## _wp14102

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---

### I. Introduction — scope and contribution
- Objective: apply a probabilistic approach to debt sustainability analysis (DSA) for HIPC/MDRI beneficiaries, following Celasun et al. (2006) and drawing on Hevia (2012).
- Empirical strategy:
  - Identify role of "deep debt determinants": real GDP growth, real interest rates, capital gains/losses on foreign-currency debt from exchange rate changes.
  - Include primary fiscal deficits and debt relief.
  - Three-step estimation and forecasting:
    - Step 1: estimate relationship between deep determinants and debt via a panel VAR (data: 2001–11; 33 countries).
    - Step 2: estimate a fiscal reaction function (System GMM) following Celasun et al. (2006).
    - Step 3: simulate random shocks for determinants and primary deficit; forecast trajectories and present probabilistic forecasts as “fan” charts for eight illustrative countries (Bolivia, Burkina Faso, Ethiopia, Ghana, Guyana, Honduras, Nicaragua, Senegal).
- Paper organization:
  - Section II: data and evolution of public debt for 33 HIPCs (2001–11).
  - Section III: panel data estimation methodology.
  - Section IV: estimation results.
  - Section V: fan chart forecasts.
  - Section VI: conclusions.

### II. Stylized facts: trends and composition of public debt in HIPCs (2001–11)
- Sample:
  - 33 HIPC beneficiaries; dataset covers 2001–11.
  - Afghanistan, The Republic of the Congo, and Liberia excluded due to limited data.
- Aggregate public debt-to-GDP (average for 33 HIPCs):
  - 2001: 120 percent of GDP
  - 2007: 54 percent of GDP
  - 2011: 41 percent of GDP
- Completion points:
  - By 2006, 21 out of 33 countries (accounting for more than three-fourths of total debt in dataset) had reached completion points and qualified for 100 percent relief on eligible debt from IMF, IDA, and AfDF.
  - Another eight HIPC/MDRI beneficiaries reached completion points between 2007 and 2010.
  - By the end of 2011, only Comoros, Côte d'Ivoire, Guinea, and Chad had not yet reached their completion points.
  - Footnote: Comoros reached the completion point in December 2012, Côte d'Ivoire in June 2012, Guinea in September 2012.
- Composition (average shares of total public debt):
  - Concessional PPG debt: 2001 = 67 percent; 2007 = 63 percent; 2011 = 53 percent.
  - Average share of public domestic debt: grew from about 10 percent to 30 percent of total.
- Domestic vs external public debt:
  - In 23 out of 33 countries, domestic public debt increased:
    - Average domestic public debt: 2001 = 5 percent of GDP; 2011 = 14 percent of GDP.
  - In the remaining 10 countries, average public debt ratio decreased from 34 percent to 10 percent of GDP.
  - Observed pattern: public external debt in percent of GDP declined in almost all countries (debt relief), while domestic public debt did not decline in step.
- Decomposition of change in total public debt (2001–11) contributors:
  - Real GDP growth: average real GDP growth was 5 percent; contributed significantly to decline in debt ratio each year.
  - Real interest rates: were negative through 2008, mostly owing to low nominal interest rate on concessional debt; negative real rates reduced debt-to-GDP ratio.
  - Real effective exchange rates: appreciated through 2010, contributing to decrease in debt-to-GDP ratio.
  - Primary fiscal deficits: increased the debt-to-GDP ratio.
  - Debt relief: notable contribution in 2006 and 2007 following MDRI launch.
  - Residual: significant contribution to debt reduction in 2006 and 2007; possible explanation includes mis-recording of debt relief after completion points (e.g., flow vs. stock relief).
- Data sources for figures and calculations: IMF, Strategy, Policy and Review Department DSA Database and own calculations.

