## 1. Global public debt-to-GDP and projected debt-at-risk

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### Overview and projections
- Public debt levels are "significantly elevated worldwide and are projected to increase further and approach 100 percent of GDP by the end of the decade (IMF 2025)".
- In a severe adverse scenario, based on the debt-at-risk framework, "global public debt could surge to approximately 117 percent of GDP by 2027."
- That 117 percent of GDP level "has not been witnessed since World War II and would be about 20 percentage points higher than current projections."
- Contextual magnitudes:
  - Global debt levels at end-2024 were 93 percent of GDP.
  - Global level of debt-at-risk at a three-year forecast horizon is estimated at around 117 percent of GDP (Furceri et al. 2025).

### Methodology and data
- Sample and data source:
  - "An unbalanced annual panel of 192 countries from 1991-2021."
  - Data from "October releases of the IMF World Economic Outlook (WEO) database."
- Fiscal shock identification:
  - Government expenditure shocks identified using "forecast errors of government spending" following approaches similar to Auerbach and Gorodnichenko (2013) and Colombo et al. (2024).
  - Use of IMF October forecasts to reduce endogeneity and account for fiscal foresight.
- Econometric framework:
  - Extension of the location-scale model (Machado and Santos Silva, 2019) into a local projection framework (Jordà, 2005).
  - Baseline local projections consider horizons ℎ = 0, 1, ⋯, 4.
  - Quantiles estimated on a grid 휏 = 0.05, 0.10, 0.15, ⋯, 0.95; the 95-th quantile is defined as debt-at-risk.
- Controls and projections:
  - Controls X푖,푡 include October forecasts for government expenditure (percent of GDP), forecasted GDP growth and inflation, and lags of debt-to-GDP, government expenditure (share of GDP), GDP growth, and inflation.
  - Formal projection residual: 푔̂푖,푡 = 푔푖,푡 − E[푔푖,푡 | X푖,푡] (Frisch-Waugh-Lovell projection residual).

### Key empirical findings
- Direction and significance:
  - "Both the location and scale parameters related to the expenditure consolidation variable are negative and statistically significant across all forecast horizons."
  - Expenditure consolidations reduce both the future level of debt-to-GDP (location) and its associated uncertainty (scale).
- Heterogeneity across the debt distribution:
  - Effects "intensify in magnitude across the quantiles of the debt-to-GDP distribution, with a more pronounced and persistent negative impact observed in the right tail."
- Magnitudes:
  - "A one percent of GDP reduction in government expenditure is found to reduce the 95th quantile of the debt-to GDP distribution (debt-at-risk) by 1 percentage point in the short term and by 1.5 percentage point in the medium term."
  - Location (mean) effects: a one percent of GDP reduction lowers the mean debt-to-GDP by about 0.7 percentage point on impact, peaking at about 0.9 percentage point after two years.
  - Standard-deviation benchmark: a one-standard deviation government expenditure consolidation shock is associated with a reduction in debt-to-GDP of about 17% of its standard deviation on impact, peaking at 23% after 2 years.
- Asymmetric quantile effects:
  - Downward trend in the response of debt-to-GDP across quantiles from the 5th to the 95th quantile at each horizon.
  - Left tail (5th quantile): effects are lower in magnitude, short-lived, and less precisely estimated; disappear two years after consolidation.
  - Right tail (95th quantile, debt-at-risk): stronger and persistent effects; one percent of GDP spending cut reduces debt-at-risk by:
    - about 1 percentage point of GDP in the short term;
    - about 1.5 percentage points after five years.
- Robustness: "These results are robust across several specifications and sensitivity tests."

### Estimation procedure (location-scale local projection - summary)
- Key estimation steps to recover location and scale parameters:
  1. Estimate location parameters (훽̂1 and 훽̂2) by regressing demeaned future debt-to-GDP on demeaned government expenditure and control variables.
  2. Compute country fixed effects of the location model and residuals r̂i,t+h.
  3. Estimate scale parameters (γ̂1 and γ̂2) by regressing demeaned absolute residuals |r̂i,t+h| on demeaned government expenditure and control variables.
  4. Estimate country fixed effects of the scale model from the absolute-residual regression.
- Quantile coefficient for a given 휏: 훽1,h + γ1,h q(휏), where q(휏) is the inverse CDF of the error term evaluated at 휏.
- Standardized residuals:
  - 휀̂푖,푡+ℎ = 푟̂푖,푡+ℎ / (훿̂푖 + γ̂1 / 푔푖,푡 + X′푖,푡 γ̂2); desired quantile of 휀̂푖,푡+ℎ indicated as 푞̂(휏).
  - One-step GMM estimation used (Stata MM-QR algorithm) with clustered standard errors at country level.

### Identification and diagnostics of fiscal shocks
- Identification rationale:
  - Government expenditure shocks preferred because they are "more feasible to identify across a large set of countries and, unlike revenues, they are not driven by cyclical forces."
  - Annual data precludes SVAR; lack of narrative documentation in many EMDEs constrains narrative approaches.
- Exogeneity checks and properties:
  - Shocks satisfy Ramey (2016) criteria: uncorrelated with current and lagged endogenous variables, unanticipated, and uncorrelated with non-fiscal shocks.
  - Regressions show no significant correlation between fiscal shocks and:
    - first principal component of selected endogenous macro variables,
    - first principal component of forecast errors,
    - World Uncertainty Index (Ahir, Bloom and Furceri 2022),
    - monetary policy shocks (Furceri, Loungani and Zdzienicka 2018).
  - Descriptive statistics of shocks:
    - Mean: -9.00e-16
    - Std. Dev.: 3.285
    - Min: -22.173
    - 25-th quantile: -1.411
    - 75-th quantile: 1.090
    - Max: 30.285
    - Interquartile range lies between -1.4% and 1.1%.

### Mechanisms and channels (location and scale effects on drivers)
- Variables analyzed as debt drivers: "government revenues and government expenditure, real GDP, inflation, and real long-term interest rates."
- Aggregate effects of spending-based consolidations:
  - Spending consolidations "lead to reductions in revenue levels (due to lower output), price levels, GDP, and interest rates."
  - Fiscal consolidation "also negatively affects the scale of many of these variables, thereby reducing both fiscal and macroeconomic uncertainty."
- Specific channel estimates and decomposition (one year after consolidation):
  - 0.55 percentage points (approximately 70%) of the 0.8 percentage point reduction in average future debt-to-GDP can be attributed to the fiscal balance improvement.
  - Remaining reduction explained by effects on other macro variables (GDP growth, interest rates) included in the debt equation.
- Fiscal balance and components:
  - Expenditure consolidations increase the future fiscal balance (share of GDP) and reduce uncertainty around it.
  - Decline in revenue accompanies expenditure cuts, potentially due to reductions in inflation and real GDP.
- Inflation:
  - Inflation declines modestly, with peak decline two years after consolidation.
  - No statistically significant effects on the scale of the inflation distribution.
- GDP (fiscal multipliers):
  - Statistically significant negative fiscal multipliers, which are lower than one.
  - Fiscal consolidation reduces the scale of the future output distribution (novel contribution).
- Long-term interest rates:
  - Statistically significant and negative effects on long-term interest rates, especially after two years.
  - Reduction in scale of the interest rate distribution, significant up to three years after consolidation.

