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---

### Data
- Sample: country-year level, annual frequency, spanning from 1980 to 2024.
- Coverage: 90 economies accounting for more than 90 percent of global sovereign debt (Table A.2).
- Preferred debt measure: ratio of (end-of-period) general government gross debt to nominal GDP.
- Key data source: April 2025 vintage of the IMF’s World Economic Outlook (WEO) database.
- Financial factors included:
  - IMF’s Financial Conditions Index (FCI).
  - Financial Stress Index (FSI) from Ahir et al. (2023).
  - Sovereign spreads: difference between 10-year government bond yields and the 10-year United States treasury yield (5-year spreads used where 10-year yields not available; for the United States, actual 10-year treasury yields used).
- Political factors included:
  - World Uncertainty Index (WUI) from Ahir, Bloom, and Furceri (2022).
  - Reported Social Unrest Index from Barrett et al. (2022).
- Note: Some dependent-variable analyses use realized future values of variables to evaluate mechanisms through which shocks affect debt risks.
- Reduced-sample restriction for comparability (2009–2024): 47 countries (24 advanced economies and 23 emerging market and developing economies), still covering more than 90 percent of global debt.

### Empirical framework (quantile regression / location-scale model)
- Target quantiles estimated: 5th, 25th, 50th, 75th, and 95th percentiles.
- Debt-at-risk defined as: the 95th predicted quantile of the forward debt-to-GDP ratio over horizons of 1 to 5 years ahead.
- Upside risk defined as: difference between the 95th and the 50th percentiles.
- Downside risk defined as: difference between the 50th and the 5th percentile.
- Baseline model (equation (2)):
  - d_{i,t+h} = α_i + X'_{i,t} β + (δ_i + X'_{i,t} γ) ε_{i,t+h}
  - α_i and δ_i: country fixed effects.
  - X_{i,t} includes conditioning variable(s) x_{i,t} and initial debt-to-GDP ratio d_{i,t}.
  - τ-th quantile (equation (3)): Q^d_{i,t+h}(τ|X_{i,t}) = (α_i + δ_i q(τ)) + X'_{i,t} β + X'_{i,t} γ q(τ), where q(τ)=F^{-1}_ε(τ).
- Interpretation:
  - Location parameter β captures effect across all quantiles (akin to OLS coefficients).
  - Scale parameter γ allows predictor effects to vary across quantiles, enabling asymmetric risk.
  - γ = 0 reduces to linear regression (no asymmetry).
  - Sign patterns of γ relative to β determine whether mean and variance move together or in opposite directions.
- Modeling approach:
  - Equation (2) estimated separately for each predictor (including initial debt) to avoid sample-size loss from overlapping data requirements and to focus on predictive roles rather than marginal/causal effects.

### Estimation procedure and smoothing/aggregation
- Three-step estimation procedure (Machado and Santos Silva (2019)):
  1. Estimate α_i and β using standard fixed effects estimator.
  2. Estimate δ_i and γ by applying fixed effects to absolute residuals from step 1 to capture conditional heteroskedasticity.
  3. Estimate q(τ) from sample quantiles of standardized residuals; inference uses two-ways clustered standard errors at the country level (Rios-Avila, Siles, and Canavire-Bacarreza (2024)).
- Smoothing quantiles into a PDF:
  - Fit predicted quantiles to the skewed t-distribution of Azzalini and Capitanio (2003) using 4 parameters: location μ, scale σ, fatness ν, and shape α.
  - Parameters chosen to minimize squared distance between predicted quantiles (5, 25, 75, 95 percentiles) and F^{-1}(τ; μ, σ, α, ν) (equation (4)).
  - Reported global debt-at-risk and its use in Section 5.1 to predict fiscal crises are obtained directly from the quantile regression prediction (Equation (3)) and are invariant to the smoothing distribution.
- Pooling individual predictor-based densities into a single pooled density:
  - f^pooled_{i,t+h}(d) = Σ_m η^m_{i,h} f^m_{i,t+h}(d), with weights η^m_{i,h} ≥ 0 and Σ_m η^m_{i,h} = 1.
  - Weights chosen to maximize out-of-sample predictive accuracy using rolling windows and prior 20 years of data, following Crump et al. (2023) and Hengge (2024) (equation (6)).
  - Optimization imposes an implicit Lasso-type penalty (nonnegative, sum-to-one constraints) that regularizes and prevents dominance by any single forecast; recursive algorithm of Conflitti, De Mol, and Giannone (2015) used to compute weights.
- Aggregation to global or group levels (equation (7) and (8)):
  - Approximate global quantile for model m via GDP-weighted average of country-level quantiles: ˆQ^d_{global,t+h}(τ) = Σ_{i=1}^I ω_{i,t} ˆQ^d_{i,t+h}(τ).
  - Re-center quantiles so median corresponds to WEO projections; fit global quantiles to skewed t-distribution to get ˆf^m_{global,t+h}(d).
  - Pooled global density: ˆf^{pooled}_{global,t+h}(d) = Σ_{m=1}^M ω^{m}_{global,h} ˆf^m_{global,t+h}(d), where ω^{m}_{global,h} = Σ_{i=1}^I ω_{i,t} η^m_{i,h}.
  - Same approach applied for aggregates of AEs and EMDEs.
- Rolling-window forecasting for weights: for each country and horizon h, out-of-sample predictions computed from prior 20 years for years 2005 onward; weights maximize cumulative predictive density scores across years 2005+h to 2024.

### Quantile regression findings (Section 4.1) — key empirical results
- Key takeaway: several economic and financial factors are consistently and asymmetrically associated with higher debt upside risks up to a forecast horizon of three years.
- Coefficients reported for 5th, 50th, and 95th percentiles; conditioning factors standardized so coefficients represent percentage point increase in a particular percentile of future debt-to-GDP associated with a one standard deviation increase in the regressor.
- Reporting cautions: estimates reflect strength of relation between current factors and future debt; not to be interpreted as causal effects.
- Financial Conditions Index (FCI):
  - FCI has statistically significant effects on the location of the future debt distribution at all horizons.
  - FCI effects on the scale of the distribution remain significant up to a three-year horizon.
  - Economic magnitude example: a one-standard deviation increase in the FCI—similar to what was experienced in Spain in 2011—correlates with a 3 percentage points of GDP increase in debt-at-risk.
- Financial Stress Index (FSI):
  - Similar directional results as FCI; effects on the scale of the distribution are weaker.
  - A one-standard deviation increase in the FSI is associated with an increase of growth-at-risk of about 0.2 percentage point (Table A.6).
- Sovereign spreads:
  - A one-standard deviation increase in sovereign spreads—similar to the spike observed in Sri Lanka in 2022—correlates with approximately a 2.6 percentage points increase in debt risks as a percentage of GDP after one year.
  - Sovereign spreads have statistically significant effects on the location of the future debt distribution at both the 1-year and 3-year horizons; effects on the scale are significant only at the 1-year horizon.
- Channels:
  - Adverse financial developments are associated with higher debt risks largely through raising growth, deficit, and interest rate risks.
  - Evidence suggests financial stress raises debt risks largely by increasing downside risk to growth (“growth-at-risk”).
- Time horizon: asymmetric associations with higher debt upside risks are most pronounced up to a three-year forecast horizon.
- Additional notes:
  - Figure 3 and Table 1 provide detailed coefficient estimates and location/scale parameter estimates.
  - Table A.6 presents results on how FCI/FSI affect growth, primary balance, interest rates, and unidentified debt; these channel results indicate financial stress increases downside growth risks which in turn raise debt risks.

