## How to Calibrate Fiscal Rules: A Primer (excerpts)

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### Stylized facts on fiscal rules and ceilings
- About 70 countries worldwide had a fiscal framework with an explicit cap on public debt as of 2015.
- Debt rules are generally set in gross rather than net terms because:
  - It is hard to determine which government assets are truly liquid, particularly in times of financial stress.
  - The concept of net debt is less transparent than gross debt and more difficult to communicate to the public.
- Gross debt ceilings frequently range between 60 percent and 70 percent of GDP.
- About three-quarters of countries with a debt ceiling are members of supranational unions:
  - European Union and Eastern Caribbean Currency Union impose a debt ceiling of 60 percent of GDP.
  - Central African Economic and Monetary Community and West African Economic and Monetary Union impose a cap of 70 percent of GDP.
  - East African Community member countries have adopted a debt ceiling of 50 percent of GDP in net present value terms during convergence toward the East African Monetary Union.
- More than 80 percent of countries with a debt ceiling also have rules imposing constraints on the (nominal or structural) budget balance; among those, almost a third also have expenditure ceilings.
- For nominal budget balance rules, the 3 percent deficit ceiling is dominant worldwide.
- Structural budget balance ceilings concentrate between 0 and 1 (percent of potential GDP) in many cases.
- Expenditure rules:
  - Real growth rules in the sample: fixed numerical ceilings in the range of 2 percent to 4 percent.
  - Expenditure-to-GDP ratio ceilings in a few emerging market economies range from 30 percent to 40 percent of GDP.

### Calibration objective and two alternative methods
- Objective: set the ceiling of the fiscal anchor (the debt ratio) so that debt is kept under control despite negative macroeconomic shocks and realization of contingent liabilities.
- Two calibration methods:
  - Method One: assumes a known maximum debt limit (MDL). Set debt rule ceiling = MDL − safety margin so debt remains below MDL with high probability using stochastic simulations.
  - Method Two: no explicit MDL. Select debt ceiling so debt can be stabilized after negative shocks without breaching a maximum feasible primary balance (MFPS).

### Method One — three-step stochastic calibration when MDL is known
- Step 1: identify the maximum debt limit.
  - IMF DSA benchmarks for Market Access Countries: 70 percent and 85 percent of GDP for emerging market economies and advanced economies, respectively.
  - For low-income countries, IMF DSA benchmarks for public debt in nominal terms range from 49 percent to 75 percent, depending on institutional quality.
  - Cecchetti, Mohanty, and Zampolli (2011) find debt becomes a drag on growth when it exceeds around 85 percent of GDP in OECD countries.
- Step 2: estimate distribution of macroeconomic and fiscal shocks and simulate potential debt trajectories over a medium-term projection horizon; summarize results in a fan chart.
  - Typical variables for simulations:
    - Advanced and emerging market economies: GDP growth, interest rates on government debt, exchange rate.
    - Low-income developing countries: add terms of trade gap and disbursements of foreign loans.
  - Use joint distribution (VAR-based draws or direct multivariate normal/Student’s t draws); generate N simulations (N large, e.g., more than 1,000).
  - Obtain debt trajectories via the debt accumulation equation and a Fiscal Reaction Function (FRF). FRF includes a fiscal shock calibrated from residuals between actual primary balances and FRF-predicted responses.
- Step 3: pick starting debt level so debt remains below MDL with chosen probability over the medium term.
  - Debt rule ceiling = MDL − safety margin.
  - Iterative percentile-based adjustment: if the 95th debt percentile is significantly off target, adjust starting debt by 0.3 percent and repeat until Debt95 ∈ [MDL − 0.4; MDL + 0.4] without significant breaches.

### Method Two — calibrating the debt ceiling when MDL is unknown (MFPS approach)
- Core assumption: the primary balance is bounded upward; there exists a maximum feasible primary balance (MFPS) above which stabilization is infeasible.
- Step 1: set the maximum feasible primary balance.
  - Illustrative benchmarks:
    - Advanced economies: MFPS ≈ 4 percent of GDP.
    - Emerging markets: MFPS ≈ 2 percent of GDP.
- Step 2: simulate macro paths and compute corresponding primary balance and debt trajectories using the debt accumulation equation and an estimated or normative FRF.
- Step 3: iteratively identify the initial debt level such that the 95th percentile of primary balances falls into pb95 ∈ [MFPS − 0.4; MFPS + 0.4] without significant breaches, adjusting starting debt by 0.3 percent until the criterion is met.
- Caution: if an estimated FRF from past undisciplined behavior is explosive, use a normative FRF; the debt level derived is not an absolute upper limit but the highest stabilizable level given the MFPS and current macro conditions.

### Fiscal Reaction Functions (FRF) — options and specifications
- Purpose: link the primary balance to macro and fiscal conditions; choice depends on calibration scope.
- Three FRF options:
  - Option 1: Estimated FRF based on past behavior (suitable when only a debt rule exists).
  - Option 2: Normative FRF — calibrate ρ so debt converges to long-term target d* in absence of shocks.
  - Option 3: Ad hoc path for the primary balance (user-prescribed mean with optional fiscal shocks).
- Option 1 — estimated FRF (advanced/emerging economies; Bohn 1998 style):
  - pb_it = α_i + β_1 pb_it−1 + β_2 ygap_it D_it + β_3 ygap_it (1 − D_it) + ρ d_it−1 + ε_it, with ε_it ∼ N(0, σ^2).
  - Allows asymmetric response to output gap (β_3 > β_2 possible).
- Option 1 — low-income country FRF:
  - pb_it = α_i + β_1 pb_it−1 + β_2 totgap_it D_it^c + β_3 totgap_it (1 − D_it^c) + β_4 extdis_it + ε_it.
  - totgap_it = deviation of terms of trade from trend; extdis_it = external public debt disbursements (ratio of GDP).
- Option 2 — normative FRF:
  - Same specification as (3.1) but calibrate ρ to ensure convergence to d* when r and g set to long-term steady state values.
  - Determination of ρ via steady-state algebra (see equation in source).
- Option 3 — ad hoc primary balance path:
  - User imposes mean pb in each period; fiscal shocks can be added to capture uncertainty.

