## 1. Countries and Default Episodes in the Full Sample

## Source details

**Canonical URL:** [1. Countries and Default Episodes in the Full Sample](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2005/_wp0542.pdf)

## Other formats

- [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2005/_wp0542.pdf.md)
- [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2005/_wp0542.pdf.json)

---

### Introduction
- Following the debt crises of the 1980s, sovereign debt defaults have become more frequent (examples cited: Russia, Ecuador, Argentina; restructurings under coercive threat: Ukraine, Pakistan, Uruguay; crises averted through large-scale IMF support: Mexico, Brazil, Turkey).
- Core questions the paper addresses:
  - What set of economic and political conditions is empirically associated with a likely occurrence of a sovereign debt crisis?
  - Can thresholds for vulnerability indicators be derived that signal higher likelihood of a sovereign debt crisis?
- Motivation includes surveillance failures where international financial institutions, rating agencies, private agents, and academics failed to correctly assess default likelihood.

### Data and Definitions
- Dataset: annual observations for 47 emerging market economies from 1970 to 2002.
- Debt crisis definition (country is in a “debt crisis” if):
  - Classified as being in default by Standard & Poor’s (failure to meet principal or interest payment on an external obligation on due date, including exchange offers, debt equity swaps, and buy back for cash), OR
  - Receives a large nonconcessional IMF loan (where “large” means in excess of 100 percent of quota).
- IMF lending considered mainly via Stand By Arrangements (SBA) and Extended Fund Facilities (EFF).

### Methodology
- Statistical approach: Binary Recursive Tree methodology (BRT), a computer-intensive data mining technique that selects explanatory variables, critical values, and interactions to identify “safe” from “crisis-prone” types.
- Software employed: CART developed by Salford Systems.
- Key BRT specifications used:
  - Criterion minimized: the Gini index.
  - A priori distribution of crises: equal to the sample distribution.
  - Cost of missing a crisis set to seven times the cost of missing a noncrisis.
  - Built-in 10-fold cross-validation used to avoid overfitting.

### Key Findings — Predictors and Thresholds
- From 50 candidate variables, 10 predictor variables are sufficient for classification and prediction:
  - total external debt/GDP ratio
  - short-term debt reserves ratio
  - real GDP growth
  - public external debt/fiscal revenue ratio
  - CPI inflation
  - number of years to the next presidential election
  - U.S. treasury bills rate
  - external financial requirements (current account balance plus short-term debt as a ratio of foreign reserves)
  - exchange rate overvaluation
  - exchange rate volatility
- A relatively “safe” country type is described by:
  - total external debt below 49.7 percent of GDP
  - short-term debt below 130 percent of reserves
  - public external debt below 214 percent of fiscal revenue
  - exchange rate overvaluation below 48 percent
- Three major types of risks identified:
  - Solvency (debt unsustainability)
    - Characterized by external debt in excess of 49.7 percent of GDP, together with monetary or fiscal imbalances and large external financing needs that signal illiquidity as an element of debt unsustainability.
  - Illiquidity (liquidity risk types)
    - Identified by moderate debt levels but with short-term debt in excess of 130 percent of reserves coupled with political uncertainty and tight international capital markets.
  - Macroexchange rate risks
    - Arise from the combination of low growth and relatively fixed exchange rates.
- First global split and conditional probabilities:
  - Split variable: total external debt (percent of GDP).
  - Threshold: more than 49.7 percent of GDP → right branch.
  - Conditional crisis probability for whole sample: 20.5 percent.
  - Conditional crisis probability for right branch (external debt > 49.7 percent of GDP): 45.4 percent.
  - Conditional crisis probability for left branch (external debt ≤ 49.7 percent of GDP): 9.7 percent.

### Empirical Tree — High-risk and Safe Nodes (selected exact figures)
- Terminal node 14:
  - Observations: 196.
  - Within-node crisis share: 66.8 percent (also reported as 66.9 percent).
  - More than half of all crisis episodes satisfy: total external debt/GDP exceeds 49.7 percent AND inflation exceeds 10.5 percent.
  - Node 14 accounts for 50.2 percent of all debt crises (Column 5 example).
- Terminal node 7 (liquidity type):
  - Node conditions include: intermediate external debt (between 49.7 percent and 19 percent of GDP), short-term debt > 130 percent of reserves, exchange-rate volatility low, years to next presidential election ≤ 5.5.
  - Observations: 130 episodes.
  - Crisis probability: 41.5 percent.
  - % of Crises: 20.7 percent.
  - % of NonCrisis: 7.5 percent.
  - Relative Safety: 0.73.
- Terminal node 3 (favorable conditions):
  - Conditions: low external debt, short-term debt to reserves below 130 percent, public external debt to revenue below 210 percent, economy not in recession.
  - Observations: 607 episodes.
  - Likelihood of crisis episode: 2.3 percent.
  - About 58.4 percent of all noncrisis episodes satisfy these conditions.
- Node 11 (safe despite high external debt):
  - Observations: 98 episodes.
  - Crisis Prob: 2.0 percent.
  - % of NonCrisis: 9.5 percent.
  - Relative Safety: 1.23.
  - Example: Algeria (1991).
- Node 2 (macro/exchange extreme):
  - Observations: 4 episodes.
  - Crisis Prob: 100.0 percent.
  - % of Crises: 1.5 percent.
  - Characterized by TotExtDebt/GDP ≤ 49.7, ShortTDebt/Res ≤ 1.3, PublicExtDebt/Revenue ≤ 2.1, GDPGrowth ≤ -5.5, EchROverV > 48 → crisis probability 100.0 percent.