### Effects of political conditions and economic drivers
- Political uncertainty:
  - Increases in the World Uncertainty Index (WUI) are linked to asymmetric rises in the future debt distribution across all horizons.
  - A one standard deviation increase in the Reported Social Unrest Index (RSUI) is associated with a statistically significant increase of approximately 1.9 percentage points of GDP in the 3-year ahead debt-at-risk; this effect is not statistically different from those observed in other quartiles of the future debt distribution (γ=0).
- Economic drivers and proximate debt dynamics:
  - Initial debt levels and lower growth rates have long-lasting and larger effects on the right tail of the distribution.
  - The scale parameter for initial debt is statistically significant across all horizons.
  - The parameter for GDP growth is modestly significant at the 3-year horizon.
  - A one-standard-deviation change in GDP growth is associated with an increase of more than 5 percentage points of GDP in the 3-year ahead debt-at-risk.
  - Higher primary balances reduce debt across all quantiles of the debt distribution.

### Aggregate debt distributions (global and by country group)
- Predicted three-years ahead global debt-to-GDP ratio distribution for 2024:
  - Global debt-at-risk is estimated at 116.6 percent of GDP.
  - Median projection (calibrated to the IMF WEO baseline) is 97.5 percent (P50).
  - Upside risks to public debt exceed downside risks of about 15 percentage points; debt risks are tilted to the upside.
- Drivers of upside risks (P95 − P50) for the global sample: the primary deficit and financial conditions are the main contributors (Figure 5).
- Time variation and heterogeneity:
  - Global debt-at-risk has steadily risen since 2009; upside risks spike during global shocks (GFC 2009, COVID-19 2020) and downside risks diminish.
  - Financial conditions were the main influence in 2009; growth and primary balance were more significant in 2020.
- Advanced vs Emerging market economies (three-year ahead, 2024):
  - Advanced economies: projected debt-at-risk = 131.1 percent of GDP.
  - Emerging market economies: projected debt-at-risk = 95.8 percent of GDP.
  - These represent increases of approximately 20 percentage points (advanced) and 17 percentage points (emerging) relative to the median projection.
  - Distributions show positive skewness: advanced economies = 0.15; emerging market economies = 0.13.
- Conditioning factor contributions differ:
  - Advanced economies: primary deficits and financial conditions are the two largest contributors to upside risks.
  - Emerging market economies: uncertainty and primary deficits are the most significant contributors.
- Post-pandemic trends:
  - Debt-at-risk for advanced economies has broadly retreated from pandemic peaks.
  - Debt-at-risk for emerging market economies has increased due to higher projected debt levels and larger risks from lower growth and higher primary deficits.

### Debt‑at‑Risk as a predictor of fiscal crises
- Fiscal crisis definition (binary, country-year; crisis = 1 if occurs over next two years) — crisis if any of four criteria met:
  1. credit event with nonrepayment or creditor losses including restructuring;
  2. exceptionally large official financing from IMF or European Union;
  3. implicit default on domestic debt with high inflation or domestic arrears;
  4. loss of market confidence (loss of market access or very large spikes in sovereign yields).
- Logit estimation findings:
  - Independent variable: upside risk to the debt projection = (P95 − P50) at one- and two-year horizons.
  - Upside risks to debt across all models are positive and statistically significantly correlated with fiscal crisis indicator.
  - Magnitude: a one percentage point of GDP increase in (P95 − P50) is associated with an 8-10 percentage points increase in the probability of a fiscal crisis within the next two years.
  - Context: current estimated gap between P95 and median for the one-year ahead global debt distribution ≈ 20 percentage points of GDP; frequency of a crisis in the sample = 5 percent.
- Bayesian Model Averaging (BMA) results:
  - Considered three predicted debt-at-risk quantile measures (P95 level; P95 − P50; interaction (P95 − P50) × P95), conditioning variables (8 variables), and 24 control variables.
  - In-sample estimation period: 1986-2024.
  - Posterior inclusion probability (PIP) for the interaction term ((P95 − P50) × P95) for each conditioning variable and the combined distribution is 1 at both a 1-year and 2-year forecast horizon.
  - Out-of-sample (20-year rolling window forecasts) yields PIPs remaining at 1 for all cases except one on inflation for the one-year ahead estimation.
- Random Forest (RF) with Boruta feature selection:
  - Predictor set reduced from nearly 780 candidate variables to 188 factors via Boruta.
  - RF uses out-of-bag permuted predictor importance (scaled 0–100) to assess variable importance.
  - Debt-at-risk is the most useful metric in predicting fiscal crises among a wide range of economic variables.
  - Key predictive measures: predicted P95 and the difference (P95 − P50) have the highest variable importance.
  - Introducing debt-at-risk reduces the relative importance of other variables such as the debt-to-GDP level.

### Extensions, sample coverage, and robustness
- Extensions presented:
  1. Evaluate effectiveness of debt-at-risk in predicting fiscal crises relative to other macro variables.
  2. Construct debt-at-risk measures using a reduced set of conditioning factors to expand sample coverage to 175 countries.
  3. Enhance the quantile regression framework to accommodate non-linear effects of conditioning factors varying by country characteristics.
- Extended sample:
  - Baseline sample includes 90 economies (restriction due to sovereign bond yield availability).
  - Using only available conditioning variables allows extension to 175 countries (many low-income).
  - PDFs for a highly indebted low-income country with only economic variables still show asymmetric debt distributions even without financial variables.
  - World simple average of debt-at-risk for the extended sample follows trends similar to the world GDP-weighted average, with upside and downside risk changes less pronounced during the GFC when financial stress measures are unavailable for several countries.
- Annual dataset: debt-at-risk measures available for 175 countries since 2009.