### State-dependency and non-linearity: fiscal rules and initial debt levels
- Fiscal rules
  - Dataset: Acalin et al. (2025) covering 106 IMF members with fiscal rules adopted at some point over 1985 to 2024.
  - Binary fiscal rule variable R푖,푡 equals one if country adopts any type of fiscal rule in a given year.
  - Extended model allows location and scale parameters to differ when R푖,푡 = 1 versus R푖,푡 = 0.
  - Main heterogeneous findings:
    - With a fiscal rule in place:
      - Fiscal consolidations reduce debt-to-GDP by more than 1.2 percentage points in the medium term.
      - Scale effects are stronger and more precisely estimated; difference compared to no-rule case statistically significant at the 5% on impact and at the 10% after four years.
      - Reduction in debt-at-risk of about 2 percentage points after four years.
    - Without a fiscal rule:
      - Reductions are modest, less than 0.5 percentage points in the medium term (0.5 percentage points not statistically significant in absence of a fiscal rule).
  - Robustness: results consistent when focusing on fiscal rules imposing targets on public debt and for other rule types (budget balance, expenditure, revenue rules).
  - Fiscal rule strength (stringency index, 훾훾 = 5):
    - Expenditure consolidation lowers debt-to-GDP by more than 1 percentage points on impact and by more than 1.5 percentage points in the medium-term when the fiscal rule is tighter.
    - Debt-at-risk (95-th quantile) drops by 2 percentage points on impact in the high fiscal rule strength regime versus 0.5 percentage points when the fiscal rule is looser.
    - In the medium term, debt-at-risk drops by about 2.5 percentage points when the fiscal rule is tighter; no effects when the fiscal rule is looser.
- Initial debt levels
  - Smooth transition model: F(z_i) = e^{−γ z_i} / (1+ e^{−γ z_i}) with γ calibrated to 5 (robust to γ = 2.5 and γ = 7).
  - Parameters capture location and scale effects for low- and high-debt regimes.
  - Empirical findings:
    - Expenditure consolidations lower debt-at-risk more substantially when debt levels are higher.
    - Effects on location are much larger when debt levels are high versus low.
    - Difference between location effects for high and low debt countries statistically different from zero from horizon 0 to horizon 3.
    - Expenditure consolidations more strongly decrease uncertainty of debt-to-GDP in the medium term when debt levels are elevated.
    - Effects of fiscal expenditure consolidations on debt-at-risk in high-debt countries are approximately four times larger than in low-debt countries; difference statistically significant at the 5% level at the 3-year horizon.

### Heterogeneity by income and other characteristics
- Income level:
  - Expenditure consolidation reduces debt risks more in Advanced Economies (AEs) than in Emerging Market and Developing Economies (EMDEs), primarily due to stronger location effects.
  - A one percentage point reduction in government expenditure-to-GDP leads to a medium-term decline in debt-at-risk of about 3 percentage points of GDP in AEs, compared to less than 1 percentage point in EMDEs.
- Other characteristics:
  - Effects marginally greater in countries with lower informality and higher institutional quality.
  - No significant differences based on trade openness or exchange rate regimes.
  - Effects do not vary significantly under different economic and financial conditions examined (business cycle, Zero Lower Bound low interest rates, financial stress).

### Robustness checks (Annex A summary)
- Robustness exercises:
  - Alternative numbers of lags: baseline uses two lags; tested 1, 3 and 4 lags — results similar.
  - Removal of extreme values: remove observations below 1st and above 99th percentiles for debt-to-GDP and/or government expenditure-to-GDP — results similar.
  - Alternative impulse variable: use government balance instead of government expenditure — location effects similar; scale effects somewhat weaker.
  - Alternative estimators: Canay (2011) two-step approach and IV quantile regression (Chernozhukov and Hansen 2006) with forecast-error instruments — qualitatively similar asymmetric effects.
  - Fiscal rule stringency index and alternative γ values for smooth transition (γ = 2.5, 5, 7) — core results robust.
- Selected p-values for tests of difference (examples, Table 1 and Tables A.1–A.2):
  - General fiscal rule (Table 1, panel (a)) p-values for difference in Location model at horizon 4: 0.0010; Debt-at-risk at horizon 4: 0.0022.
  - Table 2 (difference by debt level): Location model horizon 0 p-value 0.0050; Debt-at-risk horizon 3 p-value 0.0136.
  - Annex A p-values show numerous statistically significant differences for location effects and some scale/debt-at-risk differences across regimes and rule strengths.

### Data diagnostics (Annex C highlights)
- Coverage: 192 countries; period 1991-2021.
- Key descriptive statistics (Table C.2):
  - Debt-to-GDP (# obs.: 4369): Mean: 55.745; Std. Dev.: 43.281; Min: 0.052; Max: 600.117.
  - Government expenditure (percent of GDP) (# obs.: 4369): Mean: 31.489; Std. Dev.: 14.251; Min: 2.483; Max: 134.665.
  - Government balance (percent of GDP) (# obs.: 4367): Mean: -2.301; Std. Dev.: 6.189; Min: -55.407; Max: 125.135.
  - Government revenues (percent of GDP) (# obs.: 4367): Mean: 29.182; Std. Dev.: 14.874; Min: 0.637; Max: 166.030.
  - Forecast of government expenditure (percent of GDP) (# obs.: 4369): Mean: 1.32E+09; Std. Dev.: 7.24E+09; Min: 0; Max: 1.36E+11.
- Diagnostics for implied government expenditure shock (Table C.3):
  - Regressions show coefficients not statistically significant for correlations with principal components, World Uncertainty Index, or monetary policy shocks (examples: first principal component contemporaneous 0.358 (0.393); World Uncertainty Index 0.192 (0.362); monetary policy shocks -0.017 (0.083)).

### Conclusions and policy implications
- Summary empirical conclusions:
  - Government expenditure consolidations shift the debt-to-GDP distribution left (lower location) and reduce its scale (lower uncertainty), with asymmetric effects across quantiles.
  - Effects on the left tail are nearly negligible; effects on the right tail are highly statistically significant, economically substantial, and persistent.
  - Two main amplifiers of consolidation effectiveness in reducing debt and risks:
    - Presence of robust fiscal rules.
    - High initial public debt levels at the start of consolidation.
- Policy recommendations:
  - "Countries with very high debt ratios should consider government expenditure consolidation measures to mitigate upside risks to public debt: high public debt ratios require stabilization and consolidation of public spending is even more effective in this context."
  - "Implementing consolidations within credible medium-term frameworks guided by fiscal rules would enhance credibility and effectiveness in reducing debt risks."
- Policy magnitude implication:
  - IMF Fiscal Monitor 2024 recommends a cumulative fiscal adjustment of about 3-4.5 percent of GDP over the medium term, which implies a reduction in the debt-at-risk of approximately 4.5 to 7 percentage points after five years according to the paper’s calculation.

*Source: IMF Working Paper (chapter/section "1. Global public debt-to-GDP and projected debt-at-risk", excerpt from wpiea2025201-source-pdf).*

### 1. Global public debt-to-GDP and projected debt-at-risk  ___________________________ 27

### 1. Global public debt-to-GDP and projected debt-at-risk

### Overview and projections
- Public debt levels are "significantly elevated worldwide and are projected to increase further and approach 100 percent of GDP by the end of the decade (IMF 2025)".
- In a severe adverse scenario, based on the debt-at-risk framework, "global public debt could surge to approximately 117 percent of GDP by 2027."
- That 117 percent of GDP level "has not been witnessed since World War II and would be about 20 percentage points higher than current projections."

### Methodology and data
- Sample: "an unbalanced annual panel of 192 countries from 1991-2021."
- Data source: "October releases of the IMF World Economic Outlook (WEO) database."
- Focus of shocks: government expenditure shocks identified using "forecast errors of government spending" following an approach similar to Auerbach and Gorodnichenko (2013) and Colombo et al. (2024).
- Econometric framework:
  - Extension of the location-scale model (Machado and Santos Silva, 2019) into a local projection framework (Jordà, 2005).
  - Baseline local projections consider horizons ℎ = 0, 1, ⋯, 4.
  - Quantiles estimated on a grid 휏 = 0.05, 0.10, 0.15, ⋯, 0.95; the 95-th quantile is defined as debt-at-risk.
- Identification rationale: government expenditure shocks are preferred because they are more feasible to identify across a large set of countries and, "unlike revenues, they are not driven by cyclical forces."

### Key empirical findings
- Direction and significance:
  - "Both the location and scale parameters related to the expenditure consolidation variable are negative and statistically significant across all forecast horizons."
  - This implies expenditure consolidations reduce both the future level of debt-to-GDP (location) and its associated uncertainty (scale).
- Heterogeneity across the debt distribution:
  - Effects "intensify in magnitude across the quantiles of the debt-to-GDP distribution, with a more pronounced and persistent negative impact observed in the right tail."
- Magnitudes highlighted:
  - "A one percent of GDP reduction in government expenditure is found to reduce the 95th quantile of the debt-to GDP distribution (debt-at-risk) by 1 percentage point in the short term and by 1.5 percentage point in the medium term."
  - Effects on the left tail (the 5th quantile) are "lower in magnitude and less precisely estimated."
- Robustness: "These results are robust across several specifications and sensitivity tests."