### Classification of Crisis Types (tree-based typology)
- Liquidity Crisis-Prone (type B):
  - Nodes 7 and 10.
  - Common feature: short-term debt to reserves in excess of 130 percent.
  - Node 10: tight international monetary conditions (USTresBill > 9.7) → crisis probability 40.0 percent.
  - Liquidity type (node 7) makes up 20.7 percent of all debt crisis episodes.
- Unsustainable Debt Path / Solvency Crisis-Prone (type C):
  - Nodes 5, 12, 13, and 14.
  - Common thresholds: TotExtDebt/GDP > 49.7 or PublicExtDebt/Revenue > 215 percent.
  - Partition 14: TotExtDebt/GDP > 49.7 and Inflation > 10.5 → crisis probability 66.9 percent.
  - Nodes 13 and 12: high external debt and ExtFinReqs/Reserve > 1.4 raise default probabilities to 46.8 percent (node 13) and 40.0 percent (node 12).
  - Nodes 13 and 12 jointly account for 15.7 percent of crises.
- Exchange Rate and Macrocrisis-prone (type D):
  - Node 2 and similar partitions where low growth and large exchange-rate overvaluation coincide with crisis risk.
- Relatively Safe (Block A):
  - Nodes 1, 4, 6, 8, 9, 11, and 3.
  - Common attributes: low TotExtDebt/GDP (below 49.7 percent except partition 11), short-term debt/reserves below 130 percent, public external debt/revenue below 215 percent, inflation below 10.5 percent, growth above -5.5 percent in many partitions.

### Descriptive Statistics (selected exact means and observations from Table 2)
- Total external debt over GDP: 45.5 / 37.0 / 54.7 / 71.4 / 63.7 / 1,051
- Total external debt over exports: 290.7 / 239.3 / 359.3 / 455.9 / 350.2 / 1,053
- Short-term external debt (RM, over GDP): 10.9 / 9.4 / 15.0 / 15.1 / 15.7 / 993
- Short-term external debt (RM, over reserves): 1.7 / 1.2 / 2.9 / 2.9 / 2.2 / 948
- Public external debt over GDP: 32.2 / 25.5 / 36.4 / 53.0 / 46.5 / 1,051
- Public external debt over revenue: 1.7 / 1.3 / 1.9 / 3.0 / 2.3 / 827
- Current account balance: -2.7 / -2.7 / -4.3 / -2.6 / -1.3 / 1,231
- Inflation: 54.6 / 17.5 / 241.1 / 169.6 / 84.9 / 1,273
- Real GDP growth: 4.1 / 4.8 / 1.8 / 2.1 / 2.2 / 1,276
- Financing requirement: 1.6 / 1.3 / 3.0 / 2.4 / 1.4 / 986
- U.S. treasury bill rate: 6.4 / 6.3 / 7.8 / 6.9 / 6.3 / 1,229
- LIBOR: 9.7 / 9.5 / 10.5 / 10.5 / 9.4 / 1,229
- M2 over reserves: 5.6 / 5.3 / 7.9 / 6.2 / 6.2 / 1,185
- Reserves growth: 19.1 / 20.8 / -5.4 / 17.8 / 22.9 / 1,155
- Note: the pattern across states is deterioration before and during crises and improvement on exit; volatility measures (real exchange rate and inflation variability) rise during crises.

### Prediction Performance (exact figures from Table 4)
- Full Sample (observations 1,276):
  - Correctly called episodes: 82.8 percent.
  - Correctly called entries into default: 88.9 percent.
  - Incorrectly called entries into default: 18.5 percent.
  - Correctly called exits from default: 32.0 percent.
  - Incorrectly called exits from default: 4.8 percent.
  - Correctly called default episodes: 93.9 percent.
  - Correctly called nondefault episodes: 79.9 percent.
- 1990’s Onwards (observations 556):
  - Correctly called episodes: 78.8 percent.
  - Correctly called entries into default: 85.0 percent.
  - Incorrectly called entries into default: 15.0 percent.
  - Correctly called exits from default: 35.3 percent.
  - Incorrectly called exits from default: 64.7 percent.
  - Correctly called default episodes: 89.5 percent.
  - Correctly called nondefault episodes: 76.0 percent.
- Overall summary: tree correctly predicts 82.9 percent of states, 93.9 percent of crisis, and 79.9 percent of noncrisis states.

### Variable Importance and Robustness (selected exact figures from Table 6 and checks)
- Full Sample (1970–2002) top importance scores:
  - Tot Ext Debt/GDP: Importance 100, Rank 1.
  - Pub Ext Debt/GDP: Importance 88.349, Rank 2.
  - Tot Ext Debt/Exp: Importance 69.352, Rank 3.
  - ShTermExtDebt/Res (orig mat): Importance 34.55, Rank 5.
  - ExtFinRequir: Importance 28.233, Rank 8.
  - Nominal GDPGrowth: Importance 25.503, Rank 9.
  - Inflation: Importance 24.748, Rank 10.
- Post-1990 (1990–2002) top importance scores:
  - ExtFinRequir: Importance 100, Rank 1.
  - ShTermExtDebt/Res (o.mat): Importance 72.56, Rank 2.
  - PubExtDev/Res: Importance 71.255, Rank 3.
  - Inflation: Importance 56.891, Rank 4.
  - ShTermExtDebt/GDP(rem.mat): Importance 54.272, Rank 5.
- Correlation between full sample and post-1990 variable importance:
  - Correlation = 0.466689.
  - Rank correlation = 0.614502.
- Distributional checks (N1990/N ratios) show no discernible systematic difference between the 1990s and full sample; some nodes are over- or under-represented (examples: Node 7 liquidity 1.202039 over-represented; Node 11 safe 1.573834 over-represented).

### Policy Implications and Recommendations (exact phrasing preserved)
- Correct diagnosis matters for the appropriate policy response:
  - Solvent but illiquid countries with large short-term debt may need IMF support to avoid liquidity runs or “roll-off” crises.
  - Highly indebted countries may require strong and credible fiscal consolidation.
  - Conditionality should set targets indicating that macroeconomic fundamentals are heading towards a relatively “safe” zone.
- Unconditional thresholds (for example, debt-output ratios) are of little value per se for assessing probability of default.
- Political instability and tight monetary conditions in international financial markets aggravate liquidity problems.
- The tree approach can:
  - Derive rules of thumb or vulnerability thresholds useful for early warning.
  - Derive policy adjustment paths to reduce crisis likelihood for countries entering danger zones.
  - Be used for surveillance, crisis prevention, and crisis resolution.
- Prediction trade-offs:
  - Model predictive accuracy is high but type II errors (false alarms) are somewhat higher than desirable.
  - The tree approach allows adjustment of relative weights given to type I and type II errors; future challenge is calibrating to reduce false alarms while maintaining high predictive ratio.