### Implementation notes, simulation options, and econometric choices
- Box 2 econometric options:
  - VAR forecasts: quarterly VAR with shocks drawn from estimated residuals/variance-covariance matrix; best with quarterly data and large samples.
  - Drawing directly from joint distribution: calibrate multivariate normal/Student’s t with historical means, variances, covariances for annual data.
  - Ad hoc mean paths: user specifies mean for each macro variable each period and draws shocks around those means.
- Shocks drawn from symmetric distributions (normal or Student’s t); fan charts may be asymmetric due to nonlinear debt responses.
- Debt accumulation equation used in simulations:
  - d_t = [1 + ((r_t − g_t) / (1 + g_t))] d_{t−1} − pb_t + SFA_t
    - where d_t = debt ratio; r_t = average effective real interest rate on debt; g_t = real GDP growth rate; pb_t = primary balance ratio; SFA_t = stock-flow adjustment ratio.

### Key parameters, tolerances, and operational rules translations
- Baseline tolerated probability of breaching the debt limit in examples: 10 percent.
- Iterative adjustment step for starting debt: 0.3 percent.
- Small interval tolerance around MDL or MFPS: ±0.4 (i.e., [MDL − 0.4; MDL + 0.4] and [MFPS − 0.4; MFPS + 0.4]).
- Typical MFPS benchmarks preserved from text: 4 percent of GDP (advanced economies) and 2 percent of GDP (emerging market economies).
- Projection horizon examples: six-year horizon (N = 6) used in Method Two; six-year projection horizon used in VAR/multivariate draw simulations.

### From debt rules to operational rules — convergence approaches and calibration formulas
- Long-term convergence (Approach 1):
  - Constant balance target b* with long-term debt target d*:
    - b* = λ d*.
    - λ = −γ / (1 + γ) for overall balance target; λ = (i − γ) / (1 + γ) for primary balance target.
  - Numerical example: if nominal growth γ = 5 percent and nominal interest rate i = 3 percent, then a 60 percent of GDP debt target implies:
    - an overall deficit target of about 2.9 percent of GDP.
    - a primary deficit target of about 1.1 percent of GDP.
  - Convergence can be slow: about 15 years to complete half the distance from initial debt of 70 percent to target of 60 percent.
- Convergence by a given date (Approach 2):
  - Force debt ratio to hit d* after N years. Constant b* given by:
    - b* = λ / ((1 + λ)^N − 1) [ d_0 (1 + λ)^N − d* ].
  - Example (illustrative assumptions: γ = 5 percent; i = 3 percent; initial debt d_0 = 70 percent; d* = 60 percent; initial balance = −5%):
    - Convergence by N = 15 years (instantaneous adjustment) requires an overall deficit of 2.4 percent or a primary deficit of 0.6 percent of GDP to hit 60 percent from 70 percent within 15 years.
    - Versus convergence in the long run requiring about 2.9 percent (overall) and 1.1 percent (primary).
- Convergence with transition period (Approach 3):
  - Allow transition T years with linear annual adjustment α until year T, then maintain b*_T until year N; calibration includes additional term A(T, b_0, N, λ, d_0, d*_N).
  - Illustrative case: T = 5 and N = 15 produces backloaded adjustment (initial higher deficits, tighter policies after transition).
- Convergence with buildup of fiscal buffers (Approach 4):
  - Build buffers to accommodate expected future increases in age-related spending; balance path b_t = b* for 0 < t < P and b_t = b* − ∆A_t for P ≤ t ≤ N.
  - Main calibration: b* = λ / ((1 + λ)^(N − 1)) [ d_0 (1 + λ)^N − d*_N + S ], with S = Σ_{t=1}^N (1 + λ)^{N−t} ∆A_t.
  - Illustrative parameters: set δ ≈ 0.2 percent of GDP per year starting in year P = 6 so total increase in age-related spending would be 2 percent of GDP until year N = 15; present value S / (1 + λ)^N = 22 percent of GDP (in the illustrative simulation).
- From structural deficit ceiling sb* to nominal deficit ceiling nb*:
  - nb* = sb* − OG_max [ r (1 − η) − e (1 − κ) ].
  - Common proxy when revenue elasticity ≈ 1 and spending elasticity ≈ 0:
    - nb* = sb* + OG_max × e.
  - Example: structural deficit target sb* = −1 percent of potential GDP, spending 40 percent of GDP, output gap OG_max = −2 percent ⇒ nominal deficit ceiling nb* = 2 percent of GDP under equation (8).

### Limitations and cautions of the framework
- Framework is an approximation; does not replace full macroeconomic models; relies on simplifying assumptions:
  - Macro variables projected using simple VAR or joint normal/Student’s t draws; simulations informed by historical data and cannot capture recent or expected structural changes.
  - No structural/behavioral equations (for example, aggregate demand or monetary policy rule) used to project macro variables.
  - VAR linear structure may fail to capture nonlinearities and skewed or fat-tailed shocks.
  - Data constraints may limit precise estimation when time series are short.
  - FRF contains only a small set of independent variables and ignores nonlinearities and breaks.
  - Fiscal variables are not included in the VAR/joint distribution, so there is no feedback from fiscal policy changes to macro variables, particularly GDP.
- Calibration focuses on protection against negative shocks and does not capture trade-offs with development priorities; may bias toward austerity in some emerging and developing economies.
- For commodity exporters:
  - Gross debt rule may be less relevant where large financial assets exist; net debt or net wealth anchors may be more relevant.
  - Two approaches to calibrate net wealth targets:
    - Long-term sustainability (PIH): preserve government net wealth; allow temporary scaling-up of public investment with net wealth rules introduced after scaling-up.
    - Risk-based approach: calibrate precautionary buffers using VaR or model-based approaches; example calibrations aim to generate investment returns sufficient to cover a share of lost commodity revenue with given probabilities (for example, 75 percent to 90 percent).