### Mechanisms and channels (location and scale effects on drivers)
- Variables analyzed as debt drivers: "government revenues and government expenditure, real GDP, inflation, and real long-term interest rates."
- Aggregate effects of spending-based consolidations:
  - Spending consolidations "lead to reductions in revenue levels (due to lower output), price levels, GDP, and interest rates."
  - Novel contribution: fiscal consolidation "also negatively affects the scale of many of these variables, thereby reducing both fiscal and macroeconomic uncertainty."

### State-dependency and non-linearity
- Fiscal rules:
  - "Fiscal rules amplify the impact of fiscal consolidation on both the level of debt and its associated uncertainty."
  - Quantified comparison: "fiscal consolidations reduce the debt-to-GDP ratio by more than 1.2 percentage points in the medium term when a fiscal rule is in place, whereas they result in a medium-term decline of less than 0.5 percentage points when the fiscal rule is absent."
- Initial debt levels:
  - "Fiscal consolidations significantly reduce debt-at-risk when debt levels are elevated."
  - Mechanism: "the reduction in both the location and scale of the debt-to-GDP distribution is considerably greater at higher debt levels compared to lower ones."

### Relationship to existing literature
- Extends literature by:
  - Analyzing effects across a larger set of countries (advanced, emerging, developing).
  - Estimating causal effects using plausibly exogenous government expenditure forecast errors.
- Links and distinctions:
  - Builds on Furceri et al. (2025) on drivers of debt-at-risk but focuses on plausibly exogenous fiscal actions.
  - Connects to studies on policy shocks and distributional effects (e.g., Forni et al. 2024; Frangiamore et al. 2025; Frangiamore, Furceri and Giannone 2025).

### Estimation procedure (location-scale local projection steps, summary)
- Key estimation steps implemented to recover location and scale parameters:
  1. Estimate location parameters (훽̂1 and 훽̂2) by regressing demeaned future debt-to-GDP on demeaned government expenditure and control variables.
  2. Compute country fixed effects of the location model and residuals r̂i,t+h.
  3. Estimate scale parameters (γ̂1 and γ̂2) by regressing demeaned absolute residuals |r̂i,t+h| on demeaned government expenditure and control variables.
  4. Estimate country fixed effects of the scale model from the absolute-residual regression.
- The quantile coefficient for a given 휏 is given by 훽1,h + γ1,h q(휏), where q(휏) is the inverse CDF of the error term evaluated at 휏, allowing recovery of effects across the full distribution.

*Source: IMF Working Paper (chapter/section "1. Global public debt-to-GDP and projected debt-at-risk").*

### 5. Compute  the  standardized  residuals 휀휀̂

### 5. Compute  the  standardized  residuals 휀휀̂
푖푖,푡푡+ℎ

### Estimation of standardized residuals and quantiles
- Standardized residuals defined as:  
  휀̂푖,푡+ℎ = 푟̂푖,푡+ℎ / (훿̂푖 + γ̂1 / 푔푖,푡 + X′푖,푡 γ̂2) and the desired quantile of the distribution of 휀̂푖,푡+ℎ is indicated as 푞̂(휏).
- Estimation procedure:
  - One-step GMM estimation (specifically the Stata MM-QR algorithm).
  - Clustered standard errors at the country level to account for time within-country correlation (Rios-Avila 2022).
- Note: The horizon subscript ℎ is omitted in the text for simplicity but is considered in the estimation when moving forward after estimating contemporaneous effects.

### Identification of public expenditure (fiscal) shocks
- Methodology:
  - Approach akin to Auerbach and Gorodnichenko (2013) and Colombo et al. (2024), relying on forecast errors of government spending.
  - Use IMF forecasts of government expenditures made in October of the same year to reduce influence of potentially endogenous responses of fiscal policy.
- Rationale for this approach:
  - Annual data frequency precludes SVAR (Blanchard and Perotti 2002) which relies on quarterly data.
  - Lack of publicly available official documentation for many EMDEs prevents broad use of the narrative approach.
  - Method inherently addresses “fiscal foresight” by aligning econometrician information with agents’ information.
- Formal projection and residual:
  - 푔̂푖,푡 = 푔푖,푡 − E[푔푖,푡 | X푖,푡] (Frisch-Waugh-Lovell projection residual).
- Control variables X푖,푡 include (explicitly):
  - The forecast of government expenditure as a percentage of GDP (October forecasts).
  - Contemporaneous and lagged endogenous variables: October forecasts for GDP growth and inflation, lags of debt-to-GDP, government expenditure (share of GDP), GDP growth, and inflation.
- Exogeneity checks and properties:
  - The shocks satisfy Ramey (2016) criteria: uncorrelated with current and lagged endogenous variables, unanticipated, and uncorrelated with non-fiscal shocks.
  - Regressing fiscal shocks against the first principal component of a set of endogenous macro variables and of forecast errors shows no significant correlation (see Table C.3 in Annex C).
  - No significant correlation with the World Uncertainty Index (Ahir, Bloom and Furceri 2022) or monetary policy shocks (Furceri, Loungani and Zdzienicka 2018) (see Table C.3 in Annex C).
- Descriptive statistics:
  - Average and median of the shock are both approximately zero.
  - Interquartile range (25th to 75th percentile) lies between -1.4% and 1.1%.

### Baseline results: effects of expenditure consolidations on debt distribution
- Main result:
  - Fiscal expenditure consolidations are associated with lower future debt levels and reduced debt risks.
- Location (mean) effects:
  - A one percent of GDP reduction in government expenditure statistically significantly lowers the mean (location) of the debt-to-GDP ratio by about 0.7 percentage point on impact, with a peak response of about 0.9 percentage point after two years. The effect diminishes in the medium term but remains statistically significant.
- Scale (uncertainty) effects:
  - Spending-based consolidations significantly reduce the scale of the debt-to-GDP distribution in both the short and medium term.
  - This implies fiscal consolidation decreases uncertainty around public debt and makes very high public debt realizations less likely.
- Standard-deviation benchmark:
  - A one-standard deviation government expenditure consolidation shock is associated with a reduction in the level of debt-to-GDP of about 17% of its standard deviation on impact, peaking at 23% after 2 years.
- Asymmetric quantile effects:
  - Quantile regression effects from the location-scale model (훽1,ℎ + γ1,ℎ q(τ)) show a downward trend in the response of debt-to-GDP across quantiles from the 5th to the 95th quantile at each horizon.
  - Left tail (low debt outcomes): effects are short-lived and small, disappearing two years after consolidation.
  - Right tail (high debt outcomes): effects are stronger and persistent.
  - Debt-at-risk (95th percentile): a one percent of GDP spending cut results in:
    - Reduction in debt-at-risk of about 1 percentage point of GDP in the short term.
    - Reduction of about 1.5 percentage points after five years following the consolidation.
- Contextual magnitudes:
  - Global debt levels at end-2024 were 93 percent of GDP.
  - Global level of debt-at-risk at a three-year forecast horizon is estimated at around 117 percent of GDP (Furceri et al. 2025).
  - IMF Fiscal Monitor 2024 recommends a cumulative fiscal adjustment of about 3-4.5 percent of GDP over the medium term, which implies a reduction in the debt-at-risk of approximately 4.5 to 7 percentage points after five years according to the paper’s calculation.
- Summary statement:
  - Expenditure consolidations reduce future debt-to-GDP ratios and the uncertainty surrounding them, with stronger and more persistent effects on the right tail of the distribution.