### III. Methodology — PVARX, structural break, and fiscal reaction function
- Debt accounting identity and notation:
  - P_t: total nominal public debt in local currency at time t.
  - P_t^d and P_t^f: domestic and external public debt respectively.
  - i_t^d: implicit nominal interest rate on domestic public debt.
  - i_t^f: implicit nominal interest rate on external public debt.
  - Δe_t: rate of depreciation of nominal exchange rate.
  - PB_t: primary balance.
  - O_t: other debt-creating flows (recapitalization costs, privatization receipts, contingent liabilities, treasury float, debt relief, etc.).
  - After normalization by nominal GDP, key determinants: real GDP growth g_t, real interest rates r_t^d and r_t^f, change in real exchange rate Δq_t, primary balance, and other debt-creating flows.
- Panel VAR with exogenous regressors (PVARX) specification:
  - y_it' = (g_it, r_it^d, r_it^f, Δq_it); x_it' = (g*_t, r*_t) where g* is world real GDP growth and r* is world real interest rate; α_i is country fixed effect.
  - y_it = α_i + Σ_{k=1}^{p} Φ_k y_{i,t-k} + Σ_{k=0}^{p} Γ_k x_{i,t-k} + ϵ_{i,t}, i = 1,...,N; t = 1,...,T.
  - Error terms ϵ_{i,t} are i.i.d. with E[ϵ_{i,t}] = 0 and covariance Ω; slope coefficients Φ and Γ and covariance matrix Ω are common across countries.
- Structural break hypothesis and candidate breakpoint:
  - Hypothesis: HIPC and MDRI may have changed relationships among deep determinants and debt levels.
  - Data limitation: T = 11 annual observations per country (2001–11); completion points occur at different times across countries.
  - Candidate common break-point year chosen: 2006 (21 out of 33 HIPCs had reached completion points before 2006; Figures 1–3 indicate 2006 as intuitive candidate).
- Estimation novelties and bias correction:
  - Use of PVARX in probabilistic public DSA context.
  - Bootstrap bias correction algorithm (Pesaran and Zhao (1999); Everaert and Pozzi (2007)) implemented to address bias from fixed effects in short T dynamic panel estimation.
- Fiscal reaction function:
  - Purpose: tie primary balance to public debt to capture government adjustment behavior.
  - Estimation period for fiscal reaction function focused on 2007–11 to capture post-break dynamics.
  - General specification (Celasun et al. (2006) style):
    - Dependent variable: primary balance to GDP.
    - Key regressors: lagged public debt-to-GDP, demeaned real GDP growth, World Bank’s CPIA index (institutions), country fixed effect, and i.i.d. shocks.
  - Endogeneity addressed using Arellano and Bover (1995)/Blundell and Bond (1998) System GMM.
- Nonlinear fiscal reaction (debt spline and business cycle interaction):
  - Dummy D = 1 if lagged debt > 50 percent of GDP.
  - Dummy B = 1 if output gap is positive (expansion).
  - ρ = response of primary balance to a change in lagged debt when lagged debt < 50 percent of GDP.
  - ρ + ρ̅ = response when lagged debt > 50 percent of GDP.
  - Threshold 50 percent corresponds to the DSF’s threshold for Debt/GDP for countries with strong policy component.

### IV. Break–point testing and fiscal reaction estimation results
- Break–point testing (SS and BP tests):
  - Tests applied: “split sample” (SS) test (assumes covariance matrix of errors unchanged) and “break–point” (BP) test (allows covariance matrix to change).
  - Both SS and BP tests rejected the null hypothesis of constant parameters over 2001–11:
    - For the full sample of 33 HIPCs.
    - For the subsample of 21 HIPCs that reached their completion points before or during 2006.
  - Implication: post–2006 dynamics of debt accumulation for HIPC/MDRI beneficiaries have changed and forecasts must account for that.
- Interpretation and caveats:
  - SS and BP tests confirm a significant change in dynamics of "deep" determinants, but other factors besides HIPC/MDRI could have caused the changes; further study warranted.
  - Practical implication: forecasts should allow for changed post–2006 dynamics.
- Fiscal reaction function estimation (summary and preferred specification):
  - Estimators reported: OLS, LSDV, and multiple System GMM specifications (Table 1).
  - Sample and observations reported across specifications: 330, 330, 231, 231, 231, 198.
  - Key findings:
    - System GMM (1) produced a puzzling negative coefficient on lagged debt in one specification, suggesting nonlinearity.
    - Introducing a “debt spline” (>50 percent of GDP) (System GMM (2)) produced:
      - Positive coefficient on lagged debt for “low” debt (<50 percent of GDP).
      - Negative coefficient for the “high” debt dummy (>50 percent of GDP).
    - Testing business cycle interaction (System GMM (3)) yielded a positive output gap dummy coefficient that was not statistically significant; variable dropped.
    - Preferred specification: System GMM (4) estimated for 2007–11:
      - Coefficient on lagged debt: positive.
      - Coefficient on “high” debt dummy (>50 percent of GDP): negative.
      - Coefficient on institutional quality (CPIA): positive but statistically insignificant (limited variability may explain insignificance).
  - Diagnostics reported for System GMM specifications: Hansen test P-values, AR(1) and AR(2) P-values, and number of instruments (always less than number of cross-section units).