*Source: Box 1, Box 3, and methodological excerpts from How to Calibrate Fiscal Rules: A Primer (extracted from the supplied content).*

### Box 1. How to Calibrate Fiscal Rules in Three Steps

### Box 1. How to Calibrate Fiscal Rules in Three Steps

### Stylized Facts on Fiscal Rules and Ceilings
- As of 2015, about 70 countries worldwide had a fiscal framework with an explicit cap on public debt.
- Debt rules are generally set in gross rather than net terms because:
  - It is hard to determine which government assets are truly liquid, particularly in times of financial stress.
  - The concept of net debt is less transparent than gross debt and more difficult to communicate to the public.
- Gross debt ceilings frequently range between 60 percent and 70 percent of GDP.
- About three-quarters of countries with a debt ceiling are members of supranational unions.
  - European Union and Eastern Caribbean Currency Union impose a debt ceiling of 60 percent of GDP.
  - Central African Economic and Monetary Community and West African Economic and Monetary Union impose a cap of 70 percent of GDP.
  - East African Community member countries have adopted a debt ceiling of 50 percent of GDP in net present value terms during convergence toward the East African Monetary Union.
- More than 80 percent of countries with a debt ceiling also have rules imposing constraints on the (nominal or structural) budget balance; among those, almost a third also have expenditure ceilings.
- For nominal budget balance rules, the 3 percent deficit ceiling is dominant worldwide.
- Structural budget balance ceilings are more widely distributed; a concentration between 0 and 1 (percent of potential GDP) reflects EU medium-term budgetary objectives and specific regional practices (for example, use of average oil revenues).
- Expenditure rules are less common than debt or budget balance rules:
  - Most often an explicit cap on nominal or real expenditure growth.
  - Real growth rules in the sample: fixed numerical ceilings in the range of 2 percent to 4 percent.
  - In a few emerging market economies, expenditure-to-GDP ratio ceilings range from 30 percent to 40 percent of GDP.
  - The European Union’s expenditure benchmark caps annual growth of primary expenditure with long-term nominal GDP growth and corrects for revenue measures.

### Calibration Objective and Two Alternative Methods
- Objective: set the ceiling of the fiscal anchor (the debt ratio) so that debt is kept under control despite negative macroeconomic shocks and realization of contingent liabilities.
- Two alternative calibration methods:
  1. Method One: assumes a known maximum debt limit beyond which debt dynamics spiral out of control. The debt rule ceiling is set as the maximum debt limit minus a safety margin, chosen so debt remains below the debt limit with high probability.
  2. Method Two: does not rely on an explicit debt limit; selects the debt ceiling so that debt can be stabilized following negative shocks without breaching a policy limit (a maximum feasible level of the primary balance).

- Institutional coverage note: the debt ceiling can be based on any institutional coverage (for example, central or general government), but the same coverage must be used throughout the calibration exercise.

### Method One — Three-Step Calibration When the Maximum Debt Limit Is Known
- Overview: stochastic simulations compute a safety margin below a known debt limit; calibration in three steps:
  - Step 1: identify the maximum debt limit.
  - Step 2: estimate the distribution of macroeconomic and fiscal shocks and simulate potential debt trajectories over a medium-term projection horizon; summarize results in a fan chart.
  - Step 3: identify the debt rule ceiling as a sufficiently low starting level for debt (in the first projection year) such that debt remains below the maximum debt limit over the medium term with high probability. Debt rule ceiling = maximum debt limit − safety margin.

Step 1: Setting the Debt Limit — possible approaches
- Risk of debt distress:
  - IMF DSA benchmarks for Market Access Countries: 70 percent and 85 percent of GDP for emerging market economies and advanced economies, respectively.
  - For low-income countries, IMF DSA benchmarks for public debt in nominal terms range from 49 percent to 75 percent, depending on institutional quality.
- Risk of growth slowdown:
  - Cecchetti, Mohanty, and Zampolli (2011) find debt becomes a drag on growth when it exceeds around 85 percent of GDP in OECD countries.
- Sensitivity analysis based on alternative debt limit estimates is warranted.

Step 2: Estimating the Effect of Shocks on Debt
- Perform stochastic simulations to gauge potential impact of macroeconomic and fiscal shocks over the medium term by estimating the joint distribution of macroeconomic variables.
- Variables typically included:
  - For advanced and emerging market economies: GDP growth, interest rates on government debt, and the exchange rate.
  - For low-income developing countries: add terms of trade gap and disbursements of foreign loans.
- Multiple simulations are carried out using the joint distribution; each simulation produces a path for macroeconomic variables over the projection horizon subject to shocks.
- Medium-term debt trajectories for each simulated macro path are obtained from:
  - debt accumulation equation (government budget constraint), and
  - a Fiscal Reaction Function (FRF) in which the level of the primary balance may respond to the level of debt and realizations of macroeconomic variables.
- The FRF includes a fiscal shock realized each period; the distribution of fiscal shocks is calibrated from residuals between actual primary balances and FRF-predicted responses.
- Simulated debt trajectories are summarized in a fan chart.

Step 3: Calibrating the Debt Ceiling
- Choose an initial level of debt (first projection year) so that debt remains below the maximum debt limit with a chosen probability over the medium term despite negative shocks.

Box 2: Simulating Macroeconomic Variables — econometric options
- VAR forecasts:
  - Quarterly VAR estimated for key macro variables; shocks drawn from distribution calibrated using estimated VAR residuals and variance-covariance matrix.
  - Works best with quarterly data and large sample size.
- Drawing directly from the joint distribution:
  - For annual data, calibrate a joint normal (or Student’s t) distribution using historical means, variances, and covariances; draw shocks directly.
- Forecasting with ad hoc path:
  - User specifies a mean for each macro variable each period (e.g., own forecast); generate shocks around these means by drawing from the joint distribution.

### Fiscal Reaction Functions (FRF): Options and Specifications
- Purpose: link the primary balance to prevailing macroeconomic and fiscal conditions; choice depends on the calibration scope.
- Three FRF options:
  - Option 1: Estimated FRF based on past behavior — appropriate when the fiscal framework has only a debt rule.
  - Option 2: Normative FRF — ensures a “well behaved” policy response consistent with debt sustainability when operational rules are present; normative in that it steers debt to a long-term target d* in absence of further shocks.
  - Option 3: Ad hoc path for the primary balance over the projection period.

Option 1 — Estimated FRF Based on Past Behavior (advanced/emerging economies)
- FRF specification (Bohn 1998 style):
  pb_it = α_i + β_1 pb_it−1 + β_2 ygap_it D_it + β_3 ygap_it (1 − D_it) + ρ d_it−1 + ε_it
  where pb_it is the primary balance (ratio of GDP), d_it is debt (ratio of GDP), ygap_it is the output gap, D_it is an indicator equal to 1 when the output gap is positive, α_i is country fixed effect, ε_it ∼ N(0, σ^2).
- The FRF allows asymmetric response to the output gap (β_3 > β_2 possible).
- Output gap is projected using GDP growth forecasts from simulations combined with a Hodrick-Prescott filter to estimate potential output.