### Robustness checks
- Lag specification:
  - Baseline uses two lags of debt-to-GDP, government expenditure-to-GDP, and other macro variables.
  - Tested alternative lags: 1, 3 and 4; results similar to baseline (Figure A.1 in Annex A).
- Influence of extreme values:
  - Three exercises to test sensitivity to extremes:
    - (i) Remove observations with debt-to-GDP below 1st and above 99th percentiles.
    - (ii) Remove observations with government expenditure-to-GDP below 1st and above 99th percentiles.
    - (iii) Remove observations simultaneously with debt-to-GDP and government expenditure-to-GDP below 1st and above 99th percentiles.
  - Results remain very similar to baseline (Figure A.2 of Annex A).
- Alternative fiscal variable:
  - Use government balance instead of government expenditure; effects:
    - Location effects very similar.
    - Scale effects somewhat weaker relative to baseline (Figure A.3 of Annex A).
    - Quantile effects less asymmetric, though medium-term asymmetry persists with more pronounced effects on right tail three and four years after consolidation.
- Alternative fixed-effects and IV strategies:
  - Canay (2011) two-step approach (purge fixed effects in first stage, then quantile regressions) yields lower degree of asymmetry but overall downward-sloping trend largely confirmed (purple dashed lines, Figure A.4 of Annex A).
  - Inverse Quantile Regression (IQR) estimator (Chernozhukov and Hansen 2006) applied with Canay (2011) purging and forecast-error instruments:
    - First stage: regress forecast errors of real government expenditure growth on the baseline control set; use residuals as instruments.
    - Results (blue dashed lines, Figure A.4 of Annex A) qualitatively similar to baseline, confirming asymmetric effects across quantiles.

### Mechanisms: channels through which expenditure consolidations affect debt distribution
- Primary focus:
  - Dynamic response of primary balance, inflation, long-term government bond yields, and GDP to expenditure consolidations.
- Fiscal balance and its components:
  - Expenditure consolidations increase the future fiscal balance (share of GDP) and reduce uncertainty around it (Figure 5).
  - Effects primarily arise from reduction in fiscal expenditure, but also accompanied by a decrease in revenue.
  - Decline in revenue potentially explained by reductions in inflation and real GDP associated with consolidation.
- Inflation:
  - Inflation declines modestly, with peak decline two years after consolidation.
  - No statistically significant effects on the scale of the inflation distribution (Figure 6).
- GDP (fiscal multipliers):
  - Statistically significant negative fiscal multipliers, which are lower than one, consistent with literature (e.g., Colombo et al., 2024; Ramey and Zubairy, 2018).
  - Multipliers estimated using Ramey and Zubairy (2018) one-step estimation strategy to be consistent with literature.
  - Fiscal consolidation reduces the scale of the future output distribution (novel contribution), aligning with reduced risk to public debt outlook and credit market.
- Long-term interest rates:
  - Expenditure consolidations have statistically significant and negative effects on long-term interest rates, especially after two years.
  - Reduction in scale of the interest rate distribution, significant up to three years after consolidation.
  - Interpreted as improvement in credit market conditions and a reduction in sovereign distress risk premium.
- Quantitative decomposition (one year after consolidation):
  - 0.55 percentage points (approximately 70%) of the 0.8 percentage point reduction in average future debt-to-GDP can be attributed to the fiscal balance improvement.
  - Remaining reduction explained by effects on other macro variables (GDP growth, interest rates) included in the debt equation.

### Heterogeneous effects: role of fiscal rules
- Fiscal rules dataset:
  - Use fiscal rule dataset from Acalin et al. (2025), covering 106 IMF members that have adopted fiscal rules at some point over 1985 to 2024.
  - Binary fiscal rule variable R푖,푡 equals one if country adopts any type of fiscal rule in a given year; zero otherwise.
  - Also consider fiscal rules on public debt (and report results for rules on budget balance, government expenditure and revenues in Annex A).
- Extended model (interaction with fiscal rule dummy):
  - Equation (4) extends baseline to allow location and scale parameters to differ when R푖,푡 = 1 versus R푖,푡 = 0.
  - 훽1,ℎ,RR measures location effects when fiscal rule in place; 훽1,ℎ,NRR when no fiscal rule.
  - γ1,ℎ,RR measures scale effects when fiscal rule in place; γ1,ℎ,NRR when no fiscal rule.
  - Quantile effects for the two cases follow from the location-scale model; focus on 95th quantile (debt-at-risk).
- Main heterogeneous findings:
  - Fiscal expenditure consolidations are more effective in reducing debt risks when fiscal rules are adopted.
  - With a fiscal rule in place:
    - Fiscal consolidations reduce debt-to-GDP by more than 1.2 percentage points in the medium term.
    - Scale effects are stronger and more precisely estimated; difference compared to no-rule case statistically significant at the 5% on impact and at the 10% after four years (see Table 1, panel (a)).
    - Reduction in debt-at-risk of about 2 percentage points after four years.
  - Without a fiscal rule:
    - Reductions are modest, less than 0.5 percentage points in the medium term.
- Interpretation:
  - Fiscal rules likely improve credibility of fiscal consolidation and reduce risk of future fiscal slippages, enhancing the effectiveness of expenditure consolidations in lowering debt-at-risk.

*IMF WORKING PAPERS The effects of fiscal consolidations on the debt distribution — excerpt from wpiea2025201-source-pdf*

### 0.5 percentage points (not statistically significant) in the absence of a fiscal rule (Figure 7, panel (a)).

### wpiea2025201-source-pdf - 0.5 percentage points (not statistically significant) in the absence of a fiscal rule (Figure 7, panel (a)).

### Effects of fiscal rules on expenditure consolidations and debt
- Expenditure consolidations reduce future public debt ratios and their uncertainty more when a fiscal rule is in place.
- Difference in effects at horizon 4 between countries that adopt a fiscal rule and those that do not, as well as the difference observed at impact (horizon 0), are statistically significant at the 5% level (see Table 1, panel (a)).
- Results are consistent when focusing on fiscal rules imposing targets on public debt (Figure 7, panel (b), and Table 1, panel (b)).
- 0.5 percentage points (not statistically significant) in the absence of a fiscal rule (Figure 7, panel (a)).
- Annex evidence:
  - Figure A.5 and Table A.1 show robustness when considering other types of fiscal rules, including budget balance rules, expenditure rules, and revenue rules.
  - The difference is less pronounced for revenue rules, as IRFs exhibit wider confidence intervals for countries adopting revenue rules.
- Alternative index-based methodology (Acalin et al. 2025) applied to a subset of countries:
  - Figure A.6 and Table A.2 reveal a similar pattern: stronger fiscal rules correlate with greater effects of expenditure consolidations on public debt ratios.
- Mechanisms: larger effects with fiscal rules are primarily explained by lower costs in terms of output losses (Figures B.1, B.2 and Table B.1 for general fiscal rule; Figures B.3, B.4 and Table B.2 for debt rule, Annex B).

### Role of initial debt levels
- Initial debt level is an important determinant of exposure to public debt risks; high initial debt implies higher future debt burden absent consolidation and is a leading indicator of fiscal and sovereign distress.
- Smooth transition model specified in equation (5) uses normalized debt-to-GDP variable 푧푧푖:
  - 퐹퐹(푧푧푖) = e^{−훾훾푧푧푖} / (1+ e^{−훾훾푧푧푖}) with 훾훾> 0.
  - Parameter 훾훾 calibrated to 5 (intermediate value) following referenced literature; results robust to 훾훾=2.5 and 훾훾=7 (Figure A.7).
- Interpretation of parameters:
  - 훽훽1,ℎ,퐿퐿 and 훾훾1,ℎ,퐿퐿 capture location and scale effects of consolidations for very low public debt countries.
  - 훽훽1,ℎ,퐻퐻 and 훾훾1,ℎ,퐻퐻 measure effects when debt levels are very high.
- Empirical findings:
  - Government expenditure consolidations lower debt-at-risk more substantially when debt levels are higher.
  - Reduction in the location of the debt-to-GDP distribution is much larger when debt levels are high versus low (Figure 8).
  - Difference between location effects for high and low debt countries is statistically different from zero from horizon 0 to horizon 3 (Table 2).
  - Expenditure consolidations more strongly decrease uncertainty of debt-to-GDP in the medium term when debt levels are elevated.
  - Effects of fiscal expenditure consolidations on debt-at-risk in high-debt countries are approximately four times larger than in low-debt countries.
  - This difference is statistically significant at the 5% level (at the 3-year horizon), as indicated in Table 2.
- Alternative quartile-based methodology (IMF (2024) and Furceri et al. (2025)) yields similar conclusions (Figure A.8).
- Mechanisms: stronger reduction in output uncertainty following consolidation is the main channel for higher effects in high-debt countries (Figures B.5 and B.6 and Table B.3, Annex B).