### V. Fan charts, probabilistic DSA index, and illustrative findings
- Forecast construction steps:
  (i) Use PVARX estimates to predict trajectories of deep determinants (real GDP growth, real interest rates, capital gains/losses from exchange rate changes) assuming common distribution/co-movements and country fixed effects.
  (ii) Predict primary balances using System GMM equation (4) from Table 1; policy shocks drawn from mean-zero normal distribution with variance equal to country-specific variance of residuals.
  (iii) Combine (i) and (ii) to produce many debt paths and display them as “fan” charts.
- Fan chart display conventions (illustrative Figure 4):
  - Middle area: 20 percent confidence interval around the median projection (median shown as an orange line for 2012–16).
  - Fan cone: 80 percent confidence interval around the median projection.
  - IMF/WB baseline DSA forecasts superimposed as red lines.
- Uses and interpretation:
  - Fan charts provide probabilistic benchmarks for DSA forecasts.
  - Example policy threshold: 50 percent debt-to-GDP — in the illustrative set, Guyana, Nicaragua, and Senegal would have debt outlooks considered negative relative to that threshold.
  - For Bolivia, Burkina Faso, and Ghana the baseline DSA forecasts fall within the 20 percent confidence interval around the fan chart median in some cases.
  - Divergences between DSA and fan charts may indicate that DSA incorporates information not captured by the probabilistic model and warrant further investigation.
- Probabilistic DSA index definition (following Celasun et al. (2006)):
  - Index = Pr(d_{t+τ} < d_t) * (1 − Pr(d_{t+τ} > d_t + x)), where x is the acceptable debt increase (application uses x = 10 percent).
  - Fan charts used with fan cone = 80% confidence interval around the median; 2012 is the first year of the projection period.
  - Interpretation: index combines probabilities that debt declines from initial level and that it does not increase by more than x percentage points; higher index values indicate more desirable positions but the index itself has no direct probabilistic interpretation.
- Illustrative probabilities (selected countries; probabilities that debt ratio is lower than in 2011):
  - Bolivia: 2012 0.51; 2013 0.54; 2014 0.55; 2015 0.54; 2016 0.57
  - Burkina-Faso: 2012 0.58; 2013 0.56; 2014 0.55; 2015 0.49; 2016 0.49
  - Ethiopia: 2012 0.54; 2013 0.64; 2014 0.68; 2015 0.69; 2016 0.68
  - Ghana: 2012 0.69; 2013 0.68; 2014 0.68; 2015 0.68; 2016 0.66
  - Guyana: 2012 0.40; 2013 0.37; 2014 0.36; 2015 0.31; 2016 0.31
  - Honduras: 2012 0.58; 2013 0.57; 2014 0.58; 2015 0.57; 2016 0.57
  - Nicaragua: 2012 0.44; 2013 0.41; 2014 0.39; 2015 0.37; 2016 0.38
  - Senegal: 2012 0.48; 2013 0.41; 2014 0.40; 2015 0.31; 2016 0.33
- Probabilities that the debt ratio is more than 10 percent of GDP higher than in 2011 (selected countries):
  - Bolivia: 2012 0.09; 2013 0.14; 2014 0.18; 2015 0.21; 2016 0.22
  - Burkina-Faso: 2012 0.07; 2013 0.14; 2014 0.18; 2015 0.24; 2016 0.27
  - Ethiopia: 2012 0.08; 2013 0.12; 2014 0.13; 2015 0.13; 2016 0.15
  - Ghana: 2012 0.02; 2013 0.06; 2014 0.07; 2015 0.09; 2016 0.10
  - Guyana: 2012 0.06; 2013 0.18; 2014 0.23; 2015 0.35; 2016 0.37
  - Honduras: 2012 0.06; 2013 0.08; 2014 0.13; 2015 0.18; 2016 0.17
  - Nicaragua: 2012 0.12; 2013 0.26; 2014 0.32; 2015 0.40; 2016 0.39
  - Senegal: 2012 0.05; 2013 0.14; 2014 0.22; 2015 0.31; 2016 0.35
- Sustainability index values (selected countries):
  - Bolivia: 2012 0.47; 2013 0.47; 2014 0.45; 2015 0.43; 2016 0.45
  - Burkina-Faso: 2012 0.54; 2013 0.48; 2014 0.45; 2015 0.38; 2016 0.36
  - Ethiopia: 2012 0.49; 2013 0.57; 2014 0.59; 2015 0.60; 2016 0.58
  - Ghana: 2012 0.68; 2013 0.64; 2014 0.63; 2015 0.62; 2016 0.60
  - Guyana: 2012 0.38; 2013 0.31; 2014 0.28; 2015 0.20; 2016 0.19
  - Honduras: 2012 0.55; 2013 0.53; 2014 0.50; 2015 0.47; 2016 0.47
  - Nicaragua: 2012 0.39; 2013 0.30; 2014 0.27; 2015 0.22; 2016 0.23
  - Senegal: 2012 0.46; 2013 0.35; 2014 0.31; 2015 0.22; 2016 0.21