*Source: _wp0542 — 1. Countries and Default Episodes in the Full Sample (excerpts).*

### 1. Countries and Default Episodes in the Full Sample ..............................................................9

### 1. Countries and Default Episodes in the Full Sample

### Introduction
- Following the debt crises of the 1980s, sovereign debt defaults have become more frequent, including outright defaults (Russia, Ecuador, Argentina), restructurings under coercive threat (Ukraine, Pakistan, Uruguay), and crises averted through large-scale IMF support (Mexico, Brazil, Turkey).
- The macroeconomic and structural weaknesses leading to sovereign defaults are not fully understood; many practitioners still rely on simple rules of thumb (IMF, 2003) that may miss core elements of crises.
- The paper aims to answer:
  - What set of economic and political conditions is empirically associated with a likely occurrence of a sovereign debt crisis?
  - Can thresholds for vulnerability indicators be derived that signal higher likelihood of a sovereign debt crisis?
- Motivation includes surveillance failures where international financial institutions, rating agencies, private agents, and academics failed to correctly assess default likelihood.

### Data and Definitions
- Dataset: annual observations for 47 emerging market economies from 1970 to 2002.
- A country is defined to be in a “debt crisis” if:
  - It is classified as being in default by Standard & Poor’s (failure to meet principal or interest payment on an external obligation on due date, including exchange offers, debt equity swaps, and buy back for cash), or
  - It receives a large nonconcessional IMF loan (where “large” means in excess of 100 percent of quota).

### Methodology
- Statistical approach: Binary Recursive Tree methodology (BRT), a computer-intensive data mining technique that selects explanatory variables, critical values, and interactions to identify “safe” from “crisis-prone” types.
- Software employed: data mining software CART developed by Salford Systems.

### Key Findings
- From 50 candidate variables, 10 predictor variables are sufficient for classification and prediction:
  - total external debt/GDP ratio
  - short-term debt reserves ratio
  - real GDP growth
  - public external debt/fiscal revenue ratio
  - CPI inflation
  - number of years to the next presidential election
  - U.S. treasury bills rate
  - external financial requirements (current account balance plus short-term debt as a ratio of foreign reserves)
  - exchange rate overvaluation
  - exchange rate volatility
- A relatively “safe” country type is described by:
  - total external debt below 49.7 percent of GDP
  - short-term debt below 130 percent of reserves
  - public external debt below 214 percent of fiscal revenue
  - exchange rate overvaluation below 48 percent
- Three major types of risks identified:
  - Solvency (debt unsustainability)
    - Characterized by external debt in excess of 49.7 percent of GDP, together with monetary or fiscal imbalances and large external financing needs that signal illiquidity as an element of debt unsustainability.
  - Illiquidity (liquidity risk types)
    - Identified by moderate debt levels but with short-term debt in excess of 130 percent of reserves coupled with political uncertainty and tight international capital markets.
  - Macroexchange rate risks
    - Arise from the combination of low growth and relatively fixed exchange rates.
- Each risk type differs in its likelihood of producing a crisis.

### Policy Implications
- Correct diagnosis matters for the appropriate policy response:
  - Solvent but illiquid countries with large short-term debt may need IMF support to avoid liquidity runs or “roll-off” crises.
  - Highly indebted countries may require strong and credible fiscal consolidation.
  - Conditionality should set targets indicating that macroeconomic fundamentals are heading towards a relatively “safe” zone.
- The typology (solvency vs. illiquidity vs. macroexchange risks) informs distinct policy options for prevention and response.

### Implications for Sustainability Analysis
- Unconditional thresholds (for example, debt-output ratios) are of little value per se for assessing probability of default.
- A country may be heavily indebted but have negligible probability of default if debt burden coexists with monetary stability, a large current account surplus, and sound public finances.
- Conversely, moderate debt ratios can coincide with considerable default risk when joint effects of short maturity, political uncertainty, and relatively fixed exchange rates increase likelihood of a liquidity crisis.

### Plan of the Paper
- Section II: review of the literature.
- Section III: dataset description.
- Binary Recursive Tree methodology is reviewed (subsequent sections not reproduced here).