### Other sources of heterogeneity
- Income level:
  - Expenditure consolidation reduces debt risks more in Advanced Economies (AEs) than in Emerging Market and Developing Economies (EMDEs), primarily due to stronger location effects.
  - Scale effects are stronger in AEs versus EMDEs, but their difference is only statistically significant on impact.
  - A one percentage point reduction in the ratio of government expenditure to GDP leads to a medium-term decline in the debt-at-risk of about 3 percentage points of GDP in AEs, compared to less than 1 percentage point in EMDEs.
  - Findings align with earlier results on fiscal rules and debt; AEs generally have stronger fiscal rules and higher debt levels.
- Other characteristics:
  - Effects are marginally greater in countries with lower informality and higher institutional quality.
  - No significant differences based on trade openness or exchange rate regimes.
  - Effects do not vary significantly under different economic and financial conditions examined (business cycle, Zero Lower Bound low interest rates, financial stress).

### Conclusions and policy implications
- Dataset and methodology:
  - Unbalanced panel covering 192 countries from 1991 to 2021; panel quantile regressions applied.
- Key empirical conclusions:
  - Government expenditure consolidations shift the debt-to-GDP distribution left and reduce its scale, with asymmetric effects across quantiles.
  - Effects on the left tail are nearly negligible; effects on the right tail are highly statistically significant, economically substantial, and persistent.
  - Two main factors amplify consolidation effectiveness in reducing debt and risks:
    - Presence of robust fiscal rules.
    - High initial public debt levels at the start of consolidation.
- Policy recommendations:
  - Countries with very high debt ratios should consider government expenditure consolidation measures to mitigate upside risks to public debt: high public debt ratios require stabilization and consolidation of public spending is even more effective in this context.
  - Implementing consolidations within credible medium-term frameworks guided by fiscal rules would enhance credibility and effectiveness in reducing debt risks.

*Source: wpiea2025201-source-pdf - 0.5 percentage points (not statistically significant) in the absence of a fiscal rule (Figure 7, panel (a)).*

### References

### wpiea2025201-source-pdf - References

### Bibliographic scope and themes
- References focus on fiscal policy, fiscal consolidations, debt dynamics, debt-at-risk, fiscal rules, fiscal multipliers, quantile and location-scale econometric methods, fiscal foresight, and macroeconomic tail risk.
- Recurring authors and institutions include Alesina, Furceri, Giavazzi, Ramey, Leeper, IMF publications (Fiscal Monitor chapters 2024 and 2025), National Bureau of Economic Research, Journal of International Economics, American Economic Review, European Economic Review, Journal of Political Economy, and others.
- Methodological sources cited:
  - Quantile regression and instrumental quantile methods (e.g., Koenker & Bassett Jr, 1978; Koenker, 2004; Chernozhukov & Hansen, 2006; Canay, 2011; Kaplan & Sun, 2017).
  - Local projections and impulse response estimation (Jordà, 2005; Montiel Olea et al., 2025).
  - Location-scale modeling and quantiles via moments (Machado & Silva, 2019; Machado & Silva related methods cited).
  - Fiscal narrative and consolidation datasets (Adler et al. 2024; Davoodi et al. 2022; Ando et al. 2025).
- Empirical topics and datasets:
  - Debt dynamics and debt surges (Kose et al., 2022; Reinhart et al., 2015; Moreno Badia et al., 2022).
  - Fiscal multipliers and asymmetric effects of government expenditure (Auerbach & Gorodnichenko, 2013; Blanchard & Perotti, 2002; Ramey & Zubairy, 2018; Frangiamore et al., 2025 forthcoming).
  - Uncertainty and macro risk measures (Ahir et al., 2022; Forni et al., 2024; Loria et al., 2025).
  - Greenflation empirical evidence (Bettarelli et al., 2025).
- Datasets and working papers from IMF and NBER are heavily referenced (e.g., IMF SDN/2025/004; Furceri et al., 2025; NBER working papers).

### Figures: descriptions, models, and notes
- Figure 1: Global public debt-to-GDP and projected debt-at-risk.
  - Note: Source IMF Fiscal Monitor (2024, 2025). Panel (a) shows the time evolution of public debt-to -GDP ratios. Panel (b) shows the predicted probability density function of debt-to -GDP for 2027 conditional on various economic, financial and political factors, based on information up to December 2024 (see IMF 2025 and Furceri et al. 2025).
- Figure 2: Illustrative example on how the location and scale coefficients affect the debt-to-GDP distribution.
  - Note: Illustrative example showing 훽훽 indicates the location coefficient whereas 훾훾 the scale coefficient. 푡푡0 is before the change in the predictor while 푡푡1 indicates the period after the change in the predictor.
- Figure 3: Location and scale effects of fiscal expenditure consolidations on debt-to-GDP.
  - Note: Dynamic effects plotted with point estimates (black solid lines) and 68% and 90% confidence intervals. Estimates use sample of 192 countries over period 1991-2021. Location-scale model specified as:
    - 푑푑푖,푡+ℎ = 훼훼푖,ℎ + 훽훽1,ℎ 푔푔푖,푡 + X′푖,푡 훽훽2,ℎ + [훿훿푖,ℎ + 훾훾1,ℎ 푔푔푖,푡 + X′푖,푡 훾훾2,ℎ] 휀푖,푡+ℎ,
    - where 푑푑푖,푡+ℎ is the debt-to -GDP ratio, 훼훼푖,ℎ and 훿훿푖,ℎ are country fixed effects in the location and scale equation, 푔푔푖,푡 is the government expenditure-to -GDP ratio and X′푖,푡 is a set of control variables (see section II.3).
- Figure 4: Quantile effects of fiscal expenditure consolidation on debt-to-GDP.
  - Note: Effects across quantiles from 5-th to 95-th at different years after consolidation. Quantile function retrieved as:
    - Q_{d_{i,t+h}}(τ|g_{i,t},X_{i,t}) = (α_{i,h} + δ_{i,h} q(τ)) + β_{1,h} g_{i,t} + X′_{i,t} β_{2,h} + (γ_{1,h} g_{i,t} + X′_{i,t} γ_{2,h}) q(τ),
    - estimate of quantile coefficient is β_{1,h} + γ_{1,h} q(τ) at τ-th quantile.
- Figure 5: Location and scale effects of fiscal expenditure consolidations on selected government budgetary variables.
  - Note: Three series shown: government balance as percent of GDP, government revenues as percent of GDP, government expenditure as percent of GDP. Model form analogous to Figure 3 with dependent variable y_{i,t+h} (selected budgetary variable).
- Figure 6: Location and scale effects of fiscal expenditure consolidations on selected macroeconomic variables.
  - Note: Lines for inflation, real long-term interest rates, and GDP (reported as multipliers). Multipliers for GDP estimated using Ramey and Zubairy (2018) approach extended to the location-scale model.
- Figure 7: Effects of fiscal expenditure consolidations on debt-to-GDP and debt-at-risk in countries adopting and not adopting a general fiscal rule and a debt rule.
  - Note: Panels (a) General fiscal rule and (b) Debt rule. Dynamic effects on location, scale and 95-th quantile (debt-at-risk) plotted for adopters (blue) and non-adopters (red). Model specified with regime dummy R_{i,t} interacting with coefficients; debt-at-risk effect estimated as β_{1,h} + γ_{1,h} q(τ) for τ = 0.95.
- Figure 8: Effects of fiscal expenditure consolidations on debt-to-GDP and debt-at-risk conditional on the country’s debt levels.
  - Note: Dynamic effects shown for high (blue) and low (red) debt level countries. Regime function F(z_i) = e^{−γ z_i} / (1 + e^{−γ z_i}), with z_i the debt-to-GDP normalized as z_i = (d̅_i − d̅) / sd(d̅_i). Debt-at-risk effect estimated as β_{1,h} + γ_{1,h} q(τ) for τ = 0.95.