### VI. Conclusions, limitations, and applications
- Key conclusions:
  - Econometric evidence suggests a structural break in debt accumulation dynamics around 2006 for HIPC/MDRI beneficiaries; forecasts should therefore use post–break information to avoid biased estimates.
  - The probabilistic approach illustrates uncertainty around debt paths and can support DSA via fan charts and a probabilistic sustainability index.
- Limitations:
  - Post–HIPC/MDRI time period is too short for reliable country-by-country estimations.
  - Probabilistic forecasts are constrained by relatively short estimation sample for PVARX and fiscal reaction function (data through 2011).
  - Approach does not explicitly model links from public investment financed by debt to resulting growth and debt-servicing capacity.
- Application:
  - Fan charts serve as probabilistic benchmarks or “confidence bands” to assess standard DSA forecasts and to compute a probabilistic debt sustainability index for relative ranking.

*Source: _wp14102 (IMF staff calculations) — excerpt from provided PDF chapter/section.*

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

### _wp14102 - References

### I. Introduction

- 1996: IMF and World Bank launched the Highly Indebted Poor Countries (HIPC) Initiative to relieve external debt burdens in a number of low-income countries (LICs), mostly in Africa.
- 1999: HIPC Initiative enhanced to provide deeper, broader, and faster debt relief; enhancements aimed to strengthen links between debt relief and poverty reduction.
- 2005: HIPC Initiative supplemented by the Multilateral Debt Relief Initiative (MDRI), adding the possibility of debt relief from the IMF, the World Bank (IDA), and the African Development Fund (AfDF) for countries completing the HIPC process.
- Evaluation and critiques:
  - Review concluded DSF/DSA performed relatively well, with the DSF last reviewed in 2012 (see IMF 2003 and 2012).
  - Concerns noted about potential re-accumulation of unsustainable debt (Leo (2009)) and about quality of inputs into DSAs (Baduel and Price (2012)).
  - Alternative/probabilistic approaches advocated by Garcia and Rigobon (2005), Celasun et al. (2006); imprecision concerns highlighted by Panizza (2008), Wyplosz (2007, 2011).

- Contribution of this paper:
  - Applies the probabilistic approach to the DSA proposed by Celasun et al. (2006) and insights from Hevia (2012).
  - Econometric analysis suggests implementation of HIPC/MDRI may have coincided with a structural break in debt accumulation dynamics for HIPC/MDRI countries.
  - Empirical strategy:
    - Role of "deep debt determinants": real GDP growth, real interest rates, capital gains/losses on foreign-currency debt from exchange rate changes.
    - Includes primary fiscal deficits and debt relief.
    - First step: estimate relationship between deep determinants and debt via a panel VAR (data: 2001–11; 33 countries).
    - Second step: estimate a fiscal reaction function (following Celasun et al. (2006)).
    - Third step: simulate random shocks for determinants and primary deficit; forecast trajectories and present probabilistic forecasts as “fan” charts for eight illustrative countries (Bolivia, Burkina Faso, Ethiopia, Ghana, Guyana, Honduras, Nicaragua, Senegal).

- Paper organization:
  - Section II: data and evolution of public debt for 33 HIPCs (2001–11).
  - Section III: panel data estimation methodology.
  - Section IV: estimation results.
  - Section V: fan chart forecasts.
  - Section VI: conclusions.

### II. Stylized facts: trends and composition of public debt in HIPCs

- Sample and data period:
  - 33 HIPC beneficiaries; dataset covers 2001–11.
  - Afghanistan, The Republic of the Congo, and Liberia excluded due to limited data.

- Aggregate debt dynamics:
  - Average public debt ratio for 33 HIPCs:
    - 2001: 120 percent of GDP
    - 2007: 54 percent of GDP
    - 2011: 41 percent of GDP
  - Decline largely attributed to HIPC Initiative and MDRI debt relief.

- Completion points and coverage:
  - By 2006, 21 out of 33 countries (accounting for more than three-fourths of total debt in dataset) had reached completion points and qualified for 100 percent relief on eligible debt from IMF, IDA, and AfDF.
  - Another eight HIPIC/MDRI beneficiaries reached completion points between 2007 and 2010.
  - By the end of 2011, only Comoros, Côte d'Ivoire, Guinea, and Chad had not yet reached their completion points.
  - Footnote: Comoros reached the completion point in December 2012, Côte d'Ivoire in June 2012, Guinea in September 2012.

- Composition of public debt (average shares):
  - Concessional public and publicly guaranteed (PPG) debt:
    - 2001: 67 percent of total public debt
    - 2007: 63 percent
    - 2011: 53 percent
  - Share of non-concessional debt declined.
  - Average share of public domestic debt grew from about 10 percent to 30 percent of total.