### Heterogeneity Analysis (Section 5.3)
- Heterogeneity by initial debt level:
  - Method: interact conditioning variables and initial debt with quartile indicators; quartile thresholds correspond to debt values of 33 and 69 percent of GDP (Q1 = low initial debt; Q4 = high initial debt).
  - Main findings:
    - Location parameters of growth, primary balance, and inflation tend to be statistically significantly larger for countries with higher initial debt.
    - Scale parameters do not systematically vary with initial debt level; exceptions:
      - Spread at the one-year horizon: scale larger and more precisely estimated for higher debt levels.
      - Growth at the three-year horizon: scale larger and more precisely estimated for higher debt levels.
    - Overall implication: initial debt amplifies the impact of economic factors on the entire future debt distribution (including debt-at-risk), with effects typically not significantly different between left and right tails.
- Heterogeneity by country income group (AEs vs EMDEs):
  - Method: interact conditioning variables and initial debt with country-group indicators (AE, EMDE).
  - Main findings:
    - Location differences:
      - Sovereign spreads: statistically significant effects in AEs in the short term; in EMDEs, effects more pronounced in the medium term.
      - Uncertainty: larger and more persistent effects in EMDEs.
      - Initial debt: larger effects in AEs (reflecting higher debt values in AEs).
      - Primary balance and growth: point estimates tend to be larger in EMDEs, but differences not statistically significant.
    - Scale differences:
      - No systematic differences for most variables.
      - Exception: initial debt suggests larger asymmetric effects on the right tails for EMDEs compared to AEs.
- Implications for policy and research:
  - Elevated debt levels today amplify the negative effects of weaker growth or tighter financial conditions on future debt ratios.
  - Debt-at-risk can be used to:
    - Quantify size of debt risks in severely adverse scenarios and assess main determining factors.
    - Serve as a robust predictor of fiscal crises and an early-warning tool to monitor and prevent crises.
    - Be combined with complementary tools and predictive scenarios used in debt sustainability analyses (IMF 2022).
    - Estimate likelihoods of alternative scenarios using the predictive density (Adrian et al. (2025)).
- Suggested methodological extensions:
  - Incorporate additional conditioning factors and further interaction effects (for example, state dependencies around periods of economic downturns).
  - Combine debt-at-risk with routine predictive scenarios developed for debt sustainability analyses to provide narratives on how debt risks could evolve under alternative macro-financial conditions or fiscal consolidation paths.
  - Consider additional heterogeneity interactions for specific sub-groups within AEs and EMDEs.

### Selected quantitative tables and statistics (highlights)
- Table 1 (selected entries, Location β and Scale γ; standard errors in parentheses):
  - Financial Conditions: Location β for horizon 1 = 1.189 ∗∗∗ (0.294); Scale γ for horizon 1 = 0.314 (0.194).
  - Financial Stress: Location β for horizon 1 = 1.021 ∗∗∗ (0.241); Scale γ for horizon 1 = 0.079 (0.187).
  - Debt-to-GDP: Location β for horizon 1 = 0.901 ∗∗∗ (0.016); Scale γ for horizon 1 = 0.092 ∗∗∗ (0.026).
  - Primary Balance: Location β for horizon 1 = -2.365 ∗∗∗ (0.391); Scale γ for horizon 1 = -0.062 (0.170).
  - GDP Growth: Location β for horizon 1 = -1.816 ∗∗∗ (0.434); Scale γ for horizon 1 = -0.507 (0.404).
  - Inflation: Location β for horizon 1 = 1.235 ∗∗∗ (0.046); Scale γ for horizon 1 = 0.537 ∗∗∗ (0.055).
- Global three-year-ahead summary for 2024:
  - Global debt-at-risk = 116.6 percent of GDP.
  - Median projection (P50) = 97.5 percent of GDP.
  - Upside risks exceed downside risks by about 15 percentage points.
- Appendix summary statistics (Table A.1, selected entries):
  - Financial Conditions: Mean 0.008; Median -0.045; SD 0.694; Obs 1,262.
  - Financial Stress: Mean 0.031; Median 0.000; SD 0.098; Obs 2,844.
  - Spread: Mean 3.557; Median 1.531; SD 9.812; Obs 2,487.
  - Debt-to-GDP: Mean 55.527; Median 48.677; SD 35.728; Obs 2,876.
  - GDP Growth: Mean 3.406; Median 3.564; SD 4.311; Obs 3,721.
  - Inflation: Mean 49.931; Median 4.230; SD 1,142.129; Obs 3,719.
- Table A.6 channel example:
  - Financial Conditions Index coefficient on Primary Balance Q5 = -2.343 ∗∗∗ (0.476); on Interest Q95 = 1.465 ∗∗ (0.638).
  - Reported Social Unrest Index on Growth Q5 = -0.328 ∗∗∗ (0.104); on Primary Balance Q5 = -1.504 ∗∗∗ (0.406).

*Source: wpiea2025086-print-pdf — https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025086-print-pdf.pdf*

### Section 3 discusses the quantile regression estimation strategy.  Section 4 presents the debt-

### wpiea2025086-print-pdf - Section 3 discusses the quantile regression estimation strategy.  Section 4 presents the debt-

### Data
- Sample: country-year level, annual frequency, spanning from 1980 to 2024.
- Coverage: 90 economies accounting for more than 90 percent of global sovereign debt (Table A.2).
- Preferred debt measure: ratio of (end-of-period) general government gross debt to nominal GDP.
- Key data source: April 2025 vintage of the IMF’s World Economic Outlook (WEO) database.
- Financial factors included:
  - IMF’s Financial Conditions Index (FCI).
  - Financial Stress Index (FSI) from Ahir et al. (2023).
  - Sovereign spreads: difference between 10-year government bond yields and the 10-year United States treasury yield (5-year spreads used where 10-year yields not available; for the United States, actual 10-year treasury yields used).
- Political factors included:
  - World Uncertainty Index (WUI) from Ahir, Bloom, and Furceri (2022).
  - Reported Social Unrest Index from Barrett et al. (2022).
- Note: Some dependent-variable analyses use realized future values of variables to evaluate mechanisms through which shocks affect debt risks.
- Reduced-sample restriction for comparability (2009–2024): 47 countries (24 advanced economies and 23 emerging market and developing economies), still covering more than 90 percent of global debt.