### Tables: statistical tests and exact p-values
- Table 1: p-values for the test of significance of the difference between the effects of fiscal expenditure consolidations on debt-to-GDP in countries adopting and not adopting a fiscal rule.
  - (a) General fiscal rule
    - Horizon | Location model | Scale model | Debt-at-risk
    - 0 | 0.1051 | 0.0253 | 0.0287
    - 1 | 0.0516 | 0.4839 | 0.2055
    - 2 | 0.1511 | 0.9412 | 0.4006
    - 3 | 0.0462 | 0.9975 | 0.2775
    - 4 | 0.0010 | 0.0832 | 0.0022
  - (b) Debt rule
    - Horizon | Location model | Scale model | Debt-at-risk
    - 0 | 0.1225 | 0.0296 | 0.0364
    - 1 | 0.0458 | 0.5960 | 0.2221
    - 2 | 0.1513 | 0.9835 | 0.4095
    - 3 | 0.0727 | 0.9534 | 0.3256
    - 4 | 0.0023 | 0.0658 | 0.0034
  - Note: The null hypothesis is that the difference between the coefficients in the two regimes is zero.
- Table 2: p-values for the test of significance of the difference between the effects of fiscal expenditure consolidations on debt-to-GDP conditional on the country’s debt level.
  - Horizon | Location model | Scale model | Debt-at-risk
  - 0 | 0.0050 | 0.2001 | 0.7543
  - 1 | 0.0850 | 0.6396 | 0.7513
  - 2 | 0.0671 | 0.8592 | 0.2389
  - 3 | 0.0332 | 0.0673 | 0.0136
  - 4 | 0.7788 | 0.3703 | 0.4778
  - Note: The null hypothesis is that the difference between the coefficients in the two regimes is zero.

*Content extracted from the "References" section of the source PDF.*

### Annex A: Robustness checks

### Annex A: Robustness checks

### Data, sample, and baseline model
- Sample: 192 countries over the period 1991-2021.
- Baseline location-scale model (as presented):
  - 푑푑푖푖,푡푡+ℎ = 훼훼푖푖,ℎ + 훽훽1,ℎ 푔푔푖푖,푡푡 + 푋푋푖푖,푡푡′ 훽훽2,ℎ + [훿훿푖푖,ℎ + 훾훾1,ℎ 푔푔푖푖,푡푡 + 푋푋푖푖,푡푡′ 훾훾2,ℎ] 휀휀푖푖,푡푡+ℎ
  - 푖푖 refers to the country, 푡푡 the time, and ℎ the horizons (years) after the fiscal shock.
  - 푑푑푖푖,푡푡+ℎ is the debt-to-GDP ratio; 푔푔푖푖,푡푡 is the government expenditure-to-GDP ratio; 푋푋푖푖,푡푡′ is the set of control variables useful for the identification of the fiscal shock (see section II.3 for further details).
  - Quantile effects are estimated as 훽훽1,ℎ + 훾훾1,ℎ 푞푞(휏휏).

### Robustness checks performed (overview)
- Alternative numbers of lags (Figure A.1).
- Removal of extreme values from debt-to-GDP and government expenditure-to-GDP (Figure A.2).
- Replacing government expenditure-to-GDP with fiscal balance-to-GDP as impulse variable (Figure A.3).
- Alternative estimators for quantile effects for cross-sectional data: standard quantile regressions and IV quantile regressions (Figure A.4).
- Heterogeneity by adoption of fiscal rules: budget balance rule, expenditure rule, revenues rule (Figure A.5 and Table A.1).
- Alternative approach using fiscal rule stringency index (smooth transition model with 훾훾 = 5) and tests by high vs low rule strength (Figure A.6 and Table A.2).
- Sensitivity to smooth transition parameter 훾훾 for splitting countries by debt levels (훾훾 = 2.5 and 훾훾 = 7) (Figure A.7).
- Alternative dummy-based quartile approach for initial public debt (four dummies for quartiles; low = first quartile, high = fourth quartile) (Figure A.8).

### Key quantitative findings from robustness checks
- Sample and estimation details repeated across exercises: 192 countries, 1991-2021, standard errors clustered at the country level. Confidence intervals reported at 68% and 90% depending on panel.
- Fiscal rule strength (stringency index approach, 훾훾 = 5):
  - Expenditure consolidation lowers debt-to-GDP by more than 1 percentage points on impact and by more than 1.5 percentage points in the medium-term when the fiscal rule is tighter.
  - Debt-at-risk (95-th quantile) drops by 2 percentage points on impact in the high fiscal rule strength regime versus a 0.5 percentage points reduction when the fiscal rule is looser.
  - In the medium term, debt-at-risk drops by about 2.5 percentage points when the fiscal rule is tighter, whereas there are no effects when the fiscal rule is looser.
  - For the general fiscal rule, scale effects are significantly higher in the high rule strength regime only four years after the fiscal shock.
  - For the debt rule, scale effects are significantly higher both on impact and after four years.
  - Statistical significance: differences in effects between high and low rule strength are significant at various horizons (see Table A.2 for p-values).
- Sensitivity to smooth transition parameter 훾훾 when splitting by debt levels:
  - Effects on location, scale, and 95-th quantile (debt-at-risk) were examined for 훾훾 = 2.5 and 훾훾 = 7 (Figure A.7), showing robustness to alternative 훾훾 choices.
- Alternative estimators for quantile effects:
  - Baseline estimation retrieves the conditional quantile function Q푑푖,푡+ℎ(휏휏|푔푖,푡,푋푖,푡) = [훼푖,ℎ + 훿푖,ℎ q(휏휏)] + 훽1,ℎ 푔푖,푡 + 푋′푖,푡 훽2,ℎ + [훾1,ℎ 푔푖,푡 + 푋′푖,푡 훾2,ℎ] q(휏휏).
  - Check 1: standard quantile regressions for cross-sectional data (Canay (2011) style first-stage removal of fixed effects).
  - Check 2: IV quantile regressions for cross-sectional data where IV is constructed as forecast errors in government expenditure purged from controls (see Colombo et al. 2024).
- Robustness to removal of extreme values:
  - Three checks: removing extreme values from debt-to-GDP, from government expenditure-to-GDP, and from both variables simultaneously (Figure A.2) — results compared to baseline and represented via colored dashed lines.

### Statistical tests: p-values for differences by fiscal rule adoption (Table A.1)
- (a) Budget balance rule — p-values for test of significance of the difference between effects in countries adopting vs not adopting a budget balance rule:
  - Horizon 0: Location model 0.0467; Scale model 0.1349; Debt-at-risk 0.0670
  - Horizon 1: Location model 0.0248; Scale model 0.4770; Debt-at-risk 0.1456
  - Horizon 2: Location model 0.1353; Scale model 0.5070; Debt-at-risk 0.1748
  - Horizon 3: Location model 0.0703; Scale model 0.8483; Debt-at-risk 0.2714
  - Horizon 4: Location model 0.0040; Scale model 0.0872; Debt-at-risk 0.0054
- (b) Expenditure rule:
  - Horizon 0: Location model 0.0034; Scale model 0.4802; Debt-at-risk 0.1020
  - Horizon 1: Location model 0.0203; Scale model 0.9160; Debt-at-risk 0.4389
  - Horizon 2: Location model 0.0158; Scale model 0.5415; Debt-at-risk 0.4842
  - Horizon 3: Location model 0.0302; Scale model 0.1095; Debt-at-risk 0.8810
  - Horizon 4: Location model 0.0022; Scale model 0.1179; Debt-at-risk 0.4839
- (c) Revenues rule:
  - Horizon 0: Location model 0.7505; Scale model 0.2471; Debt-at-risk 0.4507
  - Horizon 1: Location model 0.2640; Scale model 0.7116; Debt-at-risk 0.4533
  - Horizon 2: Location model 0.3061; Scale model 0.9229; Debt-at-risk 0.5296
  - Horizon 3: Location model 0.3064; Scale model 0.9115; Debt-at-risk 0.6263
  - Horizon 4: Location model 0.0266; Scale model 0.6287; Debt-at-risk 0.2887

