## 1.   Variance Parameters for Simulations

## Source details

**Canonical URL:** [1.   Variance Parameters for Simulations](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2005/_wp05207.pdf)

## Other formats

- [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2005/_wp05207.pdf.md)
- [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2005/_wp05207.pdf.json)

---

### Scope and components
- Variance Parameters for Simulations (section header and primary content unit).
- Associated simulation outputs and sections:
  - Selected Statistics, Simulated Debt Ratios (b_J), and Levels (B_J) after 20 Periods (J=20).
  - Selected Statistics, Simulated Debt Ratios (b_J), and Levels (B_J) after 100 Periods (J=100).
  - Selected Statistics, Simulated Tax Rates, Within Period (τ_J, j=1 to J), and End-Period (τ_J), 20 periods (J=20).
  - Selected Statistics, Simulated Tax Rates, Within Period (τ_j, j=1 to J), and End-Period (τ_J), 100 Periods (J=100).
- Figures and regimes covered:
  - Figures 1a–1g: Simulated Tax Rates, Regimes TS, DT, and PR (Figures 1a through 1g).
- Key identifiers and notation (preserved exactly):
  - b_J (simulated debt ratios)
  - B_J (simulated debt levels)
  - J=20
  - J=100
  - τ_J (end-period tax rate)
  - τ_J, j=1 to J (within-period tax rates)
  - τ_j, j=1 to J (within-period tax rates)
  - Regimes: TS, DT, PR
  - Figures: 1a, 1b, 1c, 1d, 1e, 1f, 1g

### Model primitives and government decision sequence
- Output decomposition and shocks:
  - Yt = YtP + εt
  - YtP = Yt-1P + ηt
  - Shocks ηt and εt are uncorrelated at all leads and lags; only Yt observed. Signal-to-noise ratio q = var(ηt)/var(εt) known.
- Recursive estimate of permanent output:
  - ŶP t|t-1 = (1-α) ŶP t-1|t-2 + α Yt-1, with α derived from q (α = q + [q^2+4q]^0.5 / [2+q+[q^2+4q]^0.5]).
- Government budget constraint (levels and ratios):
  - Bt-1(1+r) − PSt = Bt
  - bt-1 θt − pst = bt, with θt ≡ (1+r)/(1+λt), λt ≡ (Yt/Yt-1 −1)
  - Intertemporal constraint: b-1 − Σ(ps_t / Πθ_t) = lim b_t / Πθ_t-1 (t→∞)
  - Transversality: lim b_t/(1+r)^{t-1} = 0
- Sequence of decisions:
  - Government commits spending Gt = γ ŶP t|t-1 at the beginning of period.
  - Government chooses tax rate τ on realized Yt at the beginning of period.

### Regimes and policy rules
- Tax smoothing (TS) — linear-quadratic loss:
  - Objective: minimize discounted expected Φ(τt) with Φ(τt) = φ τt^2; for φ = r, Euler implies τt = E t(τt+1).
  - TS policy rule: τt_TS = γ + r b t-1_TS, where b t-1_TS ≡ r B t-1_TS / ŶP t|t-1.
  - Debt dynamics (ex post):
    - Bt_TS = Bt-1_TS(1+r - r Ψt) + γ[ŶP t|t-1 − Yt], with Ψt ≡ Yt/ŶP t|t-1.
    - bt_TS = bt-1_TS(1+r - r Ψt) + γ[Ψt-1 − 1].
  - Key implications:
    - If perfect forecasting (Ψ = 1), debt ratio stabilizes.
    - Large var(η) relative to var(ε) ⇒ taxes and debt may drift and be volatile; Jensen’s inequality can induce upward drift in debt ratio.
    - With only temporary shocks (var(η)=0), adverse shocks increase debt ex ante permanently.
    - When both var(η) and var(ε) nonzero, long-horizon uncertainty in debt increases because agents cannot immediately distinguish shock types.
- Debt targeting (DT):
  - Rule: τt_DT = γ + r b t-1_DT + α[b t-1_DT − b*], where b* is ad hoc target (assumed b0), α is adjustment coefficient from signal-extraction.
  - Advantages: stabilizes debt ratio; reduces standard deviation and upper percentiles of terminal debt ratio bJ.
  - Drawbacks: increases tax rate volatility and causes more frequent and deeper fiscal adjustments (inhibits countercyclical borrowing); spreading adjustment over time (lower α) increases permanent debt accumulation.
- Precautionary (PR) regime — value-at-risk style objective:
  - Motive: target a primary surplus today to preclude future fiscal adjustments for all but the worst z-percent of macro circumstances.
  - Heuristic assumptions:
    - Variability of tax burden linear in debt: variability = ω b.
    - CARA/exponential loss used: Φ(τt) = -1/φ exp(-φ τt).
  - Heuristic rule result: τt_PR = γ + (r + κ ω) b t-1_PR.
  - Interpretation:
    - φ ≠ 0 introduces precautionary motive (nonzero third moment/prudence).
    - κt* ≡ κ(σt^2)* = -φ σt^2 / 2; τt* = κt* + γt + r b t-1.
    - If φ < 0, authority raises today’s primary surplus beyond TS level to reduce probability of future adjustments (value-at-risk interpretation).
  - Practical: choose prudence parameter -φ to secure debt outcomes with probability (1 − z).