### Empirical framework (quantile regression / location-scale model)
- Target quantiles estimated: 5th, 25th, 50th, 75th, and 95th percentiles.
- Debt-at-risk defined as: the 95th predicted quantile of the forward debt-to-GDP ratio over horizons of 1 to 5 years ahead.
- Upside risk defined as: difference between the 95th and the 50th percentiles.
- Downside risk defined as: difference between the 50th and the 5th percentile.
- Baseline model (equation (2)):
  - d_{i,t+h} = α_i + X'_{i,t} β + (δ_i + X'_{i,t} γ) ε_{i,t+h}
  - α_i and δ_i: country fixed effects.
  - X_{i,t} includes conditioning variable(s) x_{i,t} and initial debt-to-GDP ratio d_{i,t}.
  - τ-th quantile (equation (3)): Q^d_{i,t+h}(τ|X_{i,t}) = (α_i + δ_i q(τ)) + X'_{i,t} β + X'_{i,t} γ q(τ), where q(τ)=F^{-1}_ε(τ).
- Interpretation:
  - Location parameter β captures effect across all quantiles (akin to OLS coefficients).
  - Scale parameter γ allows predictor effects to vary across quantiles, enabling asymmetric risk.
  - γ = 0 reduces to linear regression (no asymmetry).
  - Sign patterns of γ relative to β determine whether mean and variance move together or in opposite directions.
- Estimation procedure (three-step; Machado and Santos Silva (2019)):
  1. Estimate α_i and β using standard fixed effects estimator.
  2. Estimate δ_i and γ by applying fixed effects to absolute residuals from step 1 to capture conditional heteroskedasticity.
  3. Estimate q(τ) from sample quantiles of standardized residuals; inference uses two-ways clustered standard errors at the country level (Rios-Avila, Siles, and Canavire-Bacarreza (2024)).
- Smoothing quantiles into a PDF:
  - Fit predicted quantiles to the skewed t-distribution of Azzalini and Capitanio (2003) using 4 parameters: location μ, scale σ, fatness ν, and shape α.
  - Parameters chosen to minimize squared distance between predicted quantiles (5, 25, 75, 95 percentiles) and F^{-1}(τ; μ, σ, α, ν) (equation (4)).
  - Reported global debt-at-risk and its use in Section 5.1 to predict fiscal crises are obtained directly from the quantile regression prediction (Equation (3)) and are invariant to the smoothing distribution.
- Modeling approach:
  - Equation (2) estimated separately for each predictor (including initial debt) for two reasons:
    1. Including all predictors simultaneously would substantially reduce sample size due to limited overlapping availability.
    2. Focus is on predictive role of each factor for the future debt distribution rather than marginal/causal effects.
- Pooling individual predictor-based densities into a single pooled density (equation (5)):
  - f^pooled_{i,t+h}(d) = Σ_m η^m_{i,h} f^m_{i,t+h}(d)
  - Weights η^m_{i,h} sum to one and are non-negative.
  - Weights chosen to maximize out-of-sample predictive accuracy using rolling windows and prior 20 years of data, following Crump et al. (2023) and Hengge (2024).
  - For each country and horizon h, for years T=2005,...,2024, compute out-of-sample predictive densities ˆp^m_{T+h|T}(d) and choose weights η to maximize Σ_{t=2005+h}^{2024} Σ_{m=1}^M η^m_{i,h} ˆp^m_{T+h|T}(d) subject to η≥0 and Σ η = 1 (equation (6)).
  - Optimization imposes an implicit Lasso-type penalty (nonnegative, sum-to-one constraints) that regularizes and prevents dominance by any single forecast; recursive algorithm of Conflitti, De Mol, and Giannone (2015) used to compute weights.
- Aggregation to global or group levels (section 3.1):
  - Approximate global quantile for model m via GDP-weighted average of country-level quantiles (equation (7)): ˆQ^d_{global,t+h}(τ) = Σ_{i=1}^I ω_{i,t} ˆQ^d_{i,t+h}(τ), where ω_{i,t} is country i’s nominal US dollar GDP share among in-sample countries.
  - Re-center quantiles so median corresponds to WEO projections.
  - Fit global quantiles to skewed t-distribution to get ˆf^m_{global,t+h}(d).
  - Pooled global density (equation (8)): ˆf^{pooled}_{global,t+h}(d) = Σ_{m=1}^M ω^{m}_{global,h} ˆf^m_{global,t+h}(d), where ω^{m}_{global,h} = Σ_{i=1}^I ω_{i,t} η^m_{i,h}.
  - Same approach for aggregates of advanced economies (AEs) and emerging market and developing economies (EMDEs).
  - For historical comparability, conditioning variables must be available for all countries across 2009–2024, leading to the 47-country reduced sample (still >90 percent of global debt).

### Estimation details and predictive weighting
- Rolling-window forecasting for weights: for each country and horizon h, out-of-sample predictions computed from prior 20 years for years 2005 onward.
- Weights maximize cumulative predictive density scores across years 2005+h to 2024.
- Non-negativity and sum-to-one constraints generate implicit shrinkage, improving robustness.
- Recursive algorithm of Conflitti, De Mol, and Giannone (2015) used for computational efficiency.

### Results — Quantile regression findings (Section 4.1)
- Key takeaway: several economic and financial factors are consistently and asymmetrically associated with higher debt upside risks up to a forecast horizon of three years.
- Estimated quantiles reported for the 5th, 50th, and 95th percentiles across forecast horizons (Figure 3); conditioning factors standardized so coefficients represent percentage point increase in a particular percentile of future debt-to-GDP associated with a one standard deviation increase in the regressor.
- Reporting cautions: estimates reflect strength of relation between current factors and future debt; not to be interpreted as causal effects.
- Financial Conditions Index (FCI) results:
  - FCI has statistically significant effects on the location of the future debt distribution at all horizons.
  - FCI effects on the scale of the distribution remain significant up to a three-year horizon.
  - Economic magnitude example: a one-standard deviation increase in the FCI—similar to what was experienced in Spain in 2011—correlates with a 3 percentage points of GDP increase in debt-at-risk.
- Financial Stress Index (FSI) results:
  - Similar directional results as FCI; effects on the scale of the distribution are weaker (Figure 3, Panel B).
  - A one-standard deviation increase in the FSI is associated with an increase of growth-at-risk of about 0.2 percentage point (Table A.6).
- Channels: adverse financial developments are associated with higher debt risks largely through raising growth, deficit, and interest rate risks; evidence suggests financial stress raises debt risks largely by increasing downside risk to growth (“growth-at-risk”).
- Time horizon: asymmetric associations with higher debt upside risks are most pronounced up to a three-year forecast horizon.
- Additional notes:
  - Figure 3 and Table 1 provide detailed coefficient estimates and location/scale parameter estimates.
  - Table A.6 presents results on how FCI/FSI affect growth, primary balance, interest rates, and unidentified debt; these channel results indicate financial stress increases downside growth risks which in turn raise debt risks.