Option 1 — Low-Income Country FRF
- Alternative FRF reflecting low-income country dynamics:
  pb_it = α_i + β_1 pb_it−1 + β_2 totgap_it D_it^c + β_3 totgap_it (1 − D_it^c) + β_4 extdis_it + ε_it
  where totgap_it is deviation of terms of trade from trend, D_it^c is an indicator for commodity exporters, extdis_it are disbursements of external public debt (ratio of GDP).
- Terms of trade and external financing disbursements are included in the joint distribution for simulations.

Option 2 — Normative FRF
- Uses the same specification as equation (3.1) but calibrates ρ to ensure debt converges to long-term target d* in absence of shocks.

### Implementation Notes and References to Algorithms
- More details on algorithms and formulas are provided in Appendix 1 and the manuals accompanying the note.
- The accompanying econometric files allow choice among VAR, joint-distribution drawing, or ad hoc mean paths for macroeconomic variable simulation.
- Shocks are drawn from symmetric (normal or Student’s t) distributions; fan charts may be asymmetric because the impact of shocks on debt depends on the level of debt.

*Source: Box 1. How to Calibrate Fiscal Rules in Three Steps (extracted from the supplied content).*

### Box 3. Specifying the Fiscal Reaction Function

### Box 3. Specifying the Fiscal Reaction Function

### Method 1: Calibrating a Debt Rule Ceiling with a Fan Chart
- Baseline tolerated probability of breaching the debt limit: 10 percent.
- Example: debt ceiling calibration over a six-year horizon with a safety margin below the maximum debt limit (debt projected from 2017-2022; last year before projections begin is 2016).
- Interpretation:
  - If the current debt level is above the debt ceiling, the country is not maintaining a sufficient safety margin given the degree of risk tolerance.
  - Changing the starting debt level shifts and tilts the entire fan chart.
- Key parameter for long-term target:
  - One option for advanced and emerging market economies: long-term target level at 60 percent of GDP.
- Determining the appropriate value of ρ:
  - “The appropriate setting for ρ is the value consistent with a steady state of the system of simultaneous equations formed by the debt accumulation equation and fiscal reaction function (3.1) when debt is set equal to its long-term target level d*, with growth g and the real interest rate on government debt r set to their long-term steady state values. Algebraically, the appropriate value of ρ can be expressed as
   5 (1 2 
  1
  )  
  (

  r 2 g

  ____

  1 1 g

  ) 2   
  
  i
  __
  d*
   (3.3)”
  - Long-term steady state values for g and r can be proxied by imposing steady state on an estimated VAR model (imposing that lagged values equal current values) and solving for the vector of steady state values in terms of estimated VAR coefficients. The accompanying econometric files compute these steady state values automatically.
  - If the interest-growth differential is low and the long-term debt target d* is close to the current level of debt, the value of ρ computed using the formula can be smaller (positive) than in the estimated FRF from option 1; in that case debt will converge to the long-term target (in the absence of shocks) even if fiscal policy is less responsive to debt than historically.
- Option 3 (within Box 3): Ad Hoc Primary Balance Path
  - User can impose a mean value for the primary balance in each period over the medium-term horizon (for example, the user’s baseline forecast).
  - A fiscal shock can be added to this mean value each period to capture uncertainty about future fiscal behavior.

### Factors Affecting the Safety Margin in Method 1
- The safety margin will be larger if:
  - (1) the level of risk aversion increases (debt must be lower to reduce the probability of breaching the limit);
  - (2) the amount of macroeconomic volatility implied by the estimated joint distribution increases;
  - (3) the response of the primary balance to changes in debt becomes weaker (FRF parameters reflect weaker response);
  - (4) the length of the medium-term projection horizon increases.
- Stochastic simulations:
  - Simulations are based on symmetric laws drawing positive and negative shocks, but only negative shocks matter for calibration because the debt ceiling is calibrated so debt remains below the maximum limit except in a small percentage of cases when particularly bad combinations of shocks are realized.
- Treatment of contingent liabilities:
  - Above-the-line contingent liabilities: FRF includes fiscal shocks drawn from historical distribution; shocks are affected by materialization of past contingent liabilities if recorded above the line (generally under transfers) and transmitted to the primary balance.
  - Below-the-line contingent liabilities: debt accumulation equation includes stock-flow adjustments simulated over the forecast horizon using their historical distribution; debt simulations will reflect historical pattern of contingent liability realization if recorded below the line.
  - If contingent liabilities are expected to be larger than historically experienced, manually adjust the stock-flow adjustments in the debt accumulation equation.

### Method 2: Calibrating the Debt Rule Ceiling When the Maximum Debt Limit Is Unknown
- Applicability: most suitable for advanced economies with unconstrained market access and considerable uncertainty about sustainable debt levels.
- Core assumption: the primary balance is bounded upward; a primary surplus above a certain bound may be unachievable.
- Concept: there is a level of public debt above which debt stabilization becomes “impossible” because the maximum feasible primary balance cannot be exceeded indefinitely.
- Implementation steps:
  - Step 1: Setting the Maximum Primary Balance
    - Identify the maximum feasible primary balance (country-specific or based on cross-country historical experience).
    - Illustrative values:
      - Advanced economies: maximum feasible primary surplus of about 4 percent of GDP.
      - Emerging markets: maximum feasible primary surplus of 2 percent of GDP.
    - Tailoring the choice should consider what primary balances can be sustained over a number of years in prevailing macroeconomic circumstances. Large primary surpluses may be easier to achieve than to maintain; median primary surplus measured using five-year moving averages is lower than using individual-year primary balances.
  - Step 2: Estimating the Effect of Shocks on the Primary Balance and Debt
    - Use stochastic simulations (an estimated VAR subject to shocks in the accompanying files) to forecast multiple trajectories of macroeconomic variables over a medium-term projection horizon.
    - Compute corresponding trajectories of the primary balance and debt using the simultaneous system of the debt accumulation equation and a fiscal reaction function.
    - Summarize potential trajectories for the primary balance and debt under shocks using separate fan charts.
  - Step 3: Calibrating the Debt Ceiling
    - Identify the initial debt level (debt rule ceiling) that ensures with high probability that the maximum feasible primary balance is not breached over the medium term when negative shocks occur.
    - Iterative procedure:
      - Start with a certain initial debt level and compute debt trajectories under various shocks.
      - If the primary balances required to stabilize debt breach the maximum feasible primary balance in a large number of trajectories (for example, more than 5 percent or 10 percent), lower the initial debt level and repeat until most of the primary balance fan chart falls below the feasible maximum over the projection horizon.
    - Sensitivities:
      - A lower debt ceiling is required when there is higher risk aversion.
- Cautions:
  - When using Method Two with an estimated (rather than normative) fiscal reaction function, if the estimated reaction function is explosive (reflecting undisciplined past fiscal behavior), the initial debt level computed with Method Two cannot be considered safe; in that case use a normative or calibrated reaction function.
  - The debt level backed out as consistent with the maximum feasible primary balance is not an absolute upper limit as in Method One; it is the highest debt level that can be stabilized without breaching the maximum feasible primary balance given current macroeconomic conditions. If the interest-growth differential increases, the maximum debt mechanically declines.