- Domestic vs. external public debt:
  - In 23 out of 33 countries, domestic public debt increased:
    - Average domestic public debt: 2001 = 5 percent of GDP; 2011 = 14 percent of GDP.
  - In remaining countries (10 countries), average public debt ratio decreased from 34 percent to 10 percent of GDP.
  - Observed pattern: public external debt in percent of GDP declined in almost all countries (debt relief), while domestic public debt did not decline in step.

- Decomposition of change in total public debt (2001–11) for 33 HIPCs:
  - Contributors to change in public debt-to-GDP ratio:
    - Real GDP growth: average real GDP growth was 5 percent; contributed significantly to decline in debt ratio in every year.
    - Real interest rates: were negative through 2008, mostly owing to low nominal interest rate on concessional debt (concessional debt majority of public debt); negative real rates reduced debt-to-GDP ratio.
    - Real effective exchange rates: appreciated through 2010, contributing to decrease in debt-to-GDP ratio.
    - Primary fiscal deficits: increased the debt-to-GDP ratio.
    - Debt relief: notable contribution in 2006 and 2007 following MDRI launch.
    - Residual: significant contribution to debt reduction in 2006 and 2007; possible explanation includes mis-recording of debt relief after completion points (e.g., flow vs. stock relief).

- Figures referenced (data sources):
  - Figure 1: Average Public Debt for Countries Eligible for HIPC Initiative Relief (In percent of GDP).
  - Figure 2: Public Debt in Countries Eligible for HIPC Initiative Relief: Comparison of 2001, 2006, and 2011 (In percent of GDP).
  - Figure 3: Average Change in total Public Debt for 33 HIPCs (In percent).
  - Source for figures and calculations: IMF, Strategy, Policy and Review Department DSA Database and own calculations.

### III. Overview of the methodology

- Objectives:
  - Determine whether there is a structural break in dynamics of debt accumulation for HIPC/MDRI countries.
  - Apply probabilistic approach to forecast debt paths of HIPC/MDRI beneficiaries.

- Debt accounting identity (nominal public debt evolution):
  - Notation and variables:
    - P_t: total nominal public debt in local currency at time t.
    - P_t^d and P_t^f: domestic and external public debt respectively.
    - i_t^d: implicit nominal interest rate on domestic public debt.
    - i_t^f: implicit nominal interest rate on external public debt.
    - Δe_t: rate of depreciation of nominal exchange rate.
    - PB_t: primary balance.
    - O_t: other debt-creating flows (recapitalization costs, privatization receipts, contingent liabilities, treasury float, debt relief, etc.).
  - After dividing by nominal GDP and replacing nominal by real rates and changes:
    - d_t = function of real GDP growth g_t, real interest rates r_t^d and r_t^f, change in real exchange rate Δq_t, primary balance, and other debt-creating flows.
  - Implicit interest rates computed using formula i_t = (interest payments on initial-period debt) / (initial-period debt) (symbolic formula provided in text).

- Potential structural break due to HIPC/MDRI:
  - Hypothesis: HIPC and MDRI may have changed relationships among deep determinants (g_t, r_t^d, r_t^f, Δq_t), primary balance, and debt levels.
  - Data limitation: only 11 annual observations per country (2001–11); completion points occur at different times across countries.
  - Strategy: test for an early, common break-point around granting of debt relief; chosen candidate common break-point year: 2006 (21 out of 33 HIPCs had reached completion points before 2006; Figures 1–3 indicate 2006 as intuitive candidate).

- Estimation approach:
  - Panel VAR with exogenous regressors (PVARX) used to model dynamics of “deep” determinants while addressing limited time-series dimension by exploiting cross-sectional variation across 33 countries.
  - PVARX specification (equation (1)):
    - y_it = α_i + Σ_{k=1}^{p} Φ_k y_{i,t-k} + Σ_{k=0}^{p} Γ_k x_{i,t-k} + ϵ_{i,t}, i = 1,...,N; t = 1,...,T
    - Where y_it' = (g_it, r_it^d, r_it^f, Δq_it); x_it' = (g*_t, r*_t) where g* is world real GDP growth and r* is world real interest rate; α_i is country fixed effect.
    - Error terms ϵ_{i,t} are i.i.d. with E[ϵ_{i,t}] = 0 and covariance Ω; slope coefficients Φ and Γ and covariance matrix Ω are common across countries.
  - Novel aspects:
    - Use of PVARX (rather than fully endogenous VAR) in probabilistic public DSA context.
    - Implementation of bootstrap bias correction algorithm (Pesaran and Zhao (1999); Everaert and Pozzi (2007)) to address bias from fixed effects in short T dynamic panel estimation.
  - Two-step procedure:
    - Step 1: estimate PVARX and test for structural break (common break-point, candidate year 2006).
    - Step 2: estimate fiscal reaction function and incorporate into probabilistic debt path simulations.
  - Constraints and considerations:
    - Short time dimension (T = 11) motivates pooling and panel approach.
    - Countries reached completion points at varying times; common break test mitigates heterogeneity in timing.