*Source: wpiea2025086-print-pdf — https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025086-print-pdf.pdf*

### 9. Table A.3-A.5 report the estimated regressions coefficients for one-year, three-year and five-year future

### wpiea2025086-print-pdf - 9. Table A.3-A.5 report the estimated regressions coefficients for one-year, three-year and five-year future

### Effects of financial conditions and sovereign spreads
- A deterioration in financial conditions is associated with an increase in the left tail of the future primary balance distribution and the right tail of the future interest rate distribution (financial stress makes the primary balance and interest rate distribution riskier).
- Increases in spreads are linked to asymmetric rises in debt-at-risk in the short term (Panel C, Figure 3).
  - A one-standard deviation increase in sovereign spreads—similar to the spike observed in Sri Lanka in 2022—correlates with approximately a 2.6 percentage points increase in debt risks as a percentage of GDP after one year.
- Sovereign spreads have statistically significant effects on the location of the future debt distribution at both the 1-year and 3-year horizons; their effects on the scale of the distribution are statistically significant only at the 1-year horizon (Table 1).
- Sovereign spreads contribute to an increase in future growth-at-risk and the left tail of the primary balance distribution.
- Higher spreads elevate government debt servicing costs and increase debt levels (Gourinchas, Philippon, and Vayanos 2017) and raise borrowing costs for households and firms, dampening economic activity (Arellano, Bai, and Bocola 2024).

### Political conditions and uncertainty
- Political uncertainty influences the future debt distribution.
  - Increases in the World Uncertainty Index (WUI) are linked to asymmetric rises in the future debt distribution across all horizons (Panel D, Figure 3).
  - A one standard deviation increase in the Reported Social Unrest Index (RSUI) is associated with a statistically significant increase of approximately 1.9 percentage points of GDP in the 3-year ahead debt-at-risk; this effect is not statistically different from those observed in other quartiles of the future debt distribution (γ=0; see Table 1).

### Economic drivers and proximate debt dynamics
- Initial debt levels and lower growth rates have long-lasting and larger effects on the right tail of the distribution (Panels F and H).
  - The scale parameter for initial debt is statistically significant across all horizons.
  - The parameter for GDP growth is modestly significant at the 3-year horizon (see Table 1).
  - A one-standard-deviation change in GDP growth is associated with an increase of more than 5 percentage points of GDP in the 3-year ahead debt-at-risk.
- Higher primary balances reduce debt across all quantiles of the debt distribution, highlighting the positive impact of fiscal adjustments.
- Key insight: factors influencing debt dynamics can meaningfully and persistently shift the debt distribution, often asymmetrically.

### Aggregate debt distributions (global and by country group)
- Predicted three-years ahead global debt-to-GDP ratio distribution for 2024:
  - Global debt-at-risk is estimated at 116.6 percent of GDP.
  - Median projection (calibrated to the IMF WEO baseline) is 97.5 percent (P50).
  - Upside risks to public debt exceed downside risks of about 15 percentage points; debt risks are tilted to the upside.
- Drivers of upside risks (P95 − P50) for the global sample: the primary deficit and financial conditions are the main contributors (Figure 5).
- Time variation and heterogeneity:
  - Global debt-at-risk has steadily risen since 2009 (Figure 6, Panel A); upside risks spike during global shocks (GFC 2009, COVID-19 2020) and downside risks diminish.
  - Financial conditions were the main influence in 2009; growth and primary balance were more significant in 2020.
- Advanced vs Emerging market economies (three-year ahead, 2024):
  - Advanced economies: projected debt-at-risk = 131.1 percent of GDP.
  - Emerging market economies: projected debt-at-risk = 95.8 percent of GDP.
  - These represent increases of approximately 20 percentage points (advanced) and 17 percentage points (emerging) relative to the median projection.
  - Distributions show positive skewness: advanced economies = 0.15; emerging market economies = 0.13.
- Conditioning factor contributions differ:
  - Advanced economies: primary deficits and financial conditions are the two largest contributors to upside risks.
  - Emerging market economies: uncertainty and primary deficits are the most significant contributors.
- Post-pandemic trends:
  - Debt-at-risk for advanced economies has broadly retreated from pandemic peaks.
  - Debt-at-risk for emerging market economies has increased due to higher projected debt levels and larger risks from lower growth and higher primary deficits (Figure 9).

### Debt‑at‑Risk as a predictor of fiscal crises
- Definition of fiscal crisis (binary, country-year; crisis = 1 if occurs over next two years) — crisis if any of four criteria met: (1) credit event with nonrepayment or creditor losses including restructuring; (2) exceptionally large official financing from IMF or European Union; (3) implicit default on domestic debt with high inflation or domestic arrears; (4) loss of market confidence (loss of market access or very large spikes in sovereign yields).
- Logit estimation (bivariate correlations and simple logit models):
  - Independent variable: upside risk to the debt projection = (P95 − P50) at one- and two-year horizons.
  - Upside risks to debt across all models are positive and statistically significantly correlated with fiscal crisis indicator.
  - Magnitude: a one percentage point of GDP increase in (P95 − P50) is associated with an 8-10 percentage points increase in the probability of a fiscal crisis within the next two years.
  - Context: current estimated gap between P95 and median for the one-year ahead global debt distribution ≈ 20 percentage points of GDP; frequency of a crisis in the sample = 5 percent.
- Bayesian Model Averaging (BMA) estimation:
  - Considered three predicted debt-at-risk quantile measures (P95 level; P95 − P50; interaction (P95 − P50) × P95), conditioning variables (8 variables), and 24 control variables (various functional forms, lags, historical rates).
  - In-sample estimation period: 1986-2024.
  - Posterior inclusion probability (PIP) for the interaction term ((P95 − P50) × P95) for each conditioning variable and the combined distribution is 1 at both a 1-year and 2-year forecast horizon, implying debt-at-risk measures are key predictors of impending fiscal distress.
  - Out-of-sample (20-year rolling window forecasts) yields PIPs remaining at 1 for all cases except one on inflation for the one-year ahead estimation.
- Random Forest (RF) model with Boruta feature selection:
  - Predictor set reduced from nearly 780 candidate variables to 188 factors via Boruta.
  - RF uses out-of-bag permuted predictor importance (scaled 0–100) to assess variable importance.
  - Debt-at-risk is the most useful metric in predicting fiscal crises among a wide range of economic variables.
  - Key predictive measures: predicted P95 and the difference (P95 − P50) have the highest variable importance.
  - Introducing debt-at-risk reduces the relative importance of other variables such as the debt-to-GDP level.