*Source: _wp0542 - 1. Countries and Default Episodes in the Full Sample*

### Section IV, and applied to the data in Section V. Section VI discusses a number of

### _wp0542 - Section IV, and applied to the data in Section V. Section VI discusses a number of

### Review of the literature
- Literature on sovereign debt crises grouped into four categories: (i) theoretical models of sovereign debt and default; (ii) empirical studies of the determinants of debt crisis; (iii) empirical studies of the predictive power of credit ratings; and (iv) empirical studies of the determination of sovereign spreads.
- Conceptual distinctions used in the literature:
  - Ability to pay (solvency): depends on stock of debt relative to capacity to pay (GDP, exports, government revenues). A sovereign is solvent if the discounted value of future primary balances ≥ current net public debt stock; similarly for trade balances and net external debt.
  - Flow imbalances (primary/overall fiscal deficits, trade/current account imbalances) matter because persistent flows accumulate debt and violate intertemporal budget constraint.
  - GDP growth and terms of trade shocks affect ability to pay.
  - Exchange rate regime/misalignment can cause external imbalances and balance-sheet effects when debt is in foreign currency.
  - Willingness to pay: depends on relative costs of default vs servicing—main costs include loss of access to international capital markets and output/trade costs of default. Low output growth reduces cost of being cut off from markets.
  - Openness affects costs of default (more open economies lose more from trade disruptions).
  - Policy credibility measures (low inflation, low money growth), institutional/political factors (rule of law, property rights), political regime change and elections influence willingness to pay.
  - Illiquidity: short-term debt over reserves, M2 over reserves, external financing gap, interest burden can proxy liquidity and refinancing ability.
- Definitions of debt crisis vary:
  - Detragiache and Spilimbergo (2001): arrears on external obligations > 5 percent of commercial debt outstanding or rescheduling/restructuring with commercial creditors.
  - Standard & Poor’s (2002): sovereign in default if not current on any debt obligation.
- Empirical findings summarized:
  - Measures of solvency (debt-to-GDP) and liquidity (short-term debt over reserves, debt service over reserves/exports) significant in predicting crises, alongside macro controls (real growth, inflation, exchange rate overvaluation, fiscal balance).
  - Reinhart (2002): in 84 percent of cases a debt crisis is preceded by a currency crisis.
  - Detragiache and Spilimbergo (2001): short-term debt, debt service, and reserves enter separately; ratio restrictions (e.g., short-term debt/reserves) may be rejected. Short-term debt is endogenous (difficulty borrowing long term before crises).
  - Catão and Sutton (2002): including volatility measures (terms of trade, fiscal, monetary, exchange rate volatility) markedly increases in-sample predictive power.
  - Manasse, Roubini, and Schimmelpfenning (2003): logit model for 1970–2002 predicts about three quarters of crisis entries with few false alarms.
- Conclusion for empirical model variables:
  - Use measures of solvency (public/external debt relative to capacity to pay), liquidity (short-term external debt, debt service), political/institutional variables (willingness to pay), macro variables (real growth, inflation, exchange rate), and measures of external and policy volatility.

### The Data
- Full dataset: annual information on 47 economies with market access from 1970 to 2002 (transition economies included from 1995–2002).
- Key data sources: Standard & Poor’s (debt crisis indicator), IMF lending records (nonconcessional lending), World Bank’s Global Development Finance (external/public debt), IMF World Economic Outlook and Government Finance Statistics (public finance and macro variables).
- Debt crisis definition used in study:
  - Country is in debt crisis if classified in default by Standard & Poor’s OR if it receives a large nonconcessional IMF loan defined as access in excess of 100 percent of quota.
  - Standard & Poor’s default: government fails to meet principal or interest payment on external obligation on due date (including exchange offers, debt equity swaps, buy back for cash).
  - Large IMF loan threshold: in excess of 100 percent of quota (selects top 10 percent of loans by loan-to-quota ratio).
  - Classification includes actual defaults and “incipient” defaults avoided by large IMF support; a country can be in crisis for an extended period.
- IMF lending considered mainly via Stand By Arrangements (SBA) and Extended Fund Facilities (EFF).

### Table 1 — Sample composition and crisis episodes (summary points)
- Total crisis episodes reported: 54
- Average length in crisis: 5.5 (units implied by table as years)
- Selected country entries (as reported in table format):
  - Argentina: 3 episodes, Average Length in Crisis 5.0, Years 1982–94, 1995–96, 2001–
  - Brazil: 3 episodes, Average Length in Crisis 5.3, Years 1983–95, 1998–00, 2001–
  - Chile: 1 episode, Average Length in Crisis 8.08, Years 1983–91
  - China: 0 episodes, Average Length ...0
  - Colombia: 0 episodes, Average Length ...0
  - Costa Rica: 1 episode, Average Length 10.10, Years 1981–91
  - Dominican Republic: 1 episode, Average Length 22.22, Years 1981–
  - Ecuador: 2 episodes, Average Length 8.01, Years 1982–96, 1999–01
  - (Table continues across listed countries; total Obs. and notes: transition economies included only from 1995 onwards.)

