarse asset grids produce steps in optimal borrowing and exaggerated spread movements.
- Figure 3 (Model II):
  - Using DSS with 400 vs 5000 asset points (endowment grid with 1500 points): steps in b' and q(b', g) shrink as asset grid density increases.
  - Discrete borrowing availability can preclude adjustments, exacerbating spread volatility and contaminating ρ(Rs, y).

### Computation efficiency and DSS grid design
- DSS can be improved by concentrating asset grid points in ranges where bond price is most sensitive and where states are frequently visited.
- Table 5 (Model III): comparing different DSS asset point allocations (200 evenly, 500 evenly, 500 non-evenly concentrated, Chebyshev, Spline):
  - σ(Rs): 6.20 (DSS 200) → 3.38 (DSS 500 non-evenly) → 2.71 (Chebyshev) → 2.70 (Spline)
  - Time to converge: 31” (DSS 200) → 67’:51” (DSS 500 evenly) → 68’:5” (DSS 500 non-evenly) → 2 hours (Chebyshev) → 29’:58” (Spline)
- Observation: spline interpolation time is lower than Chebyshev collocation for the three parameterizations; in some cases Chebyshev is slower than coarse DSS despite coarse DSS producing small inaccuracies.
- Trade-off: uneven DSS grids require tailoring to parameterization, complicating calibration and comparative statics.

### Robustness of interpolation methods
- Table 6: interpolation results (Chebyshev and Spline) are robust to increasing grid resolution.
  - Model I, Model II, Model III statistics remain stable across alternative (asset, endowment) grid sizes reported.
  - Example: Model I σ(y) ≈ 4.34 across Cheb (15,10), Cheb (30,20), Spline (30,15), Spline (50,30).

### den Haan and Marcet (1994) test of numerical accuracy
- Test implementation: evaluate Euler-equation residuals using 5,000 samples × 1,500 periods; remove first 10 periods and periods of exclusion except periods with default and first 10 periods after exclusion.
- Table 7: fraction of samples where test statistic is below (left 5%) or above (right 5%) χ^2 critical values:
  - Interpolation methods (Spline, Chebyshev) produce values close to 5% in lower and upper tails for many cases (e.g., Spline lower 5%: 5.54% Model I; Chebyshev lower 5%: 5.56% Model I).
  - DSS coarse and DSS finer often show large departures (e.g., DSS coarse upper 5% for Model I and Model II = 100.0%).
- Caveat: poor DSS performance in the test may be driven by approximation errors in computing q1 (derivative of the zero-profit bond price with respect to borrowing level):
  - DSS approximation q1(b_i, y_j) = [q(b_{i+Δ}, y_j) − q(b_{i−Δ}, y_j)] / (b_{i+Δ} − b_{i−Δ}) with Δ = 1.
  - Sample distribution of the test statistic is sensitive to Δ; optimal Δ depends on parameterization and grid; bond price steps make q1 highly sensitive.
  - Conclusion: cannot definitively attribute DSS poor test performance to lack of equilibrium approximation accuracy.

### Robustness of findings in Aguiar and Gopinath (2006) and Arellano (2008)
- Comparison of reported results with interpolation-based recomputations (Table 8 and Table 9):
  - Aguiar and Gopinath (2006) (Model I vs Model II):
    - Reported ρ(Rs, y): 0.51 (Model I), -0.03 (Model II).
    - Interpolation recomputation: ρ(Rs, y) = -0.59 (Model I), 0.09 (Model II).
    - Implication: their claim that Model II (growth shocks) improves fit by producing countercyclical interest rates is called into question.
  - Arellano (2008) (Table 9):
    - Arellano reported σ(Interest rate spread) = 24.3; recomputed with spline: σ(Rs) = 8.84; Chebyshev: 9.04.
    - More than half of the spread volatility in Arellano (2008) attributable to DSS grid inaccuracies.
- Counterfactual exercise (Figure 4): feeding Model III with Argentine GDP 1993–2001:
  - Original DSS grids produce a higher predicted spread prior to 2001 default than in 1995 crisis, consistent with Arellano (2008).
  - Interpolation methods and finest DSS grids produce spread behavior indistinguishable from each other and different from original DSS results.
- Optimal policies (Figure 5):
  - Optimal saving rules and implied interest rates from interpolation or dense DSS grids differ significantly from original DSS-grid-derived policies used in the cited papers.