### Extensions and sample coverage
- Extensions presented:
  1. Evaluate effectiveness of debt-at-risk in predicting fiscal crises relative to other macro variables.
  2. Construct debt-at-risk measures using a reduced set of conditioning factors to expand sample coverage to 175 countries.
  3. Enhance the quantile regression framework to accommodate non-linear effects of conditioning factors varying by country characteristics.
- Extended sample:
  - Baseline sample includes 90 economies (restriction due to sovereign bond yield availability).
  - Using only available conditioning variables allows extension to 175 countries (many low-income).
  - Example: PDFs for a highly indebted low-income country with only economic variables still show asymmetric debt distributions even without financial variables.
  - World simple average of debt-at-risk for the extended sample follows trends similar to the world GDP-weighted average, with upside and downside risk changes less pronounced during the GFC when financial stress measures are unavailable for several countries.

*Source: wpiea2025086-print-pdf - 9. Table A.3-A.5 report the estimated regressions coefficients for one-year, three-year and five-year future*

### 5.3  Heterogeneity Analysis

### 5.3 Heterogeneity Analysis

### Heterogeneity by initial debt level
- Method:
  - Modified baseline equation to interact conditioning variables x_{i,t} and current debt-to-GDP ratio d_{i,t} with indicators for quartiles of contemporaneous debt-to-GDP: X′_{it}β = Σ_{k=1}^4 β_{1,k} x_{i,t} × 1{Q(d_{i,t}=k)} + Σ_{k=1}^4 β_{2,k} d_{i,t} × 1{Q(d_{i,t}=k)}.
  - Define “low initial debt” and “high initial debt” as first and fourth quartile of the debt-to-GDP ratio, respectively.
  - Quartile thresholds correspond to debt values of 33 and 69 percent of GDP, respectively.
  - Semi-parametric approach that does not impose restrictive assumptions on functional forms of non-linearity.

- Main findings:
  - Location parameters of several economic variables—such as growth, primary balance, and inflation—tend to be statistically significantly larger for countries with higher initial debt.
  - Scale parameters do not systematically vary with initial debt level; exceptions include:
    - Spread at the one-year horizon: scale larger and more precisely estimated for higher debt levels.
    - Growth at the three-year horizon: scale larger and more precisely estimated for higher debt levels.
  - Overall implication: initial debt amplifies the impact of economic factors on the entire future debt distribution (including debt-at-risk), with effects typically not significantly different between left and right tails.

### Heterogeneity by country income group (AEs vs EMDEs)
- Method:
  - Modified specification to interact conditioning variables and initial debt with country-group indicators: X′_{it}β = Σ_{j={AE,EMDE}} β_{1,j} x_{i,t} × 1{country i ∈ j} + Σ_{j={AE,EMDE}} β_{2,j} d_{i,t} × 1{country i ∈ j}.

- Main findings:
  - Location parameter differences:
    - Sovereign spreads: statistically significant effects in AEs in the short term; in EMDEs, effects are more pronounced in the medium term.
    - Uncertainty: tends to have larger and more persistent effects in EMDEs, consistent with evidence that EMDEs are less resilient to uncertainty shocks (Ahir, Bloom, and Furceri 2022).
    - Initial debt: appears to have larger effects in AEs, likely reflecting higher debt values in AEs.
    - Primary balance and growth: point estimates tend to be larger in EMDEs, but differences relative to AEs are not statistically significantly different from zero.
  - Scale parameter differences:
    - No systematic differences between AEs and EMDEs for most variables.
    - Notable exception: initial debt suggests larger asymmetric effects on the right tails of the future distribution for EMDEs compared to AEs.

### Implications for debt-at-risk, predictive uses, and dataset scope
- Quantitative magnitudes and scope:
  - Global debt-at-risk is estimated to be nearly 20 percentage points of GDP higher three years ahead than currently projected, reaching nearly 117 percent of GDP in a severely adverse scenario.
  - Baseline sample: 90 countries since 1980.
  - Extended exercises: produce debt-at-risk measures for an expanded sample of 175 economies.
  - Annual dataset of debt-at-risk measures available for 175 countries since 2009.

- Policy and research implications:
  - Elevated debt levels today amplify the negative effects of weaker growth or tighter financial conditions on future debt ratios.
  - Debt-at-risk can be used to:
    - Quantify the size of debt risks in severely adverse scenarios and assess main determining factors.
    - Serve as a robust predictor of fiscal crises and an early-warning tool to monitor and prevent crises.
    - Be combined with complementary tools and predictive scenarios used in debt sustainability analyses (IMF 2022).
    - Estimate likelihoods of alternative scenarios using the predictive density (Adrian et al. (2025)).

- Suggested methodological extensions:
  - Incorporate additional conditioning factors and further interaction effects (for example, state dependencies around periods of economic downturns).
  - Combine debt-at-risk with routine predictive scenarios developed for debt sustainability analyses to provide narratives on how debt risks could evolve under alternative macro-financial conditions or fiscal consolidation paths.
  - Consider additional heterogeneity interactions for specific sub-groups within AEs and EMDEs.

*Source: IMF staff analysis, "5.3 Heterogeneity Analysis" (excerpt).*

### References

### References

### Key cited works
- Adrian, Tobias; Nina Boyarchenko; and Domenico Giannone. 2019. “Vulnerable Growth.” American Economic Review 109 (4): 1263–1289.
- Adrian, Tobias; Domenico Giannone; Matteo Luciani; and Mike West. 2025. “Scenario synthesis and macroeconomic risk.”
- Adrian, Tobias; Federico Grinberg; Nellie Liang; Sheheryar Malik; and Jie Yu. 2022. “The Term Structure of Growth-at-Risk.” American Economic Journal: Macroeconomics 14 (3): 283–323.
- Ahir, Hites; Nicholas Bloom; and Davide Furceri. 2022. “The World Uncertainty Index.” NBER Working Paper No. 29763.
- Ahir, Hites; Giovanni Dell’Ariccia; Davide Furceri; Chris Papageorgiou; and Hanbo Qi. 2023. “Financial Stress and Economic Activity: Evidence from a New Worldwide Index.” IMF Working Paper No. 2023/217.
- Berg, Andrew; and Catherine Pattillo. 1999. “Predicting Currency Crises: The Indicators Approach and An Alternative.” Journal of International Money and Finance 18 (4): 561–586.
- Blanchard, Olivier. 2019. “Public Debt and Low Interest Rates.” American Economic Review 109 (4): 1197–1229.
- Breiman, Leo. 2001. “Random Forests.” Machine Learning 45 (1): 5–32.
- Reinhart, Carmen M; and Kenneth S Rogoff. 2011. “From Financial Crash to Debt Crisis.” American Economic Review 101 (5): 1676–1706.
- Sturzenegger, Federico; and Jeromin Zettelmeyer. 2007. Debt Defaults and Lessons from a Decade of Crises. The MIT Press.

(Full list of references appears in the source document.)