- Simulation and forecasting:
  - After estimating PVARX and fiscal reaction function, simulate sets of random shocks for each determinant of debt and the primary deficit.
  - Forecast trajectories of deep determinants, primary balance, and debt.
  - Illustrate probabilistic forecasts using fan charts for eight selected countries: Bolivia, Burkina Faso, Ethiopia, Ghana, Guyana, Honduras, Nicaragua, Senegal.

*Source: _wp14102 - References (PDF).*

### 17.      While we are not aware of any established procedure to test for a break–point in a

### 17. While we are not aware of any established procedure to test for a break–point in a PVARX specification

### Break–point testing (SS and BP tests)
- Tests applied: “split sample” (SS) test (assumes covariance matrix of errors unchanged) and “break–point” (BP) test (allows covariance matrix to change).
- Both SS and BP tests rejected the null hypothesis of constant parameters over the whole period (2001–11):
  - For the full sample of 33 HIPCs.
  - For the subsample of 21 HIPCs that reached their completion points before or during 2006.
- Implication: post–2006 dynamics of debt accumulation for HIPC/MDRI beneficiaries have changed and forecasts must account for that.

### Interpretation and caveats on break–point results
- The SS and BP tests confirm a significant change in the dynamics of the “deep” determinants of public debt, but:
  - Other factors besides HIPC/MDRI could have caused the changes in dynamics; this is noted as an area for further study (not pursued in this paper).
- Practical implication: forecasts of debt accumulation for HIPC/MDRI beneficiaries need to allow for changed post–2006 dynamics.

### Fiscal reaction function: specification and estimation approach
- Purpose: tie the primary balance to public debt to capture how governments adjust primary balances in response to the public debt-to-GDP ratio.
- Estimation period for the fiscal reaction function focused on 2007–11 to capture post–break debt dynamics.
- General specification (as in Celasun et al. (2006)):
  - Dependent variable: primary balance to GDP, denoted ܾ݌௜௧.
  - Key regressors: lagged public debt-to-GDP (݀௜௧ିଵ), demeaned real GDP growth (݌ܽ݃ݕ௜௧), World Bank’s CPIA index (ܽ݅݌ܿ௜௧) as institutions, country fixed effect (ߤ௜), and i.i.d. shocks (ߝ௜௧).
- Endogeneity issues addressed:
  - Lagged debt may correlate with country fixed effects.
  - Output gap may correlate with contemporaneous fiscal shocks.
  - Estimator used: Arellano and Bover (1995)/Blundell and Bond (1998) System GMM.

### Nonlinear fiscal reaction specification (debt spline and business cycle interaction)
- Nonlinear specification allows differentiated response when lagged debt exceeds 50 percent of GDP and differentiated output gap response by phase of the business cycle:
  - Dummy ܦ௜௧ିଵ = 1 if lagged debt > 50 percent of GDP.
  - Dummy ܲ௜௧ = 1 if output gap is positive (expansion).
- Interpretation:
  - ρ is response of primary balance to a change in lagged debt when lagged debt < 50 percent of GDP.
  - ߩ + ߩ̅ is response when lagged debt > 50 percent of GDP.
- Note on threshold: 50 percent corresponds to the DSF’s threshold for Debt/GDP for countries with strong policy component.

### Estimation results — summary and preferred specification
- Estimators reported: OLS, LSDV, and multiple System GMM specifications (see Table 1).
- Key sample and estimation notes:
  - First five specifications estimated for 2001–11; final specification estimated for 2007–11.
  - Number of observations reported in Table 1: 330, 330, 231, 231, 231, 198 (across columns).
  - Note: * = significant at 10%; ** at 5%; *** at 1%.
- Findings and model selection:
  - System GMM (1) showed a puzzling negative coefficient on lagged debt in one specification (implying fiscal expansion with higher lagged debt), suggesting nonlinearity.
  - Introducing a “debt spline” for debt > 50 percent of GDP (System GMM (2)) produced:
    - Positive coefficient on lagged debt for countries with “low” levels of debt (<50 percent of GDP).
    - Negative coefficient for the “high” debt dummy (>50 percent of GDP).
  - Testing differential reaction over the business cycle (System GMM (3)) via a positive output gap dummy returned an estimated coefficient that was not statistically significant; the variable was therefore not retained.
  - Preferred specification: System GMM (4) estimated for 2007–11:
    - Coefficient on lagged debt: positive.
    - Coefficient on “high” debt dummy (>50 percent of GDP): negative.
    - Coefficient on institutional quality (CPIA): positive but statistically insignificant (limited variability may explain insignificance).
- Diagnostic tests reported for System GMM specifications include Hansen test (P-values reported), AR(1) and AR(2) tests (P-values reported), and number of instruments (always less than number of cross-section units).