### Simulation setup and parameter grid
- Simulations: J-period histories with J = 20 and J = 100; 1000 simulations each (1000 simulations performed in RATS).
- Common parameter choices (broadly emerging markets):
  - Std. dev. of total GDP (ε + η): 2.5
  - Periods are quarters.
  - Initial GDP = 100; initial debt = 200 (50 percent of yearly GDP).
  - Primary expenditure ratio γ = 20 percent.
  - Interest rate r = 2 percent (about 8 percent yearly).
  - Mean tax rates under TS and DT: τ0 = 24 percent.
  - Under PR: τ0 = 24.5 percent (½ percent of GDP greater than τ0 under TS/DT) — front-loaded cushion aiming to prevent debt rises in all but worst 15–20 percent of cases depending on q.
- Variance parameter grid (selected values reported):
  - Var(ε) (permanent) | Var(η) (temporary) | q | α
  - 6.25 | 0.00 | ∞ | 1
  - 5.47 | 0.78 | 7.00 | 0.88
  - 4.69 | 1.56 | 3.00 | 0.79
  - 3.13 | 3.13 | 1.00 | 0.62
  - 1.56 | 4.69 | 0.33 | 0.43
  - 0.78 | 5.47 | 0.14 | 0.31
  - 0.00 | 6.25 | 0.00 | 0.00

### Key simulation results — debt behavior across regimes
- General stylized outcomes:
  - Under TS, higher q (more permanent shocks) ⇒ mean terminal debt and standard deviation of bJ increase monotonically with q; upper-tail percentiles worsen with q.
  - Example percentiles (TS):
    - For J = 20 and q = ∞: terminal debt ratio b20 ≥ 55 percent of GDP at the 75th percentile; b20 ≥ 59 percent of GDP at the 90th percentile.
    - For J = 100 and q = ∞: terminal debt ratio b100 ≥ 62 percent of GDP at the 75th percentile; b100 ≥ 76 percent of GDP at the 90th percentile.
  - Unscaled debt levels BJ show a nonmonotonic (inverse-U) relationship with q: volatility (std. dev., 75th and 90th percentiles) of BJ peaks at intermediate q (approximately q ≈ 0.33), where both temporary and permanent shocks are important — reflecting persistent misclassification of shocks.
- DT regime:
  - Reduces standard deviations and upper percentiles of bJ dramatically relative to TS (stabilizes debt ratio); may not reduce volatility of unscaled B_J.
- PR regime:
  - Reduces mean debt level and ratios relative to TS and DT.
  - After J = 20, mean b20 drops from initial 50 percent to about 48 percent under PR.
  - After J = 100, mean b100 drops further to about 43 percent under PR.
  - PR provides less debt volatility than TS (but not as little as DT).
  - Examples (comparative percentiles):
    - J = 20: PR 75th/90th percentiles for bJ are 52 percent and 57 percent vs TS 55 percent and 59 percent.
    - J = 100: PR 75th/90th percentiles are 49 percent and 60 percent vs TS 62 percent and 76 percent.

### Key simulation results — tax-rate behavior (selected quantitative highlights)
- Baseline tax values:
  - τ0 ≈ 24 percent under TS and DT.
  - τ0 = 24.5 percent under PR.
- PR regime baseline: "24.5 percent under PR."
- Under PR:
  - After 20 periods, the tax rate τ_J falls to about 24.3 percent.
  - After 100 periods, τ_100 falls slightly below tax rates implied by TS or DT.
- Representative statistics (q→∞, J=20):
  - TS within-period mean τ_j = 0.240, SD = 0.002, 75th = 0.242, 90th = 0.243; end-period τ_20 mean = 0.241, SD = 0.005, 75th = 0.244, 90th = 0.247.
  - DT within-period mean τ_j = 0.241, SD = 0.050, 75th = 0.273, 90th = 0.300; end-period τ_20 mean = 0.243, SD = 0.052, 75th = 0.276, 90th = 0.307.
  - PR within-period mean τ_j = 0.244, SD = 0.002, 75th = 0.246, 90th = 0.247; end-period τ_20 mean = 0.244, SD = 0.006, 75th = 0.247, 90th = 0.251.
- Percentile statements preserved:
  - At J=20, q = ∞: there is a 25 percent chance (75th percentile statistic) that the tax rate τ_20 will be at least 24.4 percent; for the worst 10 percent of cases (90th percentile statistic), the tax rate τ_20 will be at least 24.7 percent.
  - At J=100, q = ∞: there is a 25 percent chance (75th percentile statistic) that the tax rate τ_100 will be at least 24.9 percent; for the worst 10 percent of cases (90th percentile statistic), tax rates τ_100 will be at least 26.1 percent.
- DT regime tax variability highlighted:
  - At J=20, q = ∞: there is a 25 percent chance (75th percentile statistic) that the tax rate τ_20 will be at least 27.6 percent; for the worst 10 percent of cases (90th percentile statistic), the tax rate τ_20 will be at least 30.0 percent.
- Short-term variability comparison (q = ∞, t = 1 through J):
  - Standard deviation of tax rates under TS and PR are about the same: 0.002.
  - Standard deviation under DT: 0.05.