*Source: Box 3. Specifying the Fiscal Reaction Function (How-to Note).*

### 1. Primary Balance2. Public Debt

### 1. Primary Balance2. Public Debt

### Method 2: Primary Balance and Debt Fan Charts
- Presents the interaction between a maximum feasible primary balance, risk tolerance, and a debt rule ceiling using fan charts (Figure 5).
- Time labels shown: 2016 17 18 19 20 21 22.
- Emphasizes that lower debt reduces the probability of exceeding the maximum primary balance.
- Notes that:
  - A lower debt ceiling is implied by higher macroeconomic volatility because the primary balance would need to be higher to stabilize debt when negative shocks are larger.
  - A lower long-term target level of debt may require a lower debt ceiling when using a normative fiscal reaction function, since the normative FRF embodies a stronger response of the primary balance to the current level of debt.

### Limitations of the Proposed Approach
- The framework is a tractable approximation and does not replace full macroeconomic models; it relies on simplifying assumptions:
  - Macroeconomic variables are projected using a simple VAR econometric model subject to shocks (drawn from a symmetric joint normal/Student’s t distribution) or by drawing directly from the joint distribution if quarterly data are unavailable; simulations are informed by historical data and cannot capture the impact of recent or expected structural changes.
  - Structural/behavioral equations from economic theory (for example, aggregate demand curve or monetary policy rule) are not used to project macroeconomic variables.
  - A VAR econometric model has a simple linear structure and may fail to capture nonlinearities among macroeconomic variables (for example, changes in the relationship between interest rates and growth throughout the business cycle).
  - The VAR and the joint normal/Student’s t distribution assume macroeconomic shocks are symmetric; in reality shocks may be skewed in the adverse direction and the distribution may not capture tail events well.
  - Data constraints may prevent precise estimation of the VAR or calibration of the joint distribution if only short time series are available.
  - The fiscal reaction function used to project fiscal variables contains only a small set of independent variables and ignores potential nonlinearities and breaks in the reaction of fiscal policy to debt.
  - Fiscal variables (for example, the primary balance or debt) are not included in the VAR or the joint distribution, so there is no feedback from fiscal policy changes to macroeconomic variables, in particular GDP.
- The calibration methods focus exclusively on protecting fiscal position against negative shocks and do not capture the trade-off with development priorities (public investment, education, health). This can bias methods toward austerity for some emerging and developing economies.
- Notes from economic theory: public investment should be primarily financed by debt issuance rather than taxes (Ostry, Ghosh, and Espinoza 2015) to smooth tax distortions and because benefits accrue to future generations.

### Framework for Commodity Exporters
- Gross debt rule may be less relevant where large financial assets exist; net debt (debt minus financial assets) or net wealth (net financial wealth plus resource wealth) can be a more relevant anchor.
- Two approaches to calibrate net wealth/net financial wealth targets:

  Long-Term Sustainability Approach
  - Fixed net wealth benchmark: Permanent Income Hypothesis (PIH) preserves government net wealth at its initial level (real terms, real per capita, or as a share of GDP), so financial savings offset depletion of resource wealth and total net wealth remains constant as composition shifts from resource wealth to financial wealth.
  - Variable net wealth path to accommodate higher investment: extensions of PIH allow temporary scaling-up of public investment; net wealth initially declines during scaling-up because saving is lower and financial assets do not rise quickly enough to offset resource wealth decline. Fiscal multipliers can mitigate this trade-off by increasing financial savings through higher GDP and nonresource revenues. Net wealth rules can be introduced after the scaling-up period, once net wealth stabilizes.

  Risk-Based Approach
  - Resource-rich countries require larger and more durable buffers because shocks can be large and persistent; required savings depend on resource dependence, level of risk, and risk tolerance.
  - Methods to compute precautionary buffers include:
    - A value-at-risk (VaR) approach and a model-based approach to estimate the minimum buffer that can absorb tail risks in resource revenue volatility (IMF 2012). The buffer should be set high enough to ensure with high probability it is not fully depleted over the forecast horizon.
    - The approach in “The Commodities Roller Coaster” (IMF 2015): calibrate financial savings so that investment returns on financial assets are sufficient to avoid large fiscal adjustment if commodity prices fall. Example: for three major oil exporters, IMF (2015) computed the level of financial assets sufficient to generate investment returns to cover half the lost revenue over five years with 75 percent to 90 percent probability.

### From the Debt Rule to the Operational Rules
- Debt is the cumulative stock of past deficit flows; the (overall) deficit captures the annual change in debt. Currency fluctuations, nondebt financing, and accumulation of financial assets can temporarily alter the one-to-one link, but debt path generally follows the deficit path.
- Any debt target implicitly constrains deficits and ultimately spending; operational rules on deficit and spending must be consistent with the debt rule.
- Time-horizon guidance:
  - Calibration of the debt ceiling is based on prudence against repeated negative shocks over the medium term; default time horizon used earlier was six years.
  - Extending the prudence horizon beyond five or six years is unwarranted because a long-term scenario of repeated negative shocks would lead to a safe debt level of zero.
  - Time horizon for operational rules is a policy decision reflecting national preferences (whether to attain the safe debt target asymptotically in the long term or over a shorter horizon).