### Methodology and data sources referenced
- Quantile regression methods and location-scale modeling: Machado, José A.F., and J.M.C. Santos Silva. 2019. “Quantiles via moments.” Journal of Econometrics 213 (1): 145–173.
- Skewed t-distribution and perturbations of symmetry: Azzalini, Adelchi, and Antonella Capitanio. 2003. “Distributions Generated by Perturbation of Symmetry with Emphasis on a Multivariate Skewt-Distribution.” Journal of the Royal Statistical Society Series B 65 (2): 367–389.
- Model averaging and Bayesian approaches: Steel, Mark F. J. 2020. “Model Averaging and Its Use in Economics.” Journal of Economic Literature 58 (3): 644–719.
- Machine learning feature selection and Random Forests: Kursa, Miron B., and Witold R. Rudnicki. 2010. “Feature Selection with the Boruta Package.” Journal of Statistical Software 36 (11): 1–13; Breiman (2001).
- Financial conditions and stress indices: Ahir et al. (2023); IMF Financial Conditions Index; Global Financial Data (Finaeon), Eikon.

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### Figures and what they depict

### Figure 1: Public Debt-to-GDP Ratio, 2000-30 (Percent of GDP)
- Plots statistics for historical and projected debt-to-GDP ratios.
- Source: IMF World Economic Outlook database.

### Figure 2: Impact of Conditioning Variable on Density Based on Location (β) and Scale (γ) Coefficients
- Panels:
  - A. β >0, γ = 0
  - B. β >0, γ >0
  - C. β >0, γ <0
- Illustrates predicted conditional debt densities when the conditioning variable increases by one standard deviation. t0 = period zero; t1 = period one. Dots indicate the predicted 95th quantile of the debt-to-GDP ratio.

### Figure 3: Quantile Regression Results: Forward Debt-to-GDP Ratio and Financial, Economic, and Political Variables
- Panels A–I correspond to: Financial Conditions; Financial Stress; Spread; World Uncertainty; Social Unrest; Debt-to-GDP; Primary Balance; GDP Growth; Inflation.
- Displays estimated quantile regression coefficients for 5th, 50th, and 95th percentiles based on panel quantile regressions (Equation (2)). Bars denote coefficients; whiskers show associated 90 percent confidence intervals.
- Coefficients refer to percentage point change in government debt-to-GDP when explanatory variable changes by one unit. All explanatory variables (except initial debt) standardized to mean zero and standard deviation one. Standard errors clustered at the country level.

### Figure 4: Global Debt-at-Risk 2027 (Probability density of three-year-ahead government debt-to-GDP ratio)
- Predicted density of three-year-ahead global debt-to-GDP ratio estimated using panel quantile regressions and fitted to a skewedt-distribution.
- Global sample comprises 47 countries. Dots indicate predicted 5th, 50th (median), and 95th quantiles.

### Figure 5: Drivers of Global Debt-at-Risk (Percent of GDP)
- Plots contributions from conditioning variables to estimated level of global debt-at-risk.
- Green bar: baseline debt projection for 2027 from WEO database.
- Yellow bars: contributions from conditioning variables.
- Red bar: value of debt-at-risk.

### Figure 6: Evolution of Global Debt and Debt Risks (Percent of GDP)
- Panel A: Debt-to-GDP and Debt-at-Risk (three-year horizon).
- Panel B: Upside and Downside Risks.
- Debt-at-risk defined as predicted 95th quantile (P95) of combined distribution.
- Upside risks = P95 - P50 (median conditional on initial debt).
- Downside risks = P50 - P5.

### Figures 7–9: Debt-at-Risk by Income Groups and Evolution by Group
- Figure 7: Predicted three-year-ahead densities for Advanced Economies (Panel A) and Emerging Market and Developing Economies (Panel B); dots indicate predicted quantiles.
- Figure 8: Contributions from conditioning variables to debt-at-risk for AEs and EMDEs; green = baseline 2027; yellow = contributions; red = debt-at-risk.
- Figure 9: Evolution panels A–D show Debt-to-GDP and Debt-at-Risk and Upside/Downside Risks separately for AEs and EMDEs.

### Figures 10–12: Crisis Prediction and Variable Importance
- Figure 10: Logistic regression coefficients from panel logit of fiscal crisis indicator against debt-at-risk. Independent variable is difference between predicted 95th quantile one-year-ahead and the 50th quantile conditional on variables; whiskers show 90 percent confidence intervals.
- Figure 11: Posterior Inclusion Probability (PIP) from a Bayesian model averaging (BMA) model of fiscal crisis—Panels A (In-Sample) and B (Out-of-Sample). Variables include predicted debt quantiles (p95, p95-p50, p95*(p95-p50)), conditional variables, one-year-ahead debt and primary balance, and additional economic controls. PIP measures likelihood a predictor is included in the “true” model.
- Figure 12: Variable importance by grouped predictors from a Random Forest model predicting fiscal crisis. Variable importance calculated with out-of-bag permuted predictor importance in R; predictors selected by Boruta. Groups enumerated (1) Debt-at-risk through (21) FX reserves; functional forms include first and second lags, weighted averages, percent changes, and standard deviations.

### Figures 13–14: Extended and Expanded Samples
- Figure 13: Debt-at-Risk 2027 for selected low income developing country in extended sample (coverage for economic variables but not financial/political).
- Figure 14: Evolution of global simple average debt-to-GDP and debt-at-risk for expanded sample of countries (Panel A) and Upside/Downside risks (Panel B); definitions of P95, P50, P5 as in earlier figures.

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### Tables and quantitative results

### Table 1: Location-Scale Coefficients (Forward Debt-to-GDP Ratio vs. Financial, Political, and Economic Variables)
- Horizon (no. of years Ahead): 1 3 5 (columns grouped by Location and Scale).
- Selected coefficient examples (standard errors in parentheses):
  - Financial Conditions: Location β for horizon 1 = 1.189 ∗∗∗ (0.294); Scale γ for horizon 1 = 0.314 (0.194).
  - Financial Stress: Location β for horizon 1 = 1.021 ∗∗∗ (0.241); Scale γ for horizon 1 = 0.079 (0.187).
  - Debt-to-GDP (economic variable): Location β for horizon 1 = 0.901 ∗∗∗ (0.016); Scale γ for horizon 1 = 0.092 ∗∗∗ (0.026).
  - Primary Balance: Location β for horizon 1 = -2.365 ∗∗∗ (0.391); Scale γ for horizon 1 = -0.062 (0.170).
  - GDP Growth: Location β for horizon 1 = -1.816 ∗∗∗ (0.434); Scale γ for horizon 1 = -0.507 (0.404).
  - Inflation: Location β for horizon 1 = 1.235 ∗∗∗ (0.046); Scale γ for horizon 1 = 0.537 ∗∗∗ (0.055).
- Notes: Coefficients from panel quantile regressions (Equation (2)). All explanatory variables (except initial debt) standardized. Standard errors clustered at country level. Significance: ***, **, * at 1%, 5%, 10%.