### Selected quantitative table excerpts — debt ratios and levels (examples)
- J=20, q→∞, α=1:
  - TS: b_20 mean = 0.51, SD = 0.07, 75th = 0.55, 90th = 0.59; B_20 mean = 200.0, SD = 2.7, 75th = 201.9, 90th = 203.5.
  - DT: b_20 mean = 0.50, SD = 0.00, 75th = 0.50, 90th = 0.50; B_20 mean = 200.0, SD = 22.6, 75th = 215.4, 90th = 228.9.
  - PR: b_20 mean = 0.48, SD = 0.06, 75th = 0.52, 90th = 0.57; B_20 mean = 190.3, SD = 2.7, 75th = 192.1, 90th = 193.7.
- J=100, q→∞, α=1:
  - TS: b_100 mean = 0.55, SD = 0.24, 75th = 0.62, 90th = 0.76; B_100 mean = 200.0, SD = 6.0, 75th = 204.2, 90th = 207.7.
  - DT: b_100 mean = 0.50, SD = 0.00, 75th = 0.50, 90th = 0.50; B_100 mean = 200.0, SD = 50.0, 75th = 234.6, 90th = 264.9.
  - PR: b_100 mean = 0.43, SD = 0.19, 75th = 0.49, 90th = 0.60; B_100 mean = 155.9, SD = 5.4, 75th = 159.7, 90th = 162.6.

### Regime trade-offs and policy implications
- Trade-offs summarized:
  - TS: allows countercyclical borrowing but can produce unpredictable and potentially upward-drifting taxes and debt when permanent shocks are significant.
  - DT: stabilizes debt ratios but leads to higher tax volatility and may force procyclical adjustments during recessions.
  - PR: raises taxes modestly up-front to reduce long-run risk of large debt increases; smoother tax path than DT when permanent shocks are important.
- Policy recommendations and practical guidance:
  - Use signal-extraction quantified by q to inform regime choice: higher q suggests stronger precautionary elements or explicit debt targets.
  - Consider small front-loaded adjustments (example: ½ percent of GDP) to materially reduce probability of extreme debt outcomes.
  - Complement quantitative rules with stress-test/VaR analyses that account for variance and tail risks in fiscal plans.
  - Precautionary policy (PR) can lower long-run average debt and reduce probability of extreme debt outcomes while offering smoother tax paths than DT in high-q scenarios.

*Source: _wp05207 - 1.   Variance Parameters for Simulations (excerpted table of contents and selected results).*

### 1.   Variance Parameters for Simulations...........................................................................15

### 1.   Variance Parameters for Simulations

### Scope and Components
- Variance Parameters for Simulations (section header and primary content unit).

### Associated Simulation Outputs and Sections
- Selected Statistics, Simulated Debt Ratios (b_J), and Levels (B_J) after 20 Periods (J=20).
- Selected Statistics, Simulated Debt Ratios (b_J), and Levels (B_J) after 100 Periods (J=100).
- Selected Statistics, Simulated Tax Rates, Within Period (τ_J, j=1 to J), and End-Period (τ_J), 20 periods (J=20).
- Selected Statistics, Simulated Tax Rates, Within Period (τ_j, j=1 to J), and End-Period (τ_J), 100 Periods (J=100).

### Figures and Regimes Covered
- Figures 1a–1g: Simulated Tax Rates, Regimes TS, DT, and PR (Figures 1a through 1g with page references as listed).

---

### Key Identifiers and Notation (preserved exactly)
- b_J (simulated debt ratios)
- B_J (simulated debt levels)
- J=20
- J=100
- τ_J (end-period tax rate)
- τ_J, j=1 to J (within-period tax rates)
- τ_j, j=1 to J (within-period tax rates)
- Regimes: TS, DT, PR
- Figures: 1a, 1b, 1c, 1d, 1e, 1f, 1g

*Source: _wp05207 - 1.   Variance Parameters for Simulations (excerpted table of contents).*

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

### References (wp05207) — Key findings and policy implications

### Introduction and objectives
- Fiscal programs often feature erroneous (often optimistic) GDP forecasts (IMF/IEO; International Monetary Fund, 2003).
- Goal: design fiscal policy that is sustainable in the long run but not excessively harsh during temporary downturns; use of automatic stabilizers and borrowing over the business cycle is desirable.
- Problem emphasized: government cannot directly observe permanent (YtP) and transitory (εt) components of output and faces a signal-extraction problem; only the signal-to-noise ratio q = var(ηt)/var(εt) is known.
- Main question: implications for debt outcomes when permanent output is difficult to forecast; compare alternative fiscal rules in an environment with a permanent (random-walk) component to output shocks.