### Descriptive statistics (Table 2 — selected means and patterns)
- Explanatory variables grouped into: (i) macroeconomic fundamentals; (ii) variability indicators; (iii) political economy variables.
- General pattern of means across states (No crisis → Year before entry → In-crisis → Year before exit): expected deterioration before and during crises and improvement on exit.
- Selected exact means from Table 2 (current-year, All / No crisis / No crisis (year before entry?) / Crisis / Crisis (year before exit?) / Obs. where provided):
  - Total external debt over GDP: 45.5 / 37.0 / 54.7 / 71.4 / 63.7 / 1,051
  - Total external debt over exports: 290.7 / 239.3 / 359.3 / 455.9 / 350.2 / 1,053
  - Short-term external debt (RM, over GDP): 10.9 / 9.4 / 15.0 / 15.1 / 15.7 / 993
  - Short-term external debt (RM, over reserves): 1.7 / 1.2 / 2.9 / 2.9 / 2.2 / 948
  - Interest on short-term external debt over GDP: 0.5 / 0.5 / 0.8 / 0.6 / 0.7 / 754
  - Debt service on short-term external debt over GDP: 5.3 / 4.8 / 6.9 / 6.4 / 7.1 / 1,050
  - Public external debt over GDP: 32.2 / 25.5 / 36.4 / 53.0 / 46.5 / 1,051
  - Public external debt over revenue: 1.7 / 1.3 / 1.9 / 3.0 / 2.3 / 827
  - Consolidated central government debt: 47.5 / 46.4 / 38.2 / 57.3 / 54.0 / 462
  - Current account balance: -2.7 / -2.7 / -4.3 / -2.6 / -1.3 / 1,231
  - Reserves growth: 19.1 / 20.8 / -5.4 / 17.8 / 22.9 / 1,155
  - Export growth: 12.0 / 13.8 / 4.9 / 6.4 / 7.8 / 1,270
  - M2 over reserves: 5.6 / 5.3 / 7.9 / 6.2 / 6.2 / 1,185
  - Financing requirement: 1.6 / 1.3 / 3.0 / 2.4 / 1.4 / 986
  - LIBOR: 9.7 / 9.5 / 10.5 / 10.5 / 9.4 / 1,229
  - U.S. treasury bill rate: 6.4 / 6.3 / 7.8 / 6.9 / 6.3 / 1,229
  - Inflation: 54.6 / 17.5 / 241.1 / 169.6 / 84.9 / 1,273
  - Nominal GDP growth: 55.9 / 22.8 / 249.4 / 148.6 / 96.0 / 1,276
  - Real GDP growth: 4.1 / 4.8 / 1.8 / 2.1 / 2.2 / 1,276
  - REER growth: 121.5 / 124.3 / 139.7 / 111.0 / 109.4 / 937
  - Import growth: 10.0 / 12.3 / 5.3 / 4.8 / 6.9 / 902
  - FDI over GDP: 1.7 / 1.9 / 1.1 / 1.0 / 1.5 / 1,024
  - Openness: 71.2 / 71.3 / 64.1 / 72.1 / 72.5 / 903
  - Overall balance: -4.3 / -4.4 / -6.3 / -3.8 / -4.1 / 1,013
  - Primary balance: 0.6 / 0.3 / -0.9 / 2.0 / 1.5 / 616
  - Primary gap: 6.6 / 5.7 / 23.1 / 59.0 / -15.0 / 122
  - G.Gov revenue over GDP: 24.3 / 25.4 / 22.7 / 20.1 / 24.2 / 1,013
  - Overall Balance/GDP variability: 1.92 / -2.53 / -1.77 / 19.42 / -1.70 / 824
  - Inflation variability: 120.43 / 10.80 / 125.37 / 549.81 / 83.03 / 1,082
  - Real Exchange Rate Variability: 2.87 / 1.49 / 3.76 / 6.34 / 3.39 / 753
  - Terms of trade variability: 2.24 / 2.28 / 2.32 / 2.23 / 1.33 / 1,088
  - Index of political rights: 3.52 / 3.63 / 3.44 / 3.18 / 3.20 / 1,130
  - Index of civil liberties: 3.77 / 3.85 / 3.75 / 3.44 / 3.65 / 1,130
  - Index of freedom status: 1.21 / 1.16 / 1.21 / 1.39 / 1.39 / 1,130
  - Index of political constraints III: 0.24 / 0.24 / 0.23 / 0.26 / 0.31 / 1,229
  - Index of political constraints V: 0.35 / 0.33 / 0.35 / 0.37 / 0.43 / 1,229
  - Year of parliamentary elections: 0.19 / 0.18 / 0.24 / 0.23 / 0.20 / 1,276
  - Year of presidential elections: 0.09 / 0.06 / 0.11 / 0.18 / 0.15 / 1,276
  - Years to next parliamentary elections: 2.73 / 2.94 / 2.67 / 2.06 / 2.16 / 1,071
  - Years to next presidential elections: 5.02 / 6.47 / 3.15 / 2.18 / 2.39 / 660
  - Electoral system (mode): 1 / 1 / 1 / 1 / 1 / 1,210
- Patterns noted:
  - External debt measures low in no-crisis years, increase in year before crisis, increase further within crisis, and drop before exit but remain above pre-crisis levels.
  - Macroeconomic variables worsen in run-up and within crisis (current account deficits widen, real growth falters, inflation spikes, fiscal balances deteriorate).
  - LIBOR and U.S. treasury bill rate increase in years preceding crises, suggesting tighter external monetary conditions can contribute to debt servicing difficulty.
  - Volatility measures (real exchange rate and inflation) rise in wake of and during crises, falling on verge of exit.
  - Political economy indices show little systematic difference in means across in/out crisis episodes except timing of elections: upcoming elections on average associated with entry and staying in crisis.

### Methodology — Binary Recursive Tree (BRT)
- BRT overview:
  - Binary recursive partitioning algorithm developed by Breiman, Friedman, Olshen, and Stone (1984).
  - Designed to uncover nonlinear structures, interactions, and heterogeneity in complex datasets (high dimensionality, mixed data types, nonhomogeneity).
  - Splitting: parent node split into exactly two child nodes via yes/no questions; algorithm evaluates all possible splits across variables and ranks by a quality-of-split criterion (default: Gini rule).
  - Recursion continues until splitting impossible or stopped according to cost–benefit criterion (reduction in misclassification weighed against cost proportional to number of nodes).
  - Terminal node classification can use plurality rule, can adjust for misclassification costs, priors, and sampling imbalances.
- Properties and advantages:
  - Discovers context dependence, interactions, heterogeneity; robust to outliers (outliers isolated into nodes).
  - No model-specification search required; nonparametric; invariant to monotone transformations of independent variables.
  - Handles missing predictor values via surrogate splits; missing values replaced by best surrogate.
  - Built-in cross-validation (10-fold) to avoid overfitting; can perform well on fresh data without separate test sample.
- Shortcomings:
  - Individual marginal contribution of each variable to probability cannot be ascertained (same probability for all cases in a node).
  - Not well suited for uncovering relationships that hold uniformly across whole sample; may produce many local patterns.
  - Masking: a slightly better splitting variable may prevent inclusion of another relevant variable; solution: compute variable "importance" by potential classification effect (robust to masking).
  - Since nonparametric, no confidence intervals for threshold values; testing possible via logit with node dummies (authors can provide results on request).

### The empirical tree (selected results)
- BRT selected 10 variables out of 50 candidates (as listed in Table 2):
  - total external debt in percent of GDP;
  - short-term debt on a remaining maturity basis to foreign reserves;
  - public external debt to government revenue;
  - real GDP growth;
  - inflation;
  - the U.S. treasury bill rate;
  - exchange rate overvaluation;
  - exchange rate volatility;
  - the ratio of external financing requirements to foreign reserves;
  - number of years before a presidential election.
- First split (threshold and conditional probabilities reported):
  - Split variable: total external debt (percent of GDP).
  - Threshold: more than 49.7 percent of GDP → right branch.
  - Conditional crisis probability for whole sample: 20.5 percent.
  - Conditional crisis probability for right branch (external debt > 49.7 percent of GDP): 45.4 percent.
  - Conditional crisis probability for left branch (external debt ≤ 49.7 percent of GDP): 9.7 percent.
- Subsequent split on high-debt branch: episodes with high debt (>49.7 percent of GDP) are further split by inflation (larger/smaller than threshold reported in Figure 1 of source).