### Conclusions (section VII)
- Using DSS with inappropriate grid specifications introduces approximation errors that materially affect results in Aguiar and Gopinath (2006) and Arellano (2008):
  - Misleading inference that shock-to-growth specification (Model II) improves model fit for interest rate-current account co-movement.
  - More than half of reported spread volatility in those studies arises from approximation errors.
- Interpolation methods (Chebyshev collocation, cubic spline) can be significantly more efficient and more accurate than evenly spaced DSS for these default models.
- DSS efficiency can be improved by:
  - Using a one-loop algorithm.
  - Concentrating asset grid points where bond price is most sensitive and where states are observed frequently in simulations.
- Practical recommendation: prefer interpolation methods or carefully designed DSS grids (wider endowment range, more endowment grid points, denser asset grids or concentrated grids) to mitigate approximation errors and obtain reliable policy and moment implications.

*Italic: Source — _wp10100 - 3. A bond price functionq(b (extracted content).*

### References

### _wp10100 - References

### References
- Aguiar, M. and Gopinath, G. (2006). ‘Defaultable debt, interest rates and the current account’.Journal of International Economics, volume 69, 64–83.
- Aguiar, M. and Gopinath, G. (2007). ‘Emerging markets businesscycles: the cycle is the trend’.Journal of Political Economy, volume 115, no. 1, 69–102.
- Arellano, C. (2008). ‘Default Risk and Income Fluctuations inEmerging Economies’.American Economic Review, volume 98(3), 690–712.
- Athreya, K. (2002). ‘Welfare Implications of the Bankruptcy Reform Act of 1999’. Journal of Monetary Economics, volume 49, 1567–1595.
- Benjamin, D. and Meza, F. (2009). ‘Total Factor Productivity and Labor Reallocation: The Case of the Korean 1997 Crisis’.The B.E. Journal of Macroeconomics (Advances), volume 9, Article 31.
- Chatterjee, S., Corbae, D., Nakajima, M., and R ́ıos-Rull, J.-V. (2007). ‘A Quantitative Theory of Unsecured Consumer Credit with Risk of Default’. Econometrica, volume 75, 1525–1589.
- Cuadra, G., Sanchez, J. M., and Sapriza, H. (forthcoming). ‘Fiscal policy and default risk in emerging markets’.Review of Economic Dynamics.
- Cuadra, G. and Sapriza, H. (2008). ‘Sovereign default, interest rates and political uncertainty in emerging markets’.Journal of International Economics, volume 76, 78–88.
- de Boor, C. (1977).A Practical Guide to Splines. Springer-Verlag.
- den Haan, W. J. and Marcet, A. (1994). ‘Accuracy in simulations’.Review of Economic Studies, volume 61, 3–17.
- Eaton, J. and Gersovitz, M. (1981). ‘Debt with potential repudiation: theoretical and empirical analysis’.Review of Economic Studies, volume 48, 289–309.
- Hatchondo, J. C. and Martinez, L. (2009). ‘Long-duration bonds and sovereign defaults’.Journal of International Economics, volume 79, 117–125.
- Hatchondo, J. C., Martinez, L., and Sapriza, H. (2007). ‘Quantitative Models of Sovereign Default and the Threat of Financial Exclusion’.Economic Quarterly, volume 93, 251–286. No. 3.
- Hatchondo, J. C., Martinez, L., and Sapriza, H. (2009). ‘Heterogeneous borrowers in quantitative models of sovereign default.’International Economic Review. Forthcoming.
- Li, W. and Sarte, P.-D. (2006). ‘U.S. consumer bankruptcy choice: The importance of general equilibrium effects’.Journal of Monetary Economics, volume 53, 613–631.
- Livshits, I., MacGee, J., and Tertilt, M. (2008). ‘Consumer Bankruptcy: A Fresh Start’.American Economic Review, volume 97, 402–418.
- Neumeyer, P. and Perri, F. (2005). ‘Business cycles in emergingeconomies: the role of interest rates’.Journal of Monetary Economics, volume 52, 345–380.
- Schmitt-Groh ́e, S. and Uribe, M. (2003). ‘Closing small open economy models’. Journal of International Economics, volume 61, 163–185.
- Uribe, M. and Yue, V. (2006). ‘Country spreads and emerging countries: Who drives whom?’Journal of International Economics, volume 69, 6–36.