### From the Public Debt Ceiling to the Deficit Ceiling
- Four flexible approaches to derive a deficit ceiling from a debt ceiling:
  1. A constant balance target that guides debt to its ceiling in the long term.
  2. A constant balance target that guides debt to its ceiling by a given date.
  3. A constant balance target that guides debt to its ceiling by a given date following a transition period.
  4. A constant balance target that guides debt to its ceiling by a given date while creating space for long-term increases in age-related government spending.

Approach 1: Convergence in the Long Term
- Basic calibration formula (derived in Appendix 2) for the overall or primary balance target as a share of GDP ( b* ) for a given debt-to-GDP target ( d* ) and parameter λ:
  - b* 5 d*
- Notes on λ:
  - λ is alternatively equal to − γ ___ 1 + γ in the case of an overall balance target, and i − γ ___ 1 + γ in the case of a primary balance target, where γ stands for the nominal GDP growth over the long term and i is the nominal interest rate paid on public debt.
- Numerical example:
  - If nominal growth γ = 5 percent and nominal interest rate i = 3 percent, then a 60 percent of GDP debt target would imply:
    - an overall deficit target of about 2.9 percent of GDP
    - a primary deficit target of about 1.1 percent of GDP
  - Convergence can be slow: it would take about 15 years for debt to complete half the distance from an initial debt ratio of 70 percent to a target of 60 percent.
- Caveats:
  - Formula assumes λ is constant over the long term; in reality growth and the interest–growth differential vary with the level of public debt.
  - A constant primary balance rule can imply explosive debt paths (diverging to +/-∞) except when (1) GDP grows faster than the interest rate paid on debt or (2) the starting debt level equals the target d0 = d*. A constant overall balance rule places debt on a convergent path given positive nominal growth γ.

Approach 2: Convergence by a Given Date
- Calibrate the constant balance rule b* so that debt ratio hits target d* after N years. Equation (2):
  - b* 5  ________ (1 1 )N 2 1 [d0 (1 1 )N 2 d*] (2)

*Excerpt from HOW TO CALIbRATE FISCAL RuLES: A PRIMER, International Monetary Fund | December 2017.*

### 2. Debt Path

### 2. Debt Path

### Illustrative assumptions and example case
- Assumptions: nominal growth, 5%; interest rate, 3%; initial debt, 70%; debt target, 60%; initial balance, –5%.
- Figure 6 illustrates debt and overall-balance paths under alternative calibration approaches using the above assumptions (transition/convergence horizons and buffer assumptions differ across approaches).
- Example calibration results:
  - To ensure the hypothetical country hits the 60 percent target within a 15-year span starting from an initial debt of 70 percent of GDP:
    - Approach "convergence by a given date" (instantaneous adjustment) requires maintaining an overall deficit of 2.4 percent or a primary deficit of 0.6 percent of GDP.
    - Versus the previous approach (convergence in the long run) requiring 2.9 percent and 1.1 percent, respectively.

### Approach 1: Convergence in the long run
- Characteristic: balance adjusts to the debt-stabilizing value and debt converges to target in the long run (no forced finite-date convergence).
- Lambda (λ) definitions:
  - λ = −γ / (1 + γ) in the case of an overall balance target.
  - λ = (i − γ) / (1 + γ) in the case of a primary balance target.

### Approach 2: Convergence by a given date (instantaneous adjustment)
- Characteristic: forces convergence to a debt target by a chosen horizon N (e.g., N = 15 years) via an instantaneous adjustment of the fiscal balance to its target.
- Main appeal: forces convergence to be as quick as desired but typically requires a larger fiscal effort than approach 1.
- Lambda (λ) definitions as under Approach 1.

### Approach 3: Convergence by a given date following a transition period
- Feature: allows an initial transition period of T years during which the balance converges gradually (linear adjustment by a constant amount α each year) to a target b*T, then maintains b*T until year N (N ≥ T).
- Balance path (special case linear adjustment):
  - b_t = α t + b_0, when 0 < t < T
  - α_T + b_0 = b*_T, when T ≤ t ≤ N
- Equation (4) main calibration (introduces additional term A to capture delay effect):
  - b*_T = [λ / (1 + λ)^(N − 1)] [d_0 (1 + λ)^N − d*_N + A(T, b_0, N, λ, d_0, d*_N )]
- Practical implications:
  - With T = 5 and N = 15 (illustrative), the adjustment is backloaded: initially higher deficits and debt, followed by tighter policies after the transition to hit the debt target by year N.
  - The additional term A captures the higher fiscal position needed after the transition period when adjustment is gradual rather than instantaneous.
- Note: a linear adjustment schedule is not crucial; what matters is total adjustment αT at end of year T.

### Approach 4: Convergence by a given date following the buildup of fiscal buffers
- Purpose: accommodate expected future increases in spending (e.g., age-related costs) by frontloading fiscal effort to build buffers.
- Setup:
  - Short-to-medium term: age-related costs assumed stable; government can target fixed balance b*.
  - Long term: age-related spending increases, deteriorating fiscal balance; assume unchanged policies so fiscal balance excluding incremental age-related spending is constant.
- Balance path (Equation (5)):
  - b_t = b*, when 0 < t < P
  - b_t = b* − ∆A_t, when P ≤ t ≤ N
  - ∆A_t = A_t − A_0 denotes incremental aging costs; A_0 = A_{P−1}
- Main calibration formula (Equation (6)):
  - b* = [λ / (1 + λ)^(N − 1)] [d_0 (1 + λ)^N − d*_N + S]
  - S = Σ_{t=1}^N (1 + λ)^{N−t} ∆A_t denotes the value in year N of cumulative future increases in long-term age-related spending through N.
- Illustrative simulation parameters and outcome:
  - Assume long-term age-related costs defined as A_t = δ (t + 1 − P) + A_0 for t ≥ P, with P = 6.
  - In the simulation δ is set at about 0.2 percent of GDP per year starting in year P = 6, so total increase in age-related spending would be 2 percent of GDP until year N = 15 (within a 10-year horizon).
  - In present value terms, the cumulative increase in age-related spending through year N would be S / (1 + λ)^N = 22 percent of GDP.
- Practical implications:
  - Early years: lower deficits than the "convergence by a given date" scenario — buffers are built.
  - As age-related costs increase from year P onward, fiscal balance deteriorates progressively while still hitting the debt target by year N.
  - Approach is usually suited to much longer-term perspectives (decades), but N = 15 used here for comparability.
- Lambda (λ) definitions as under Approach 1.