### Economic environment (model primitives)
- Output decomposition:
  - Yt = YtP + εt
  - YtP = Yt-1P + ηt
- Shocks ηt and εt are uncorrelated at all leads and lags; only Yt observed. Signal-to-noise ratio q known.
- Recursive estimate of permanent output:
  - ŶP t|t-1 = (1-α) ŶP t-1|t-2 + α Yt-1, with α derived from q (α = q + [q^2+4q]^0.5 / [2+q+[q^2+4q]^0.5]).
- Government budget constraint (levels and ratios):
  - Bt-1(1+r) − PSt = Bt
  - bt-1 θt − pst = bt, with θt ≡ (1+r)/(1+λt), λt ≡ (Yt/Yt-1 −1)
  - Intertemporal constraint: b-1 − Σ(ps_t / Πθ_t) = lim b_t / Πθ_t-1 (t→∞)
  - Transversality: lim b_t/(1+r)^{t-1} = 0
- Sequence of decisions: government commits spending Gt = γ ŶP t|t-1 at the beginning of period; chooses tax rate τ on realized Yt at beginning of period.

### Tax smoothing (TS) regime — linear-quadratic loss
- Objective: minimize discounted expected Φ(τt) with Φ(τt) = φ τt^2; for φ = r, Euler implies τt = E t(τt+1).
- TS policy rule (with expenditure share γ constant):
  - τt_TS = γ + r b t-1_TS, where b t-1_TS ≡ r B t-1_TS / ŶP t|t-1
- Debt dynamics under TS (ex post level and ratio):
  - Bt_TS = Bt-1_TS(1+r - r Ψt) + γ[ŶP t|t-1 − Yt], with Ψt ≡ Yt/ŶP t|t-1
  - bt_TS = bt-1_TS(1+r - r Ψt) + γ[Ψt-1 − 1]
- Key implications:
  - If perfect forecasting (Ψ = 1), debt ratio stabilizes.
  - If permanent component large (var(η) large relative to var(ε)), taxes and debt may drift and be volatile; debt ratio can exhibit upward drift due to Jensen’s inequality effects on forecast errors.
  - Even with only temporary shocks (var(η)=0), adverse shocks increase debt ex ante permanently.
  - Cases with both var(η) and var(ε) nonzero produce larger long-horizon uncertainty in debt because agents cannot immediately distinguish shock types.

### Debt targeting (DT) regime
- Rule:
  - τt_DT = γ + r b t-1_DT + α[b t-1_DT − b*], where b* is ad hoc target (assumed b0), α is adjustment coefficient from signal-extraction.
- Advantages:
  - Stabilizes the debt ratio; reduces standard deviation and upper percentiles of terminal debt ratio bJ.
- Drawbacks:
  - Increases tax rate volatility and causes more frequent and deeper fiscal adjustments (inhibits countercyclical borrowing).
  - Spreading adjustment over time (lower α) increases permanent debt accumulation.

### Precautionary (PR) fiscal regime — value-at-risk style objective
- Motive: target a primary surplus today to preclude future fiscal adjustments for all but the worst z-percent of macro circumstances.
- Heuristic modeling assumptions:
  - Variability of tax burden is linear in debt: variability = ω b.
  - Use CARA/exponential loss as a tractable precautionary objective: Φ(τt) = -1/φ exp(-φ τt).
- Resulting heuristic rule:
  - τt_PR = γ + (r + κ ω) b t-1_PR
- Interpretation:
  - φ ≠ 0 introduces a precautionary motive (nonzero third moment/prudence).
  - κt* ≡ κ(σt^2)* = -φ σt^2 / 2; τt* = κt* + γt + r b t-1
  - If φ < 0, authority raises today’s primary surplus beyond TS level to reduce probability of future adjustments (value-at-risk interpretation).
- Practical interpretation: choose prudence parameter -φ to secure debt outcomes with probability (1 − z).

### Simulation setup (methods and parameters)
- Simulations: J-period histories with J = 20 and J = 100; 1000 simulations each.
- Common parameter choices (broadly emerging markets):
  - Std. dev. of total GDP (ε + η): 2.5
  - Periods are quarters.
  - Initial GDP = 100; initial debt = 200 (50 percent of yearly GDP).
  - Primary expenditure ratio γ = 20 percent.
  - Interest rate r = 2 percent (about 8 percent yearly).
  - Mean tax rates under TS and DT: τ0 = 24 percent.
  - Under PR: τ0 = 24.5 percent (½ percent of GDP greater than τ0 under TS/DT) — front-loaded cushion aiming to prevent debt rises in all but worst 15–20 percent of cases depending on q.
  - 1000 simulations performed in RATS.
- Variance parameter grid (selected values reported):
  - Var(ε) (permanent) | Var(η) (temporary) | q | α
  - 6.25 | 0.00 | ∞ | 1
  - 5.47 | 0.78 | 7.00 | 0.88
  - 4.69 | 1.56 | 3.00 | 0.79
  - 3.13 | 3.13 | 1.00 | 0.62
  - 1.56 | 4.69 | 0.33 | 0.43
  - 0.78 | 5.47 | 0.14 | 0.31
  - 0.00 | 6.25 | 0.00 | 0.00