### Putting it all together — Fan charts forecasts (methodology and illustrative results)
- Forecast construction steps:
  (i) Use PVARX estimates to predict trajectories of the “deep” determinants (real GDP growth, real interest rates, capital gains/losses from exchange rate changes) assuming common distribution/co-movements and country fixed effects.
  (ii) Predict primary balances using System GMM equation (4) from Table 1; policy shocks (߮௜௧ாఛ) drawn from mean-zero normal distribution with variance equal to country-specific variance of residuals.
  (iii) Combine (i) and (ii) to produce many debt paths and display them as “fan” charts.
- Fan chart display conventions (illustrative Figure 4):
  - Middle area: 20 percent confidence interval around the median projection (median shown as an orange line for 2012–16).
  - Fan cone: 80 percent confidence interval around the median projection.
  - IMF/WB baseline DSA forecasts superimposed as red lines.
- Illustrative country example (HIPCs that reached completion points in 2006 or before): Bolivia, Burkina Faso, Ethiopia, Ghana, Guyana, Honduras, Nicaragua, Senegal.
- Uses and interpretation:
  - Fan charts serve as a probabilistic benchmark or “confidence interval” for DSA forecasts.
  - Example policy threshold: 50 percent of debt to GDP — in the illustrative set, Guyana, Nicaragua, and Senegal would have debt outlooks considered negative relative to that threshold.
  - In some cases (Bolivia, Burkina Faso, Ghana) the baseline DSA forecasts fall within the 20 percent confidence interval around the fan chart median, indicating overlapping median expectations.
  - Divergences between DSA and fan charts may indicate that DSA incorporates information not captured by the fan charts and warrant further investigation.
- Limitations:
  - Probabilistic forecasts constrained by relatively short estimation sample for PVARX and fiscal reaction function (data through 2011).
  - Approaches do not explicitly model links from public investment financed by debt to resulting growth and debt-servicing capacity.

### Probabilistic Debt Sustainability Index (application)
- Probabilistic forecasts can be interpreted as probabilities of specified events (e.g., debt ratio exceeding a threshold in a specific year).
- Celasun et al. (2006) index formula (textual description preserved):
  - Index constructed as product of:
    - Probability that public debt-to-GDP falls below its initial level ߬ periods ahead.
    - Probability that the ratio of public debt-to-GDP does not exceed the starting ratio by more than x percentage points in ߬ periods ahead.
  - In the application x is set at 10 percent.
  - The index itself has no direct probabilistic interpretation; it combines informational content of the probabilistic forecasts where higher index values indicate more desirable positions (high probability of debt decline and high probability that increase does not exceed the cap).

*Source: Excerpt from the PDF chapter/section provided.*

### 33.      We follow Celasun et al. (2006) and calculate an index by multiplying the calculated

### _wp14102 - 33.      We follow Celasun et al. (2006) and calculate an index by multiplying the calculated

### Probabilistic DSA methodology and index definition
- The study follows Celasun et al. (2006) and constructs an index by multiplying the calculated values for two probabilistic forecasts of debt developments to enable relative comparisons and ranking of countries.
- 1/ The index is calculated as follows:Pr ሺ݀ ௧ାఛ ݀൏ ௧ ሻ ∗ ሾ 1െPr ሺ݀ ௧ାఛ ݀൐ ௧ ሻݔ൅ ሿ , where x is a debt increase considered to be acceptable within the projection period (see above).
- Fan charts are used to illustrate public debt-to-GDP ratio projections, with a fan cone = 80% confidence interval around the median. 2012 is the first year of the projection period (source: IMF staff calculations).