### From the structural deficit ceiling to the nominal deficit ceiling
- Purpose: derive a nominal balance target nb* consistent with a structural balance target sb* while allowing automatic stabilizers to operate in downturns without permitting discretionary expansions that violate structural ceilings.
- Equation (7):
  - nb* = sb* − OG_max [r(1 − η) − e(1 − κ)]
  - OG_max is the maximum output gap during a typical downturn (negative number); r and η are the revenue ratio and its elasticity relative to output; e and κ are the spending ratio and its elasticity relative to output.
- Common proxy (when revenue elasticity ≈ 1 and spending elasticity ≈ 0):
  - Equation (8): nb* = sb* + OG_max × e
- Example:
  - For a government targeting a structural deficit of 1 percent of potential GDP, spending 40 percent of GDP on average, and facing an output gap of −2 percent, equation (8) implies a constant nominal deficit ceiling of 2 percent of GDP.

### From the structural balance ceiling to the expenditure ceiling
- Assumptions:
  - No cyclical component to expenditure (expenditure does not automatically respond to economic conditions).
  - Structural tax ratio r_s is given and constant unless tax policy changes.
- Relationship:
  - e_s = r_s − sb (structural expenditure ratio as percentage of potential GDP).
  - ∆e_s = ∆r_s − ∆sb  (equation (9)).
- Interpretation when ∆r_s = 0 and the country complies with the structural balance rule (sb = sb* and ∆sb = 0):
  - Structural balance rule can be implemented as:
    - A constant ratio of spending-to-potential GDP (∆e_s = 0), or
    - A rule where spending growth equals potential GDP growth (nominal spending grows at nominal potential GDP rate).
- Numerical example:
  - A country collecting 40 percent of potential GDP in revenues with nominal potential GDP growing 4 percent a year and targeting a structural deficit of 1 percent of potential GDP:
    - Consistent options: (1) an expenditure-ratio ceiling of 41 percent of potential GDP; or (2) a ceiling of 4 percent applied to nominal expenditure growth.
- Extensions:
  - Account for structural changes in revenue mobilization (∆r_s ≠ 0): adjust expenditure ceiling upward or downward to keep sb* and debt path consistent.
  - Adapt to transition periods where sb ≠ sb*: temporarily set spending growth below or above trend to steer the structural balance to sb*; calibrate the wedge between spending growth and trend growth accordingly.
- Note: these extensions are featured in the “expenditure benchmark” implemented in the European Union.

### Appendix 1 — Deriving the debt rule threshold: Method One (simulation-based calibration)
- Required macro variables for projections:
  - Advanced and emerging market economies: growth, average interest rate on debt, exchange rate.
  - Low-income countries: also require terms of trade and external financing disbursements.
- Two simulation approaches:
  1. Indirect estimation with a VAR (when quarterly data available)
     - Estimate unrestricted VAR: X_t = A_0 + Σ_{j=1}^p A_j X_{t−j} + ε_t, with ε_t ∼ N(0, Ω).
     - Use estimated variance-covariance matrix Ω̂ to generate N sequences of macro shocks ε_t over six-year projection horizon (N large, e.g., more than 1,000).
     - For each simulation, forecast X_t over six years adding generated shocks each year.
  2. Direct calibration of a multivariate distribution (when only annual data available)
     - Calibrate a multivariate normal (or Student’s t) distribution x ∼ N_k(μ, Σ) using historical means, variances, covariances; draw N sequences of six-year projections by repeated sampling.
- Calibrating the debt ceiling (simulation steps):
  1. Forecast a set of macroeconomic variables over a six-year projection horizon N times using either the VAR-based simulation or draws from the calibrated multivariate distribution.
  2. For each of the N macro projections, generate corresponding primary-balance trajectories using a fiscal reaction function (FRF) and prior-year debt. In the accompanying econometric programs:
     - Annual changes in the primary balance implied by the FRF are constrained based on historical experience to keep projected primary balances realistic.
     - Fiscal shocks can be added directly in the FRF; the distribution of fiscal shocks is calibrated from estimated deviations between actual primary balances and FRF predictions.
  3. Obtain N corresponding debt trajectories (starting at the current debt level) using the system composed of the debt accumulation equation (government budget constraint) and the FRF.
     - Debt accumulation equation:
       - d_t = [1 + ((r_t − g_t) / (1 + g_t))] d_{t−1} − pb_t + SFA_t
       - where d_t is debt (ratio to GDP), r_t is average effective real interest rate on debt, g_t is real GDP growth rate, pb_t is primary balance (ratio to GDP), and SFA_t is stock-flow adjustment (ratio to GDP).
     - The debt accumulation equation includes a constant stock-flow adjustment each period to account for potential realization of contingent liabilities.

*HOW TO CALIBRATE FISCAL RULES: A PRIMER, International Monetary Fund | December 2017*

### 4. If the 95th debt percentile (or other chosen percen-

### howtonote1808 - 4. If the 95th debt percentile (or other chosen percen-

### Calibrating the debt ceiling using fan charts and percentiles
- Iterative adjustment rule:
  - If the 95th debt percentile (or other chosen percentile, given risk tolerance) of the debt ratio distribution is significantly below the maximum debt limit (MDL) in all years of the projection horizon [or is significantly above the MDL in at least one year], the starting level of debt is increased [decreased] by a small amount (0.3 percent), and steps 1–3 are repeated based on the new starting level.
  - Steps 1–4 are repeated until the 95th percentile of the debt level falls into a small interval around the MDL in at least one year of the medium-term projection horizon: Debt95 ∈ [MDL − 0.4; MDL + 0.4], without significantly breaching the MDL in any year.
  - The starting level of debt satisfying this criterion is called the debt ceiling; that is, the level of debt from which its projection does not exceed the MDL with 95 percent likelihood over the medium-term projection horizon.
  - The safety margin is computed as the MDL minus the debt ceiling.
- Use of fan charts:
  - Fan charts can determine the probability of breaching the maximum debt limit, conditional on any starting level (for example, using the current debt level to determine the probability that debt will exceed the debt limit in all years over the projection horizon).