### Table 2 and Table 3: Location and Scale Coefficients Interacted with Initial Debt Quartiles
- Horizons: 1, 3, 5 years ahead; Q1 and Q4 reported with Diff = Q4 - Q1.
- Selected examples:
  - Financial Conditions location (horizon 1): Q1 = 1.570 ∗∗∗ (0.302); Q4 = 2.055 ∗∗ (0.840); Diff = 0.485 (0.829).
  - Financial Stress location (horizon 1): Q1 = 6.849 (4.253); Q4 = 7.553 ∗∗ (3.363); Diff = 0.704 (5.167).
  - World Uncertainty location (horizon 1): Q1 = 2.654 (1.731); Q4 = 10.613 ∗∗ (4.583); Diff = 7.959 (5.446).
  - Debt-to-GDP location (horizon 1): Q1 = 0.783 ∗∗∗ (0.067); Q4 = 0.884 ∗∗∗ (0.022); Diff = 0.102 ∗∗ (0.049).
- Scale coefficients (Table 3) also reported by quartile with standard errors; includes notable entries such as Spread, Yield, and World Uncertainty with interacting effects.

### Table 4 and Table 5: Location and Scale Coefficients Interacted with Country Group (AE vs EM)
- Horizons: 1, 3, 5 years ahead; AE and EM reported with Diff = AE - EM.
- Selected examples:
  - Financial Conditions location (horizon 1): AE = 2.181 ∗∗∗ (0.479); EM = 1.518 ∗ (0.790); Diff = 0.663 (0.924).
  - Financial Stress location (horizon 1): AE = 13.755 ∗∗∗ (1.750); EM = 1.737 (4.965); Diff = 12.019 ∗∗ (5.265).
  - Debt-to-GDP location (horizon 1): AE = 0.965 ∗∗∗ (0.009); EM = 0.844 ∗∗∗ (0.024); Diff = 0.121 ∗∗∗ (0.026).
  - Scale coefficient example: Debt-to-GDP scale AE (horizon 1) = 0.030 ∗∗∗ (0.009); EM = 0.140 ∗∗∗ (0.026); Diff = -0.110 ∗∗∗ (0.027).

### Appendix Tables: Summary Statistics, Coverage, and Regression Results
- Table A.1: Summary statistics (Mean, Median, SD, Obs) for conditioning variables:
  - Financial Conditions: Mean 0.008; Median -0.045; SD 0.694; Obs 1,262.
  - Financial Stress: Mean 0.031; Median 0.000; SD 0.098; Obs 2,844.
  - Spread: Mean 3.557; Median 1.531; SD 9.812; Obs 2,487.
  - Yield: Mean 7.878; Median 6.279; SD 9.872; Obs 2,487.
  - World Uncertainty: Mean 0.159; Median 0.121; SD 0.144; Obs 3,417.
  - Social Unrest: Mean 102.294; Median 64.681; SD 134.142; Obs 3,033.
  - Debt-to-GDP: Mean 55.527; Median 48.677; SD 35.728; Obs 2,876.
  - Primary Balance: Mean -0.496; Median -0.581; SD 3.660; Obs 2,922.
  - GDP Growth: Mean 3.406; Median 3.564; SD 4.311; Obs 3,721.
  - Inflation: Mean 49.931; Median 4.230; SD 1,142.129; Obs 3,719.
- Table A.2: Variable Coverage by country and by sample (Global: 47 countries; Regression: 90 countries; Extended: 175 countries). “✓” indicates inclusion/coverage. (Extensive country-by-country listing provided in the source.)
- Tables A.3–A.5: Quantile regression results for 1-, 3-, and 5-year ahead debt-to-GDP ratios (Q5, Q50, Q95) with panels for Financial, Political, and Economic variables. Selected coefficient magnitudes and significance levels are reported for each horizon (see tables for full numeric detail).
- Table A.6: Conditioning variables and three-year-ahead quantiles for Growth (Q5), Primary Balance (Q5), Interest (Q95), and Unidentified Debt-at-Risk (Q95). Examples:
  - Financial Conditions Index coefficient on Growth Q5 = -0.020 (0.091); on Primary Balance Q5 = -2.343 ∗∗∗ (0.476); on Interest Q95 = 1.465 ∗∗ (0.638); on Unidentified Debt Q95 = 0.809 (0.653).
  - Reported Social Unrest Index on Growth Q5 = -0.328 ∗∗∗ (0.104); on Primary Balance Q5 = -1.504 ∗∗∗ (0.406); on Interest Q95 = -0.865 (0.582); on Unidentified Debt Q95 = -0.849 ∗∗∗ (0.286).
- Table A.7: Variables used in the BMA model:
  - Panel A: DaR Measurements: Predicted debt quantiles (Q95, Q95-Q50, Q95*(Q95-Q50)) conditional on conditioning factors and combined distribution.
  - Panel B: Conditioning Factors: Financial stress index; Spread; World uncertainty index; Social unrest index; Initial debt; Primary deficit; GDP growth; Inflation.
  - Panel C: Additional Variables: Total debt; Public debt measures; Public debt service measures; Fiscal projection (one-year-ahead debt-to-GDP and primary balance-to-GDP). Functional forms include contemporaneous values, lags, and growth rates.
- Table A.8: Variables in the Random Forest model by category, number of factors, total permutations, and number chosen in Boruta selection. Total entries: 757 (total number of factors), 791 (total number of permutations on factors), 188 (number of factors chosen in Boruta selection).

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### Key analytical constructs and definitions (from captions and tables)
- Debt-at-risk: Predicted 95th quantile (P95) of the combined distribution of government debt-to-GDP at a forecast horizon (commonly three-year ahead).
- Upside risk: P95 - P50 (difference between predicted 95th quantile of combined distribution and predicted 50th quantile conditional on initial debt).
- Downside risk: P50 - P5 (difference between predicted median conditional on initial debt and predicted 5th quantile of combined distribution).
- Conditioning variables: Financial Conditions Index, Financial Stress Index, Sovereign Spread, World Uncertainty Index, Reported Social Unrest Index, Debt-to-GDP, Primary Balance, GDP Growth, Inflation (see Table A.1 for definitions and sources).
- Model approaches employed: panel quantile regressions (Equation (2)); location-scale parameterization (β and γ coefficients); interactions with initial debt quartiles and country group (AE vs EM); Bayesian model averaging (BMA); Random Forest with Boruta feature selection.

*Source: Debt-at-Risk, Working Paper No. WP/2025/086 — References, Figures, and Tables sections from the PDF chapter.*

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_Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025086-print-pdf.pdf_