### Appendix. Computational strategy
- Model assumptions and scope
  - Discussion assumes that shocks affect only the endowment level and not the growth rate of the endowment.
  - Value functions V0 and V1 are approximated across (b,y)∈[b
 ̄
b]×[y ̄y].
  - When b or y takes values outside [b
 ̄
b]×[y ̄y], value functions are approximated using a first-order Taylor approximation evaluated at the closest point in [b
 ̄
b]×[y ̄y].

- Chebyshev collocation implementation
  - V0 and V1 are approximated as a weighted sum of Chebyshev polynomials for all (b,y)∈[b
 ̄
b]×[y ̄y].
  - Outside the approximation domain, first-order Taylor approximation at the closest point is used.

- Cubic spline interpolation implementation
  - Evenly distributed grids of asset positions and endowment shocks are defined.
  - Grid vectors and matrices of values for V0 and V1 are used to compute breakpoints and coefficients for the piecewise cubic representation using routine CSDEC from the IMSL library (based on de Boor (1977), chapter 4).
  - Evaluation procedure for V0 at a point (b,y) in [b
 ̄
b]×[y
 ̄y]:
    - Interpolate over asset positions to compute the vector (V0(b,y1),...V0(b,yNy)), where Ny denotes the number of grid points for endowment shocks.
    - Interpolate over endowment positions to compute V0(b,y).
  - When asset position or endowment shock falls outside minimum or maximum grid values, evaluate value functions using a first-order Taylor approximation.
  - The not-a-knot condition is used to determine the value of the derivatives at the end points.

- Algorithm to solve for equilibrium with interpolation methods
  1. Initial guesses
     - Use continuation values at the last period of the finite-horizon version of the model as initial guesses:
       - For (bi,yj) on the grid: V(0)0(bi,yj) = u(yj+bi)
       - V(0)1(yj) = u(yj−φ(yj))
  2. Optimization on grid
     - For each grid point of asset levels and endowment shocks, solve the optimization problem defined in equations V-V0.
     - Find a candidate for the optimal borrowing level using a global search procedure.
     - Use the candidate as an initial guess in the optimization routine UVMIF from the IMSL library (quasi-Newton method) to find the maximum.
     - Each evaluation of the borrower’s objective computes the expectation E(V(b′,y′|y)) using Gauss-Legendre quadrature points and weights, and using V(0)0 and V(0)1 to approximate next-period continuation values.
     - The bond price function q(b,y) is evaluated using the optimal default decision derived from V(0)0 and V(0)1.
     - Solutions at each grid point produce new continuation values V(1)0 and V(1)1.
  3. Convergence criterion
     - Evaluate whether the maximum absolute deviation between the new and previous continuation values is below 10
−6.
     - If below 10
−6, a solution has been found.
     - If not, repeat the optimization using V(1)0 and V(1)1 to compute expected value functions at each grid point and to evaluate q(b,y).
     - Repeat until the maximum absolute deviation between new and previous continuation values is below 10
−6.

- Additional notes on solution properties
  - The algorithm only imposes differentiability on V0 and V1.
  - The algorithm may capture discontinuities in the optimal saving rule (as illustrated in Figure 5) or kinks in the bond price function (as illustrated in Figure 2).
  - Note 10: For that reason, when solving Model III, we partition the grid for endowment shocks. The output cost assumed in Arellano (2008) displays a kink at y=λ, which generates a kink in the function V1.

*Source: _wp10100 - References*

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