### Simulation results — debt behavior across regimes
- General stylized outcomes:
  - Under TS, higher q (more permanent shocks) ⇒ greater difficulty stabilizing debt ratio: mean terminal debt and standard deviation of bJ increase monotonically with q; upper-tail percentiles worsen with q.
  - Example percentiles (TS):
    - For J = 20 and q = ∞ (all shocks permanent): terminal debt ratio b20 ≥ 55 percent of GDP at the 75th percentile; b20 ≥ 59 percent of GDP at the 90th percentile.
    - For J = 100 and q = ∞: terminal debt ratio b100 ≥ 62 percent of GDP at the 75th percentile; b100 ≥ 76 percent of GDP at the 90th percentile.
  - Unscaled debt levels BJ show a nonmonotonic (inverse-U) relationship with q: volatility (std. dev., 75th and 90th percentiles) of BJ peaks at intermediate q (approximately q ≈ 0.33), where both temporary and permanent shocks are important — reflecting persistent misclassification of shocks.
- DT regime:
  - Achieves objective of debt ratio stabilization: standard deviations and upper percentiles of bJ fall dramatically in all cases relative to TS (note: not necessarily reducing volatility of unscaled B_J).
- PR regime:
  - Reduces mean debt level and ratios relative to TS and DT.
  - After J = 20, mean b20 drops from initial 50 percent to about 48 percent under PR.
  - After J = 100, mean b100 drops further to about 43 percent under PR.
  - PR provides less debt volatility than TS (but not as little as DT).
  - PR raises the cushion in worst-case percentiles when permanent shocks matter (examples: for J = 20, PR 75th/90th percentiles for bJ are 52 percent and 57 percent vs TS 55 percent and 59 percent; for J = 100, PR 75th/90th percentiles are 49 percent and 60 percent vs TS 62 percent and 76 percent).

### Simulation results — tax-rate behavior (summary)
- Tax rates matter directly for welfare via collection costs.
- Baseline tax: τ0 ≈ 24 percent under TS and DT; τ0 = 24.5 percent under PR (front-loaded precaution).
- Trade-offs:
  - DT: stabilizes debt ratios but leads to higher tax volatility (more frequent and deeper adjustments).
  - PR: raises taxes modestly up-front to reduce long-run risk of large debt increases; smoother tax path than DT when permanent shocks are important.
  - TS: allows countercyclical borrowing but can produce unpredictable and potentially upward-drifting taxes and debt when permanent shocks are significant.

### Policy implications and recommendations (from results)
- Where permanent components of output shocks are important (high q):
  - Pure tax smoothing (TS) risks larger and persistent debt buildups and volatile tax paths; markets may perceive “debt intolerance” even at modest debt levels.
  - Debt targeting (DT) can stabilize debt ratios but increases tax-rate volatility and may force procyclical adjustments during recessions.
  - Precautionary policy (PR) combining a modest upfront surplus with sensitivity to variance can lower long-run average debt and reduce probability of extreme debt outcomes while offering smoother tax paths than DT in high-q scenarios.
- Practical guidance:
  - Use signal-extraction quantified by q to inform regime choice: higher q suggests stronger precautionary elements or explicit debt targets.
  - Consider small front-loaded adjustments (example: ½ percent of GDP) to materially reduce probability of extreme debt outcomes.
  - Complement quantitative rules with stress-test/VaR analyses that account for variance and tail risks in fiscal plans.

*Source: IMF working paper content unit _wp05207 — References.*

### 24.5 percent under PR. Note that under PR (and unlike TS or DT) taxes fall over time. As

### _wp05207 - 24.5 percent under PR. Note that under PR (and unlike TS or DT) taxes fall over time. As

### Summary of simulation setup and scenarios
- Simulations take 1000 draws and examine three regimes: TS (tax smoothing), DT (debt targeting), and PR (precautionary).
- Horizons analyzed: J=20 and J=100 periods.
- Alternative values of q considered: q→∞, q=8, q=3, q=1, q=0.33, q=0.14, q=0.
- Key reported statistics: mean, SD, 75th percentile, 90th percentile for within-period tax rates (τ_j, j=1 to J) and end-period tax rates (τ_J), and selected statistics for simulated debt ratios (b_J) and levels (B_J).

### Key quantitative findings — tax rates and dynamics
- PR regime baseline: "24.5 percent under PR."
- Under PR:
  - After 20 periods, the tax rate τ_J falls to about 24.3 percent.
  - After 100 periods, τ_100 falls slightly below tax rates implied by TS or DT.
- For q→∞ (all shocks permanent), J=20:
  - TS within-period mean τ_j = 0.240, SD = 0.002, 75th = 0.242, 90th = 0.243; end-period τ_20 mean = 0.241, SD = 0.005, 75th = 0.244, 90th = 0.247.
  - DT within-period mean τ_j = 0.241, SD = 0.050, 75th = 0.273, 90th = 0.300; end-period τ_20 mean = 0.243, SD = 0.052, 75th = 0.276, 90th = 0.307.
  - TP (PR) within-period mean τ_j = 0.244, SD = 0.002, 75th = 0.246, 90th = 0.247; end-period τ_20 mean = 0.244, SD = 0.006, 75th = 0.247, 90th = 0.251.
- Percentile statements emphasized in text (preserving original phrasing):
  - At J=20, q = ∞: there is a 25 percent chance (75th percentile statistic) that the tax rate τ_20 will be at least 24.4 percent; for the worst 10 percent of cases (90th percentile statistic), the tax rate τ_20 will be at least 24.7 percent.
  - At J=100, q = ∞: there is a 25 percent chance (75th percentile statistic) that the tax rate τ_100 will be at least 24.9 percent; for the worst 10 percent of cases (90th percentile statistic), tax rates τ_100 will be at least 26.1 percent.
- DT regime tax variability highlighted:
  - At J=20, q = ∞: there is a 25 percent chance (75th percentile statistic) that the tax rate τ_20 will be at least 27.6 percent; for the worst 10 percent of cases (90th percentile statistic), the tax rate τ_20 will be at least 30.0 percent.
  - Similar elevated variability for J=100 under DT (tables show much larger SDs and high-end percentiles for debt levels and tax rates under DT).
- Short-term variability comparison (q = ∞, t = 1 through J):
  - Standard deviation of tax rates under TS and PR are about the same: 0.002.
  - Standard deviation under DT: 0.05.