*Sources: IMF, Standard & Poor’s, World Bank, and authors’ calculations.*

### 10.5 percent). The former incur the largest default risk, 66.8 percent: see terminal node 14.

### _wp0542 - 10.5 percent). The former incur the largest default risk, 66.8 percent: see terminal node 14.

### Empirical decision tree — key findings
- More than half of all the crisis episodes in the sample satisfy the two simple conditions: total external debt/GDP exceeds 49.7 percent and inflation exceeds 10.5 percent.
- Terminal node 14: Observations: 196; within-node crisis share: 66.8 percent.
- Terminal node 7: Despite intermediate external debt levels (between 49.7 percent and 19 percent of GDP), the joint effect of short-term debt exceeding 130 percent of reserves, relatively rigid exchange rates (low volatility), and political uncertainty (less than five years to the next presidential elections) raises the crisis probability to 41 percent.
- Terminal node 3 (favorable conditions): low external debt, short-term debt to reserves below 130 percent, public external debt to revenue below 210 percent, and the economy not being in recession — likelihood of crisis episode: 2.3 percent. About 58.4 percent of all noncrisis episodes satisfy these conditions.
- The classification threshold for labeling a node “crisis-prone” is 11 percent (this is below the sample proportion of crises, 20.5 percent). Any threshold between 2.3 percent and 40 percent would not affect the classification in Table 3, Column 4.
- Unconditional debt-to-GDP thresholds are of limited value alone for assessing default probability: e.g., node 7 (debt between 49.7 percent and 19 percent) has a 41 percent crisis probability, while node 11 (debt at least 49.7 percent) has just a 2 percent crisis probability — other factors (short maturity, political uncertainty, exchange-rate rigidity, monetary stability, current account surplus, fiscal revenues) drive the difference.

### Classification tree mechanics and example application
- The tree was obtained with these specifications:
  - criterion minimized: the Gini index;
  - a priori distribution of crises: equal to the sample distribution;
  - cost of missing a crisis set to seven times the cost of missing a noncrisis.
- Classification rule: observations in a node are classified as crisis-prone if the within-node share of crisis observations is higher than in the total sample (with the 11 percent threshold given the asymmetric cost).
- Illustration — Colombia, 2004 (Arias, 2004):
  - Total external debt/GDP = 48.6 percent → answer “no” to exceeding 49.7 percent → move left.
  - Short-term debt over reserves = 98 percent → answer “no” to exceeding 130 percent → move left.
  - Public external debt to revenue = 100 percent → answer “no” to exceeding 215 percent → move left.
  - Real GDP growth = 3.13 percent → answer “Yes” to growth being above -5.45 → move right to terminal node 3.
  - Result: Colombia in 2004 is not crisis-prone; crisis probability = 2.3 percent.

### Interpretation of nodes and typologies
- Nodes regrouped into four blocks identifying four typologies:
  - Block A — “(relatively) safe fundamentals” (noncrisis-prone): Nodes 1, 4, 6, 8, 9, 11, and 3 (in descending order of safety). Common attributes: low total external debt (below 49.7 percent) except partition 11; low short-term debt over reserves (below 130 percent); low public external debt over revenue (below 215 percent); low inflation (below 10.5 percent); growth above -5.5 percent in many partitions.
  - Blocks B, C, D — potential defaulters (crisis-prone): observations in these blocks are classified as crisis-prone and present distinct risk combinations (e.g., high debt plus high inflation; moderate debt with short maturities, political uncertainty, and low exchange-rate volatility).
- Node-level statistics interpretation (described for Table 3):
  - Column 3: number of observations satisfying node inequalities (example: node 14 → 196 observations out of 1,276).
  - Column 4: within-node probability of crisis (example: node 14 → 66.9 percent).
  - Column 5: share of all crises that fall into the node (example: node 14 → 50.2 percent of all debt crises satisfy the debt and inflation criteria).
  - Column 6: share of no-crisis observations that fall into the node (example: node 14 → 6.4 percent of no-crisis episodes characterized by large external debt and inflation) — interpretable as a “type II error” for crisis nodes.
  - Column 7: normalized safety index (divide node’s share by node’s share of total observations, e.g., divide by 607/1276 for node 3); index > 1 implies safer than overall sample; index < 1 implies riskier. This index is perfectly (negatively) correlated with the crisis probability in Column 4.
  - Column 8: predicted state based on the classification (crisis = 1, no crisis = 0).

### Examples of crisis entries by nodes (illustrative list from the tree figure)
- Crisis entries listed in the figure include episodes such as:
  - El Salvador 1981; Venezuela 1983; Guatemala 1986; Pakistan 1998; Algeria 1991; Chile 1993; Jamaica 1981, 1987; Uruguay 1983; South Africa 1976; Paraguay 1986; Mexico 1982, 1995; Tunisia 1991; Egypt 1984; Russia 1998; Argentina 1982; Panama 1983; Bolivia 1986; Argentina 1995; Thailand 1997; Peru 1978; Indonesia 1997; Ecuador 1982, 1999; Philippines 1983; Brazil 1998, 2001; Argentina 2001; Indonesia 2002; Jamaica 1978; Jordan 1989; Turkey 1978, 2000; Morocco 1983, 1986; Korea 1997; Costa Rica 1981; South Africa 1985; Uruguay 1990; Ukraine 1998; Dominican Rep 1981; Venezuela 1990, 1995.
- Note: the figure associates these crisis episodes with various terminal nodes (see empirical tree).

*Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2005/_wp0542.pdf*

### 5.45 percent). Node 11 comes next in terms of observations. Node 11 shows that low external

### _wp0542 - 5.45 percent). Node 11 comes next in terms of observations. Node 11 shows that low external

### Classification of crisis types
- The tree classification suggests an intuitive split into solvency (B), liquidity (C), and macro/exchange rate (D) risks.
- Liquidity Crisis-Prone, type (B):
  - Described by nodes 7 and 10.
  - Common feature: short-term debt to reserves in excess of 130 percent.
  - Node 7: political uncertainty (YearNextPresElect ≤ 5.5) and low exchange rate volatility (EchRVolatil ≤ 27.8) → crisis probability about 41.5 percent.
  - Node 10: tight international monetary conditions (USTresBill > 9.7) → crisis probability 40.0 percent.
  - Solvency risks of this first type (node 7) make up more than one-fifth (20.7 percent) of all debt crisis episodes.
- Unsustainable Debt Path (Solvency) Crisis-Prone, type (C):
  - Identified by nodes 5, 12, 13, and 14.
  - Common thresholds: TotExtDebt/GDP > 49.7 or PublicExtDebt/Revenue > 215 percent (notationally PublicExtDebt/Revenue > 3.1 in table units).
  - Partition 14: TotExtDebt/GDP > 49.7 and Inflation > 10.5 → crisis probability 66.9 percent (node 14).
  - Node 5: solvency crises signaled by public external debt and high inflation; accounts for 4.2 percent of total.
  - Nodes 13 and 12: high external debt and high ExtFinReqs/Reserve (> 1.4) raise default probabilities to 46.8 percent (node 13) and 40.0 percent (node 12).
  - Nodes 13 and 12 jointly account for another 15.7 percent of crises.
  - Large external financing need interpreted as illiquidity: high current account deficit and/or large amounts of debt coming due make debt path nonfinanceable even if not strictly insolvent.
- Exchange Rate and Macrocrisis-prone, type (D):
  - Node 2: TotExtDebt/GDP ≤ 49.7, ShortTDebt/Res ≤ 1.3, PublicExtDebt/Revenue ≤ 2.1, GDPGrowth ≤ -5.5, EchROverV > 48 → crisis probability 100.0 percent; represents 1.5 percent of crises.
  - Examples: El Salvador (1981), Uruguay (1983), Venezuela (1984).
- Relatively Safe classification:
  - Characterized by combinations of low TotExtDebt/GDP, low ShortTDebt/Res, low PublicExtDebt/Revenue, and non-excessive EchROverV.
  - Node 11: TotExtDebt/GDP > 49.7, Inflation ≤ 10.5, ExtFinReqs/Reserve ≤ 1.4, PublicExtDebt/Revenue ≤ 3.1 → crisis probability 2.0 percent; 98 episodes; example Algeria (1991).
  - Node 3: TotExtDebt/GDP ≤ 49.7, ShortTDebt/Res ≤ 1.3, PublicExtDebt/Revenue ≤ 2.1, GDPGrowth > -5.5 → crisis probability 2.3 percent; 607 episodes; includes Venezuela (1983), South Africa (1976), Russia (1998), Argentina (1995), Thailand (1981).
- Outliers:
  - Node 1 observations (13 cases) where exchange rate not overvalued, external debt small, no liquidity problems and no crises despite strong recession.

### Node-level key statistics (selected exact figures from Table 3)
- Total observations: 1276.
- Node 7 (Liquidity): 130 episodes; Crisis Prob 41.5; % of Crises 20.7; % of NonCrisis 7.5; Relative Safety 0.73; Classified Crisis (1).
- Node 10 (Liquidity): 10 episodes; Crisis Prob 40.0; % of Crises 1.5; % of NonCrisis 0.6; Relative Safety 0.75; Classified Crisis (1).
- Node 13 (Solvency): 79 episodes; Crisis Prob 46.8; % of Crises 14.2; % of NonCrisis 4.1; Relative Safety 0.67; Classified Crisis (1).
- Node 14 (Solvency): 196 episodes; Crisis Prob 66.9; % of Crises 50.2; % of NonCrisis 6.4; Relative Safety 0.42; Classified Crisis (1).
- Node 2 (Macro/Exchange): 4 episodes; Crisis Prob 100.0; % of Crises 1.5; % of NonCrisis 0.0; Relative Safety 0.00; Classified Crisis (1).
- Node 11 (Safe despite high external debt): 98 episodes; Crisis Prob 2.0; % of Crises 0.8; % of NonCrisis 9.5; Relative Safety 1.23; Classified NonCrisis (0).

### Historical case mapping (examples preserving source labels)
- Illiquidity node 7: Korea (1997), Mexico (1995), Brazil (1998, 2001), Pakistan (1998), Ukraine (1998), Thailand (1997 inactive default on most external debt), Pakistan forced to restructure external sovereign debt.
- Insolvency node 14: Ecuador (1999), Indonesia (2002), Turkey (2000 borderline insolvent but avoided default with IMF program).
- Noncrisis-prone node 3 but crisis occurred: Russia (1998), Argentina (1995) — indicating contagion or capital flight factors not captured by fundamentals in tree.

### Prediction performance (exact figures from Table 4)
- Full Sample (observations 1276):
  - Correctly called episodes: 82.8 percent.
  - Correctly called entries into default: 88.9 percent.
  - Incorrectly called entries into default: 18.5 percent.
  - Correctly called exits from default: 32.0 percent.
  - Incorrectly called exits from default: 4.8 percent.
  - Correctly called default episodes: 93.9 percent.
  - Correctly called nondefault episodes: 79.9 percent.
- 1990’s Onwards (observations 556):
  - Correctly called episodes: 78.8 percent.
  - Correctly called entries into default: 85.0 percent.
  - Incorrectly called entries into default: 15.0 percent.
  - Correctly called exits from default: 35.3 percent.
  - Incorrectly called exits from default: 64.7 percent.
  - Correctly called default episodes: 89.5 percent.
  - Correctly called nondefault episodes: 76.0 percent.
- Overall statements:
  - Tree correctly predicts 82.9 percent of states, 93.9 percent of crisis, and 79.9 percent of noncrisis states (as summarized in prose).