### Statistical tests: p-values for differences by fiscal rule strength (Table A.2)
- (a) General fiscal rule:
  - Horizon 0: Location model 0.0170; Scale model 0.3594; Debt-at-risk 0.1205
  - Horizon 1: Location model 0.0368; Scale model 0.9846; Debt-at-risk 0.2687
  - Horizon 2: Location model 0.0122; Scale model 0.8632; Debt-at-risk 0.0987
  - Horizon 3: Location model 0.0255; Scale model 0.9969; Debt-at-risk 0.2968
  - Horizon 4: Location model 0.0030; Scale model 0.0861; Debt-at-risk 0.0122
- (b) Debt rule:
  - Horizon 0: Location model 0.0061; Scale model 0.0956; Debt-at-risk 0.0259
  - Horizon 1: Location model 0.0418; Scale model 0.7360; Debt-at-risk 0.1836
  - Horizon 2: Location model 0.0044; Scale model 0.6617; Debt-at-risk 0.0479
  - Horizon 3: Location model 0.0121; Scale model 0.5791; Debt-at-risk 0.1157
  - Horizon 4: Location model 0.0021; Scale model 0.0215; Debt-at-risk 0.0024

### Alternative modeling choices and parameter values
- Smooth transition function for fiscal rule stringency index:
  - Normalized stringency index: 푧푧푖푖,푡푡 = (푠푠푖푖,푡푡 − 푠푠̅) / ssd(푠푠푖푖,푡푡).
  - Transition function: 퐹퐹(푧푧푖푖,푡푡) = e−훾훾푧푧푖푖,푡푡 / (1 + e−훾훾푧푧푖푖,푡푡).
  - 훾훾 set to an intermediate value equal to 5 in the main stringency exercise.
- Alternative 훾훾 values for splitting by debt levels:
  - 훾훾 = 2.5 and 훾훾 = 7 examined (Figure A.7).
- Dummy-based quartile approach for initial debt:
  - Four dummies ퟏퟏ[Q(d푖,푡)=k] for k = 1, 2, 3, 4; low initial debt = k = 1; high initial debt = k = 4.
  - Extended baseline model including interaction terms between these dummies and regressors (Figure A.8).

### Interpretation and robustness conclusions (from the exercises)
- The main qualitative findings are robust to:
  - Different numbers of lags.
  - Removal of extreme values in debt-to-GDP and/or government expenditure-to-GDP.
  - Using fiscal balance-to-GDP as the impulse variable instead of government expenditure-to-GDP.
  - Alternative quantile estimation strategies for cross-sectional data (standard and IV quantile regressions).
  - Alternative parameterizations of the smooth transition function (different 훾훾 values).
  - Alternative classification of low versus high initial debt (quartile dummies).
- Fiscal rules matter:
  - Stronger fiscal rules (higher stringency index) amplify the debt-reducing effects of expenditure consolidations on both the mean/location and on debt-at-risk (95-th quantile).
  - Statistical tests (Tables A.1 and A.2) indicate significant differences in many horizons, especially for location effects and for debt rule-related comparisons.

*Source: Annex A: Robustness checks, IMF Working Paper (figures and tables as presented in the source PDF).*

### Annex B: Additional results

### Annex B: Additional results

### Location and scale effects of fiscal expenditure consolidations by fiscal-rule adoption
- Figures B.1 and B.2 present dynamic effects of fiscal expenditure consolidations on the location and scale of distributions for:
  - Government budgetary variables: government balance (percent of GDP), government revenues (percent of GDP), government expenditure (percent of GDP).
  - Macroeconomic variables: inflation, real long-term interest rates, and GDP multipliers (GDP reported as multipliers using the Ramey and Zubairy (2018) approach extended to the location-scale model).
- Estimation sample and model:
  - Sample: 192 countries over the period 1991-2021.
  - Model: location-scale model of the form y_{i,t+h} = α_{i,h} + R_{i,t} × [β_{1,h,R} g_{i,t} + X'_{i,t} β_{2,h,R}] + (1−R_{i,t}) × [β_{1,h,NR} g_{i,t} + X'_{i,t} β_{2,h,NR}] + δ_{i,h} + R_{i,t} × [γ_{1,h,R} g_{i,t} + X'_{i,t} γ_{2,h,R}] + (1−R_{i,t}) × [γ_{1,h,NR} g_{i,t} + X'_{i,t} γ_{2,h,NR}] ε_{i,t+h}, where:
    - i: country, t: time, h: horizons (years) after the fiscal shock.
    - y_{i,t+h}: selected government budgetary or macroeconomic variable.
    - R_{i,t}: dummy that takes value 1 if the fiscal rule is in place.
    - g_{i,t}: government expenditure-to-GDP ratio.
    - X_{i,t}': set of control variables for identification of the fiscal shock (see section II.3).
    - α_{i,h} and δ_{i,h}: country fixed effects in location and scale equations.
- Confidence intervals:
  - Shaded areas and dashed lines in the figures are 90% confidence intervals based on standard errors clustered at the country level.
- Visual comparison:
  - Blue lines: countries adopting a general fiscal rule.
  - Red lines: countries not adopting a general fiscal rule.

### Statistical tests: p-values for differences by general fiscal rule (Table B.1)
- (a) Government budgetary variables — p-values for tests of difference between adopters and non-adopters

  Gov. Balance (Location model / Scale model)
  - Horizon 0: 0.6016 / 0.9335
  - Horizon 1: 0.9522 / 0.3535
  - Horizon 2: 0.6564 / 0.4458
  - Horizon 3: 0.5556 / 0.0005
  - Horizon 4: 0.3288 / 0.0931

  Gov. Revenues (Location model / Scale model)
  - Horizon 0: 0.6016 / 0.9335
  - Horizon 1: 0.8330 / 0.9120
  - Horizon 2: 0.6663 / 0.3990
  - Horizon 3: 0.7269 / 0.0003
  - Horizon 4: 0.0463 / 0.0017

  Gov. Expenditure (Location model / Scale model)
  - Horizon 1: 0.9777 / 0.6851
  - Horizon 2: 0.8793 / 0.1492
  - Horizon 3: 0.2602 / 0.0217
  - Horizon 4: 0.9145 / 0.7170

- (b) Macroeconomic variables — p-values for tests of difference between adopters and non-adopters

  Inflation (Location model / Scale model)
  - Horizon 0: 0.8517 / 0.9109
  - Horizon 1: 0.3435 / 0.3211
  - Horizon 2: 0.0614 / 0.0859
  - Horizon 3: 0.7086 / 0.4842
  - Horizon 4: 0.0941 / 0.3252

  Interest Rates (Location model / Scale model)
  - Horizon 0: 0.6789 / 0.9769
  - Horizon 1: 0.2306 / 0.5899
  - Horizon 2: 0.1638 / 0.8141
  - Horizon 3: 0.9972 / 0.4838
  - Horizon 4: 0.6634 / 0.7999

  Multipliers (Location model / Scale model)
  - Horizon 0: 0.2249 / 0.0049
  - Horizon 1: 0.3419 / 0.0540
  - Horizon 2: 0.2411 / 0.1233
  - Horizon 3: 0.4442 / 0.0935
  - Horizon 4: 0.2408 / 0.1567

### Location and scale effects by debt-rule adoption
- Figures B.3 and B.4 replicate the analysis substituting a debt rule for a general fiscal rule:
  - Blue lines: countries adopting a debt rule.
  - Red lines: countries not adopting a debt rule.
  - Same variables, sample (192 countries, 1991-2021), model structure, confidence intervals, and GDP multiplier estimation approach as above.