### Regime comparisons and behavior across q
- TS regime:
  - For high q (e.g., q = ∞), TS yields a tax rate that rises gradually over time (upward drift).
  - As q falls, the upward drift under TS becomes less pronounced; for q = 0 the tax rate under TS appears roughly constant over time.
  - Tax rate variability under TS is generally low (e.g., within-period SD ≈ 0.001–0.002 for many q values).
- DT regime:
  - Taxes under DT are more volatile and do not exhibit the same upward drift as TS; DT stabilizes debt but with substantially more tax rate variability.
  - As q falls, behavior of DT becomes more similar to TS; for q = 0, TS and DT are identical.
  - End-period SDs and high percentiles under DT can be substantially higher (examples above: SD = 0.052 and 90th percentile = 0.307 for τ_20 at q = ∞).
- PR regime:
  - Tax rates are initially higher than under TS or DT but fall over time (PR taxes fall over time, unlike TS or DT).
  - PR acts as a compromise between TS and DT: it attenuates debt volatility while avoiding the additional tax rate volatility of DT.
  - If the authority keeps with PR, both the mean and the variability of tax rates will fall over time.
  - For J=100 and q ≥ 8, PR and TS have similar SDs, 75th, and 90th percentile statistics; PR appears especially appealing when shocks are mainly permanent.
  - At the end of J=100 the mean end-period tax rate τ_J under PR is less than under TS/DT; also SD, 75th, and 90th percentiles are all less under PR than either TS or DT for all values of q.

### Selected quantitative highlights for debt ratios and levels (examples from Tables 2–3)
- J=20, q→∞, α=1:
  - TS: debt ratio b_20 mean = 0.51, SD = 0.07, 75th = 0.55, 90th = 0.59; debt level B_20 mean = 200.0, SD = 2.7, 75th = 201.9, 90th = 203.5.
  - DT: b_20 mean = 0.50, SD = 0.00, 75th = 0.50, 90th = 0.50; B_20 mean = 200.0, SD = 22.6, 75th = 215.4, 90th = 228.9.
  - TP: b_20 mean = 0.48, SD = 0.06, 75th = 0.52, 90th = 0.57; B_20 mean = 190.3, SD = 2.7, 75th = 192.1, 90th = 193.7.
- J=100, q→∞, α=1:
  - TS: b_100 mean = 0.55, SD = 0.24, 75th = 0.62, 90th = 0.76; B_100 mean = 200.0, SD = 6.0, 75th = 204.2, 90th = 207.7.
  - DT: b_100 mean = 0.50, SD = 0.00, 75th = 0.50, 90th = 0.50; B_100 mean = 200.0, SD = 50.0, 75th = 234.6, 90th = 264.9.
  - TP: b_100 mean = 0.43, SD = 0.19, 75th = 0.49, 90th = 0.60; B_100 mean = 155.9, SD = 5.4, 75th = 159.7, 90th = 162.6.

### Policy implications and conclusions (preserving original framing)
- Tax smoothing under uncertainty can be perilous: both debt levels and tax rates are difficult to stabilize and may drift upwards.
- Debt targeting stabilizes debt but increases tax rate volatility and may inhibit countercyclical borrowing.
- A precautionary regime links the optimal primary surplus to the volatility of the debt ratio (unlike tax smoothing which links primary surplus to the debt ratio only).
- Over time, a precautionary regime will reduce the debt and, for many values of q (especially when shocks are mainly permanent), reduce both mean and variability of tax rates relative to TS/DT.
- The precautionary objective is to guard against future tax and debt increases for all but the worst macroeconomic circumstances; it provides an explicitly articulated objective function to bolster credibility.
- The paper’s ultimate goal: to foster good, clear, and understandable fiscal policies.