### Illustrative findings from Figures and Table 2
- Table 2 reports the probabilistic debt sustainability index and component probabilities for a selection of post–HIPCs.
- The authors note: "Table 2 shows that Bolivia, Ethiopia, Ghana, and Honduras would breach a threshold in the final year of the projection period, when a threshold of 0.4 is chosen for illustrative purposes."
- Probabilities that the debt ratio is lower than in 2011 (by year):
  - Bolivia: 2012 0.51; 2013 0.54; 2014 0.55; 2015 0.54; 2016 0.57
  - Burkina-Faso: 2012 0.58; 2013 0.56; 2014 0.55; 2015 0.49; 2016 0.49
  - Ethiopia: 2012 0.54; 2013 0.64; 2014 0.68; 2015 0.69; 2016 0.68
  - Ghana: 2012 0.69; 2013 0.68; 2014 0.68; 2015 0.68; 2016 0.66
  - Guyana: 2012 0.40; 2013 0.37; 2014 0.36; 2015 0.31; 2016 0.31
  - Honduras: 2012 0.58; 2013 0.57; 2014 0.58; 2015 0.57; 2016 0.57
  - Nicaragua: 2012 0.44; 2013 0.41; 2014 0.39; 2015 0.37; 2016 0.38
  - Senegal: 2012 0.48; 2013 0.41; 2014 0.40; 2015 0.31; 2016 0.33
- Probabilities that the debt ratio is more than 10 percent of GDP higher than in 2011 (by year):
  - Bolivia: 2012 0.09; 2013 0.14; 2014 0.18; 2015 0.21; 2016 0.22
  - Burkina-Faso: 2012 0.07; 2013 0.14; 2014 0.18; 2015 0.24; 2016 0.27
  - Ethiopia: 2012 0.08; 2013 0.12; 2014 0.13; 2015 0.13; 2016 0.15
  - Ghana: 2012 0.02; 2013 0.06; 2014 0.07; 2015 0.09; 2016 0.10
  - Guyana: 2012 0.06; 2013 0.18; 2014 0.23; 2015 0.35; 2016 0.37
  - Honduras: 2012 0.06; 2013 0.08; 2014 0.13; 2015 0.18; 2016 0.17
  - Nicaragua: 2012 0.12; 2013 0.26; 2014 0.32; 2015 0.40; 2016 0.39
  - Senegal: 2012 0.05; 2013 0.14; 2014 0.22; 2015 0.31; 2016 0.35
- Sustainability index values (by year):
  - Bolivia: 2012 0.47; 2013 0.47; 2014 0.45; 2015 0.43; 2016 0.45
  - Burkina-Faso: 2012 0.54; 2013 0.48; 2014 0.45; 2015 0.38; 2016 0.36
  - Ethiopia: 2012 0.49; 2013 0.57; 2014 0.59; 2015 0.60; 2016 0.58
  - Ghana: 2012 0.68; 2013 0.64; 2014 0.63; 2015 0.62; 2016 0.60
  - Guyana: 2012 0.38; 2013 0.31; 2014 0.28; 2015 0.20; 2016 0.19
  - Honduras: 2012 0.55; 2013 0.53; 2014 0.50; 2015 0.47; 2016 0.47
  - Nicaragua: 2012 0.39; 2013 0.30; 2014 0.27; 2015 0.22; 2016 0.23
  - Senegal: 2012 0.46; 2013 0.35; 2014 0.31; 2015 0.22; 2016 0.21

### Simulation design and econometric approach
- The authors estimate relationships between determinants and debt and also estimate a generic fiscal reaction function to account for fiscal behavior.
- They simulate sets of random shocks for determinants of debt and the primary deficit, plug these into the estimated equations, and construct future trajectories for determinants, the primary balance, and the debt ratio.
- Probabilistic forecasts are used as illustrative "confidence bands" to assess DSA forecasts and to construct a probabilistic debt sustainability index.

### Key conclusions (Section VI)
- Challenges identified:
  - The post–HIPC/MDRI time period is too short for country-by-country estimations.
  - Econometric evidence suggests a break–point in debt accumulation dynamics around 2006; forecasts therefore use post–break–point information to avoid biased coefficient estimates.
- The probabilistic approach can illustrate uncertainty around debt paths and support debt sustainability analysis via fan charts and a probabilistic sustainability index.

### Appendix: econometric results and tests (high-level)
- Table 1 and Table 2 present PVARX estimates for samples of HIPCs (33 HIPCs and 21 HIPCs with completion points pre-2006 and in 2006) for 2007-2011. Standard errors in parentheses; * = significant at 10%; ** at 5%; *** at 1%.
- Chow tests (Table 3) report test statistics and p-values rejecting the null hypothesis of no structural break in 2006:
  - 95% Quantile values shown: 186.1; 203.6; 133.2; 151.0 (as presented in the source)
  - Statistics: 227.8; 363.4; 188.6; 266.8
  - P-values: 0.000; 0.000; 0.000; 0.000
- Note: Source for econometric inputs is "IMF, Strategy, Policy and Review Department DSA Database and own calculations."

*Source: IMF staff calculations as presented in the provided content.*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2014/_wp14102.pdf_