### Method Two: calibrating the debt ceiling via primary balance trajectories
- Procedure (calibration steps):
  1. A set of macroeconomic variables is forecast over a six-year projection horizon N times using an estimated VAR, including shocks each period (the VAR estimation is similar to that in Method One).
  2. The N sets of forecasts are used to generate N trajectories of the primary balance, using either an estimated or normative fiscal reaction function (see Box 3 in the text).
  3. The N corresponding trajectories of debt (starting at the current debt level) are obtained by the system of simultaneous equations formed by the debt accumulation equation (government budget constraint) and the fiscal reaction function (which depends on the lagged value of debt).
  4. If the 95th percentile of primary balances (or other chosen percentile, given risk tolerance) is significantly below the maximum feasible primary surplus (MFPS) (that is, falls below the interval pb95 ∈ [MFPS − 0.4; MFPS + 0.4]) in all years of the projection horizon [or is significantly above the MFPS in any one year], the starting level of debt is increased [decreased] by 0.3 percent, and steps 1–3 are repeated based on the new starting level.
- Iteration stopping rule and definitions:
  - Steps 1–4 are repeated until the 95th percentile of the primary balance falls into the small interval around the MFPS in at least one year of the medium-term projection horizon: pb95 ∈ [MFPS − 0.4; MFPS + 0.4], without significantly breaching the MFPS in any year.
  - The starting level of debt satisfying this criterion is called the debt ceiling.
- Typical MFPS benchmarks:
  - For advanced economies a typical MFPS would be about 4 percent of GDP.
  - For emerging market economies it would be 2 percent of GDP (see Escolano and others 2014).

### Debt dynamics framework and operational translations
- Debt dynamics (nominal and as share of GDP) as stated:
  - D_t = (1 + i_t) D_{t−1} − PB_t ⇒ d_t = (1 + i_t)/(1 + γ_t) d_{t−1} − pb_t
  - More generally written as d_t = (1 + λ_t) d_{t−1} − b_t  (A.2.1)
    - with 1 + λ_t = 1/(1 + γ_t) when b_t = ob_t (overall balance)
    - and 1 + λ_t = (1 + i_t)/(1 + γ_t) when b_t = pb_t (primary balance)
- Long-term constant balance ratio to stabilize debt:
  - b* = λ d*  (A.2.2)
  - Specifically,
    - ob* = (−2 γ)/(1 + γ) d*
    - pb* = (i − γ)/(1 + γ) d*
- Convergence to a target by finite horizon N (time-invariant λ):
  - d_N = d_0 (1 + λ)^N − ∑_{t=1}^N (1 + λ)^{N−t} b_t  (A.2.3)
  - From this, the constant balance ratio b* that achieves target d_N* in N periods:
    - b* = λ/( (1 + λ)^N − 1 ) ( d_0 (1 + λ)^N − d_N* )  (A.2.4)
  - Decomposition of adjustment:
    - b* − b_0 = [λ d_0 − b_0] + ( d_0 − d_N* ) / ( ((1 + λ)^N − 1)/λ )  (A.2.5)
- Convergence with transition period T and linear annual adjustment α:
  - Balance path: b_t = α t + b_0 for 0 < t < T; α_T + b_0 = b_T* for T ≤ t ≤ N  (A.2.6)
  - Resulting α_T expression decomposes into components:
    - α_T = [λ d_0 − b_0] + ( α ∑_{t=1}^T (1 + λ)^{N−t} (T − t) ) / ( ((1 + λ)^N − 1)/λ ) + ( d_0 − d_N* ) / ( ((1 + λ)^N − 1)/λ )  (A.2.7)
  - Interpretation: total adjustment b_T* − b_0 equals sum of (I) gap to debt-stabilizing balance ratio at initial debt, (II) additional term from gradual vs instantaneous adjustment, and (III) extra adjustment due to distance from debt target. As T increases, annual α falls.
- Convergence allowing buildup of fiscal buffers (age-related costs ∆A_t):
  - Balance path: b_t = b* − ∆A_t  (A.2.8)
  - Solving yields:
    - b* = λ/( (1 + λ)^N − 1 ) ( d_0 (1 + λ)^N − d_N* + S )  (A.2.9)
    - with S = ∑_{t=1}^N (1 + λ)^{N−t} ∆A_t  (value in year N of cumulative increase in long-term age-related costs through N)
  - Decomposition: b* − b_0 = [λ d_0 − b_0] + ( d_0 − d_N* ) / ( ((1 + λ)^N − 1)/λ ) + 1/S term reflecting additional upfront adjustment due to aging costs.
  - Special case: if age-related costs increase linearly by annual δ starting from year P, an analytical solution for S is provided in the note.
- From structural deficit ceiling to nominal deficit ceiling:
  - Definitions:
    - OG = (Y − Y^s)/Y^s
    - Structural revenues: R^s/R = (Y^s/Y)^η
    - Structural expenditures: E^s/E = (Y^s/Y)^κ
  - Structural balance ratio (sb) expression:
    - sb = r (1 + (1 − η) OG ) − e (1 + (1 − κ) OG ) = nb + OG [ r (1 − η) − e (1 − κ) ]  (A.2.11)
      - where r and e denote revenue and expenditures as shares of GDP, nb = r − e
  - Mapping between structural balance target sb* and nominal balance target nb* (medium-term averages):
    - nb* = sb* − OG [ r (1 − η) − e (1 − κ) ]

### Operational parameters and tolerances preserved from text
- Iteration adjustment step: 0.3 percent change in starting debt level when percentile is significantly off target.
- Small interval around MDL and MFPS defined as ±0.4 (i.e., [MDL − 0.4; MDL + 0.4] and [MFPS − 0.4; MFPS + 0.4]).
- Typical MFPS values: 4 percent of GDP (advanced economies) and 2 percent of GDP (emerging market economies).
- Projection horizon examples: six-year horizon used in Method Two (N = 6 in the methodological description).

*International Monetary Fund | How to Calibrate Fiscal Rules: A Primer (excerpts from howtonote1808 - 4. If the 95th debt percentile (or other chosen percen-)*

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_Source: https://www.imf.org/-/media/files/publications/howtonotes/howtonote1808.pdf_