*Source: _wp05207 - 24.5 percent under PR. Note that under PR (and unlike TS or DT) taxes fall over time.*

### REFERENCES

### REFERENCES

### Fiscal policy, debt sustainability, and fiscal rules
- Aiyagari, S. Rao, Albert Marcet, Thomas J. Sargent, and Juha Seppälä, 2002, “Optimal Taxation without State-Contingent Debt,” Journal of Political Economy, Vol. 110, No. 6 (December), pp. 1220–54.
- Barro, Robert J., 1979, “On the Determination of the Public Debt,” Journal of Political Economy, Vol. 87, No. 5 (Part 1, October), pp. 940–71.
- Bayoumi, Tamim, and Barry Eichengreen, 1995, “Restraining Yourself: The Implications of Fiscal Rules for Economic Stabilization,” International Monetary Fund Staff Papers Vol. 42, No. 1 (March), pp. 32–48.
- Blanchard, O.J. and others., 1990, “The Sustainability of Fiscal Policy: New Answers to an Old Question,” OECD Economic Studies, 15 (Autumn), pp. 7–34.
- Burnside, C., 2005, “Some Tools for Fiscal Sustainability Analysis,” in Craig Burnside, ed. Fiscal Sustainability in Theory and Practice (forthcoming; Washington: World Bank).
- Chalk, Nigel, and Richard Hemming, 2000, “Assessing Fiscal Sustainability in Theory and Practice,” IMF Working Paper No. 00/81 (Washington: International Monetary Fund)
- Croce, Enzo, and V. Hugo Juan-Ramón, 2003, “Assessing Fiscal Sustainability: A Cross-Country Analysis,” Working Paper in process (Washington: International Monetary Fund).
- Garcia, Márcio and Roberto Rigobon, 2004, “A Risk Management Approach to Emerging Market's Sovereign Debt Sustainability with an Application to Brazilian Data,” National Bureau of Economic Research Working Paper No. 10336 (Cambridge, Massachusetts: National Bureau of Economic Research).
- Hoffmaister, A. W., M. Rojas, M. Saenz, M. Segura, and E. Tenorio, 2001, “Solvency of the Overall Public Sector: A preliminary empirical study for Costa Rica,” Central Bank of Costa Rica, Economics Division, Research Note 04-01 (San José, Costa Rica: Central Bank).
- Hostland, Douglas and Philippe Karam, 2005, “Assessing Debt Sustainability in Emerging Markets using Stochastic Simulation Methods,” IMF Working Paper Forthcoming (Washington: International Monetary Fund).
- International Monetary Fund, 2003a, Evaluation Report, Fiscal Adjustment in IMF-Supported Programs, Independent Evaluation Office (Washington D.C., International Monetary Fund); available by internet: http://www.imf.org/external/np/ieo/2003/fis/index.htm
- International Monetary Fund, 2003b, “Sustainability Assessments—Review of Application and Methodological Refinements.” Available via internet: http:/ /www.imf.org /external/np/pdr/sustain/2003/061003.pdf
- Kopits, George, and Theodore F. Barnhill, 2003, “Assessing Fiscal Sustainability Under Uncertainty,” IMF Working Paper 03/79 (Washington: International Monetary Fund).
- Mendoza, Enrique G., and P. Marcelo Oviedo, 2004, “Public Debt, Fiscal Solvency and Macroeconomic Uncertainty in Latin America: The Cases of Brazil, Colombia, Costa Rica, and Mexico,” NBER Working Paper No. w10637 (Washington: National Bureau of Economic Research).
- Reinhart, Carmen, Kenneth Rogoff, and Miguel Savastano, 2003, “Debt Intolerance,” NBER Working Paper w9908 (Washington: National Bureau of Economic Research).
- Santaella, Julio, 2000, “The Fiscal Policy Possibilities in Mexico 2000–25” (August) (Unpublished: ITAM, Mexico).
- Talvi, Ernesto, and Carlos Végh, eds., 2000, “Cómo Armar el Rompecabezas Fiscal: Nuevos Indicadores de Sostenibilidad” (Washington: Inter-American Development Bank).
- Kumhof, Michael, and Evan Tanner, 2005, “Government Debt: A Key Role in Financial Intermediation,” IMF Working Paper 05/57 (Washington: International Monetary Fund).