### Statistical tests: p-values for differences by debt-rule adoption (Table B.2)
- (a) Government budgetary variables — p-values for tests of difference between debt-rule adopters and non-adopters

  Gov. Balance (Location model / Scale model)
  - Horizon 0: 0.8668 / 0.9467
  - Horizon 1: 0.9568 / 0.4340
  - Horizon 2: 0.6029 / 0.3085
  - Horizon 3: 0.6139 / 0.0012
  - Horizon 4: 0.4240 / 0.0628

  Gov. Revenues (Location model / Scale model)
  - Horizon 0: 0.8668 / 0.9467
  - Horizon 1: 0.9721 / 0.7970
  - Horizon 2: 0.5709 / 0.1815
  - Horizon 3: 0.7525 / 0.0001
  - Horizon 4: 0.0555 / 0.0014

  Gov. Expenditure (Location model / Scale model)
  - Horizon 1: 0.8609 / 0.7356
  - Horizon 2: 0.9332 / 0.2162
  - Horizon 3: 0.3566 / 0.0377
  - Horizon 4: 0.7627 / 0.9924

- (b) Macroeconomic variables — p-values for tests of difference between debt-rule adopters and non-adopters

  Inflation (Location model / Scale model)
  - Horizon 0: 0.9657 / 0.8509
  - Horizon 1: 0.3268 / 0.2635
  - Horizon 2: 0.0826 / 0.2436
  - Horizon 3: 0.6197 / 0.4072
  - Horizon 4: 0.1308 / 0.4473

  Interest Rates (Location model / Scale model)
  - Horizon 0: 0.6746 / 0.0870
  - Horizon 1: 0.9525 / 0.0399
  - Horizon 2: 0.2638 / 0.0616
  - Horizon 3: 0.1729 / 0.0747
  - Horizon 4: 0.8764 / 0.7855

  Multipliers (Location model / Scale model)
  - Horizon 0: 0.3517 / 0.0312
  - Horizon 1: 0.6202 / 0.1751
  - Horizon 2: 0.5732 / 0.2613
  - Horizon 3: 0.8345 / 0.1758
  - Horizon 4: 0.5798 / 0.2620

### Conditional results by country debt levels
- Figures B.5 and B.6 present dynamic location and scale effects conditional on the country’s debt level:
  - Comparison between countries with high debt levels (blue lines) and low debt levels (red lines).
  - Model uses a smooth transition function F(z_i) = e^{−γ z_i} / (1 + e^{−γ z_i}) with z_i the normalized debt-to-GDP (z_i = (d̄_i − d̄) / sd(d̄_i)).
  - Same sample, horizons, variables, estimation approach, and 90% confidence intervals as above.
  - GDP multipliers estimated using the Ramey and Zubairy (2018) approach extended to the location-scale model.

### Statistical tests: p-values for differences by debt levels (Table B.3)
- (a) Government budgetary variables — p-values for tests of difference between high- and low-debt countries

  Gov. Balance (Location model / Scale model)
  - Horizon 0: 0.9592 / 0.1116
  - Horizon 1: 0.9887 / 0.9146
  - Horizon 2: 0.6445 / 0.3055
  - Horizon 3: 0.7212 / 0.9143
  - Horizon 4: 0.5749 / 0.1799

  Gov. Revenues (Location model / Scale model)
  - Horizon 0: 0.8609 / 0.0697
  - Horizon 1: 0.7881 / 0.0261
  - Horizon 2: 0.2103 / 0.0375
  - Horizon 3: 0.9138 / 0.9445
  - Horizon 4: 0.3874 / 0.2919

  Gov. Expenditure (Location model / Scale model)
  - Horizon 1: 0.6676 / 0.2526
  - Horizon 2: 0.7852 / 0.8864
  - Horizon 3: 0.5967 / 0.3644
  - Horizon 4: 0.6533 / 0.4850

- (b) Macroeconomic variables — p-values for tests of difference between high- and low-debt countries

  Inflation (Location model / Scale model)
  - Horizon 0: 0.2324 / 0.2178
  - Horizon 1: 0.1436 / 0.2443
  - Horizon 2: 0.0335 / 0.8011
  - Horizon 3: 0.5723 / 0.3192
  - Horizon 4: 0.2710 / 0.6994

  Interest Rates (Location model / Scale model)
  - Horizon 0: 0.4011 / 0.1214
  - Horizon 1: 0.1591 / 0.0838
  - Horizon 2: 0.9673 / 0.8043
  - Horizon 3: 0.3886 / 0.9667
  - Horizon 4: 0.4072 / 0.2726

  Multipliers (Location model / Scale model)
  - Horizon 0: 0.1062 / 0.0029
  - Horizon 1: 0.2385 / 0.0170
  - Horizon 2: 0.3961 / 0.2385
  - Horizon 3: 0.4206 / 0.4256
  - Horizon 4: 0.3583 / 0.5040

---

### Annex C: Descriptive statistics and data diagnostics

### Coverage: country list
- Table C.1 lists the 192 countries included in the dataset (country enumeration provided in the source).

### Key descriptive statistics (Table C.2)
- Number of observations and summary statistics:

  - Debt-to-GDP
    - # obs.: 4369
    - Mean: 55.745
    - Std. Dev.: 43.281
    - Min: 0.052
    - Max: 600.117

  - Government expenditure (percent of GDP)
    - # obs.: 4369
    - Mean: 31.489
    - Std. Dev.: 14.251
    - Min: 2.483
    - Max: 134.665

  - Log of real GDP
    - # obs.: 4369
    - Mean: 6.141
    - Std. Dev.: 3.560
    - Min: -8.062
    - Max: 16.219

  - Log of real government expenditure
    - # obs.: 4369
    - Mean: 4.890
    - Std. Dev.: 3.456
    - Min: -9.178
    - Max: 14.532

  - Inflation (percent)
    - # obs.: 4369
    - Mean: 5.523
    - Std. Dev.: 13.788
    - Min: -129.936
    - Max: 648.424

  - Real interest rate
    - # obs.: 4148
    - Mean: -0.318
    - Std. Dev.: 39.114
    - Min: -99.955
    - Max: 2459.488

  - Government balance (percent of GDP)
    - # obs.: 4367
    - Mean: -2.301
    - Std. Dev.: 6.189
    - Min: -55.407
    - Max: 125.135

  - Government revenues (percent of GDP)
    - # obs.: 4367
    - Mean: 29.182
    - Std. Dev.: 14.874
    - Min: 0.637
    - Max: 166.030

  - Forecast of government expenditure (percent of GDP)
    - # obs.: 4369
    - Mean: 1.32E+09
    - Std. Dev.: 7.24E+09
    - Min: 0
    - Max: 1.36E+11

  - Forecast of inflation (percent)
    - # obs.: 4369
    - Mean: -1.461
    - Std. Dev.: 92.175
    - Min: -2222.67
    - Max: 2137.346

  - Forecast of log of real GDP
    - # obs.: 4369
    - Mean: 10.779
    - Std. Dev.: 9.805
    - Min: -18.804
    - Max: 36.732

### Diagnostics for the implied government expenditure shock
- Table C.3: Regressions of the implied government expenditure shock from baseline local projections on selected variables (coefficients with robust standard errors in parentheses):
  - First principal component of selected macro-variables (contemporaneous): 0.358 (0.393)
  - First principal component of the first lag of selected macro-variables: -0.371 (0.271)
  - First principal component of forecast errors of selected macro-variables: 0.060 (0.107)
  - World Uncertainty Index (Ahir, Bloom and Furceri 2022): 0.192 (0.362)
  - World Uncertainty Index - Fiscal Policy (Ahir, Bloom and Furceri 2022): 0.494 (1.681)
  - World Uncertainty Index - Political (Ahir, Bloom and Furceri 2022): -0.042 (0.607)
  - Monetary policy shocks (Furceri, Loungani and Zdzienicka 2018): -0.017 (0.083)
  - Note: Country and time fixed effects are controlled for but not reported. Selected macro-variables include inflation, output, government expenditure, short-term interest rates, and exchange rates (national currency per U.S. dollar). Forecast errors are those of output, inflation and interest rates.

- Table C.4: Descriptive statistics of the government spending shock
  - Mean: -9.00e-16
  - Std. Dev.: 3.285
  - Min: -22.173
  - 25-th quantile: -1.411
  - 75-th quantile: 1.090
  - Max: 30.285
  - Note: The government spending shock is defined as the innovation implied by the local projections (the residual from projecting g_{i,t} onto the set of control variables X_{i,t}, i.e., ĝ_{i,t} = g_{i,t} − E[g_{i,t} | X_{i,t}]).

- Figure C.1:
  - Displays the distribution of government expenditure shocks (as defined above).

*IMF Working Papers — Annex B and Annex C, "The effects of fiscal consolidations on the debt distribution."*

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_Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025201-source-pdf.pdf_