### Optimal fiscal and monetary policy; rules and credibility
- Angeletos, George Marios, 2003, “Comments on Benigno and Woodford’s Optimal Monetary and Fiscal Policy,” in Mark Gertler and Kenneth Rogoff, ed., NBER Macroeconomics Annual 2003 (Cambridge, MA: National Bureau of Economic Research).
- Benigno, Pierpaolo, and Michael Woodford, 2003, “Optimal Monetary and Fiscal Policy: A Linear Quadratic Approach,” in Mark Gertler and Kenneth Rogoff, eds., NBER Macroeconomics Annual 2003 (Cambridge, Massachusetts: National Bureau of Economic Research).
- Chari, V., Lawrence J. Christiano, and Patrick J. Kehoe, 1994, “Optimal Fiscal Policy in a Business Cycle Model,” Journal of Political Economy, Vol. 102, No. 4, pp. 617–52.
- Kydland, Finn, and Edward Prescott, 1977, “Rules Rather than Discretion: The Inconsistency of Optimal Plans,” Journal of Political Economy, Vol. 85, No. 3 (June), pp. 473–91.
- Lucas, Robert E. Jr., 1976, “Econometric Policy Evaluation: A Critique,” in Karl Brunner and A.H. Meltzer, eds., The Phillips Curve and Labor Markets (Amsterdam: North-Holland).
- Lucas, Robert E. Jr., and Nancy Stokey, 1983, “Optimal Fiscal and Monetary Policy in an Economy without Capital,” Journal of Monetary Economics, Vol. 12 (July), pp. 55–93.
- Rogoff, Kenneth, 2003, “The IMF Strikes Back,” Foreign Policy (January/February).
- Sargent, Thomas J., 1987, Macroeconomic Theory, 2nd Edition (Orlando, Florida: Academic Press).
- Schmitt-Grohe, Stephanie, and Martin Uribe, 2002, “Optimal Fiscal and Monetary Policy Under Sticky Prices,” NBER Working Paper w9220 (Washington: National Bureau of Economic Research).
- Siu, Henry E., 2004, “Optimal Fiscal and Monetary Policy with Sticky Prices,” Journal of Monetary Economics, Vol. 51, No. 3 (April), pp. 575–607.
- Woodford, Michael, 2001, “Fiscal Requirements for Price Stability,” Journal of Money, Credit, and Banking, Vol. 33, No. 3 (August), pp. 669–728.
- McCallum, Bennett T., 2000, “The Present and Future of Monetary Policy Rules,” International Finance, Vol. 3, No. 2 (July), pp. 273–86.
- Cochrane, John, 1998, “A Frictionless model of U.S. Inflation,” in Ben S. Bernanke and Julio J. Rotemberg, eds., NBER Macroeconomics Annual 1998 (Cambridge, Massachusetts: MIT Press), pp. 323–384.

### Consumption, savings behavior, and precautionary motives
- Carroll, Christopher D., and Miles S. Kimball, 1996, “On the Concavity of the Consumption Function,” Econometrica, Vol. 64, No. 4. (July), pp. 981–992.
- Caballero, Ricardo, 1990, “Consumption Puzzles and Precautionary Savings,” Journal of Monetary Economics, Vol. 25, pp. 113–16.
- Deaton, Angus, 1992, Understanding Consumption, (Oxford: Clarendon Press).
- Dréze, Jacques, and Franco Modigliani, 1972, “Consumption Decisions Under Uncertainty,” Journal of Economic Theory, Vol. 5, pp. 308–335.
- Kimball, Miles, 1990, “Precautionary Savings in the Small and in the Large,” Econometrica, Vol. 58, No. 1, pp. 53–74.
- Talmain, Gabriel, 1998, “An Analytical Approximate Solution to the Problem of Precautionary Savings,” Journal of Economic Dynamics and Control, Vol. 23, pp. 113–124.
- Hall, George J., and Stefan Krieger, 2000, “The Tax Smoothing Implications of the Federal Debt Paydown,” Brookings Papers on Economic Activity, No. 2 (2000), pp. 253–84.

### Forecasting, econometric methods, and empirical accuracy
- Harvey, A. C., 1989, Forecasting, Structural Time Series Models and the Kalman Filter, (Cambridge, Massachusetts: Cambridge University Press).
- Muth, John, 1960, “Optimal Properties of Exponentially Weighted Forecasts,” Journal of the American Statistical Association.
- Juhn, Grace, and Prakash Loungani, 2002, “Further Cross-Country Evidence on the Accuracy of the Private Sector’s Output Forecasts,” IMF Staff Papers, Vol. 49, No. 1 (Washington: International Monetary Fund), pp. 49–64.
- Miniane, Jacques, 2004, “Productivity Shocks, Learning, and Open Economy Dynamics,” IMF Working Paper 04/88 (Washington: International Monetary Fund).
- Kopits, George, and Theodore F. Barnhill, 2003, “Assessing Fiscal Sustainability Under Uncertainty,” IMF Working Paper 03/79 (Washington: International Monetary Fund).

### Regional and applied studies
- Celasun, Oya, Xavier Debrun, and Jonathan Ostry, 2005, “Primary Surplus Behavior and Risks to Fiscal Sustainability in Emerging Market Countries: A "Fan-Chart" Approach,” Mimeo, Research Department, International Monetary Fund (Washington D.C., International Monetary Fund)
- Hoffmaister, A. W., M. Rojas, M. Saenz, M. Segura, and E. Tenorio, 2001, “Solvency of the Overall Public Sector: A preliminary empirical study for Costa Rica,” Central Bank of Costa Rica, Economics Division, Research Note 04-01 (San José, Costa Rica: Central Bank).
- Mendoza, Enrique G., and P. Marcelo Oviedo, 2004, “Public Debt, Fiscal Solvency and Macroeconomic Uncertainty in Latin America: The Cases of Brazil, Colombia, Costa Rica, and Mexico,” NBER Working Paper No. w10637 (Washington: National Bureau of Economic Research).
- Garcia, Márcio and Roberto Rigobon, 2004, “A Risk Management Approach to Emerging Market's Sovereign Debt Sustainability with an Application to Brazilian Data,” National Bureau of Economic Research Working Paper No. 10336 (Cambridge, Massachusetts: National Bureau of Economic Research).

*Source: _wp05207 - REFERENCES*

---


_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2005/_wp05207.pdf_