### Were the 1990s different? (distributional checks and robustness)
- Measure: N(t)1990/N(t) expressed as ratio of N1990/N; value equals one if 1990s observations proportionate.
- Selected node ratios (from Table 5):
  - Node 1 Safe: 0.936654
  - Node 3 Safe: 0.785689
  - Node 7 Liquidity: 1.202039 (over-represented)
  - Node 8 Safe: 1.383694 (over-represented)
  - Node 9 Safe: 0.119377 (under-represented)
  - Node 11 Safe: 1.573834 (over-represented)
  - Node 14 Solvency: 0.828334
- Conclusion: no discernible pattern of systematic difference; solvency crises appear in similar proportions in the 1990s as in whole sample.
- Variable importance correlation between full sample and post-1990 sample:
  - Correlation = 0.466689.
  - Rank correlation = 0.614502.
- Seven (eight counting different short-term debt measures) out of the ten most important variables for the full sample also appear among first ten variables in restricted sample.

### Variable importance (top ranks and exact importance scores from Table 6)
- Full Sample (1970–2002) top importance and ranks:
  - Tot Ext Debt/GDP: Importance 100, Rank 1.
  - Pub Ext Debt/GDP: Importance 88.349, Rank 2.
  - Tot Ext Debt/Exp: Importance 69.352, Rank 3.
  - ShTermExtDebt/Res (orig mat): Importance 34.55, Rank 5.
  - PubExtDev/Res: Importance 33.707, Rank 6.
  - ExtFinRequir: Importance 28.233, Rank 8.
  - Nominal GDPGrowth: Importance 25.503, Rank 9.
  - Inflation: Importance 24.748, Rank 10.
- Post-1990 (1990–2002) top importance and ranks:
  - ExtFinRequir: Importance 100, Rank 1.
  - ShTermExtDebt/Res (o.mat): Importance 72.56, Rank 2.
  - PubExtDev/Res: Importance 71.255, Rank 3.
  - Inflation: Importance 56.891, Rank 4.
  - ShTermExtDebt/GDP(rem.mat): Importance 54.272, Rank 5.
  - Pub Ext Debt/GDP: Importance 46.717, Rank 7.
  - IntShTermExtDebt/Res: Importance 42.618, Rank 8.
  - Nominal GDPGrowth: Importance 38.491, Rank 9.
  - ShTermExtDebt/GDP(or.mat): Importance 50.347, Rank 6.
- Correlation and rank-correlation repeated: Correlation=0.466689; Rank correlation=0.614502.

### Policy-relevant findings and implications (drawn exactly from source phrasing)
- Unconditional thresholds (e.g., debt to output ratios in isolation) are of little value per se; the particular combination of different types of vulnerability matters for default probability.
- Political instability and tight monetary conditions in international financial markets aggravate liquidity problems.
- The tree approach can:
  - Derive rules of thumb or vulnerability thresholds useful for early warning.
  - Derive policy adjustment paths to reduce crisis likelihood for countries entering danger zones.
  - Be used for surveillance, crisis prevention, and crisis resolution.
- Prediction trade-offs:
  - Model predictive accuracy is high but type II errors (false alarms) are somewhat higher than desirable.
  - The tree approach allows adjustment of relative weights given to type I and type II errors; future challenge is calibrating to reduce false alarms while maintaining high predictive ratio.

### Extensions and robustness checks (exact statements)
- A constructed indicator of (bad) “reputation” (incremental values each time a new default occurred) did not affect the classification tree.
- Distinguishing four states (staying in/out a crisis, getting into/exiting from a crisis) has not proved particularly effective so far; however preliminary conclusions:
  - Exits seem typically associated with (regained) fiscal solvency and floating exchange rates.
  - Entries seem typically associated with liquidity and monetary problems.
- Masking problem noted: variable selection in optimal tree may be sensitive to small sample changes; CART accounts for actual and potential effects of variables even if masked by first-choice split.

*Source: Author’s Calculations, _wp0542 (excerpts provided).*

### References

### _wp0542 - References

### Selected Bibliographic Entries
- Arias, Andres F., 2004, “Comments on the paper ‘Assessing Fiscal Sustainability;’ by Enrique G. Mendoza and Pedro Marcelo Oviedo.” Available via the Internet http://www.iadb.org/res/centralbanks/publications/cbm35_199.ppt
- Breiman, Leo, Jerome H. Friedman, Richard A. Olshen, and Charles J. Stone, 1984, Classification and Regression Trees, (London: Chapman & Hall).
- Beers, David T., and Ashok Bhatia, 1999, “Sovereign Defaults: History,” in Standard & Poor’s Credit Week, December 22.
- Beers, David T., and John Chambers, 2002, “Sovereign Defaults: Moving Higher Again in 2003?,” Standard & Poor’s (September). Available via the Internet http://www.emta.org/keyper/sov2003.pdf
- Beim, David O., and Charles W. Calomiris, 2001, Emerging Financial Markets, (New York: McGraw-Hill/Irwin).
- Cantor, Richard, and Frank Packer, 1996, “Determinants and Impact of Sovereign Credit Ratings,” Federal Reserve Bank of New York Policy Review (October), pp. 37–52.
- Catão, Luis, and Bennett Sutton, 2002, “Sovereign Defaults: The Role of Volatility,” IMF Working Paper 02/149 (Washington: International Monetary Fund).
- Corsetti, Giancarlo, Bernardo Guimarães, and Nouriel Roubini, 2003, “The Tradeoff Between an International Lender of Last Resort to Deal with Liquidity Crisis and Moral Hazard Distortions: A Model of the IMF’s Catalytic Finance Approach,” IMF Seminar Series Research Paper 03/149 (Washington: International Monetary Fund).
- Cottarelli, Carlo, and Curzio Giannini, 2002, “Bedfellows, Hostages, or Perfect Strangers? Global Capital Markets and the Catalytic Effect of IMF Crisis Lending,” IMF Working Paper 02/193 (Washington: International Monetary Fund).
- Dell’Arricia, Giovanni, Isabel Schnabel, and Jeromin Zettelmeyer, 2002, “Moral Hazard and International Crisis Lending: A Test,” IMF Working Paper 02/181 (Washington: International Monetary Fund).
- Detragiache, Enrica, and Antonio Spilimbergo, 2001, “Crises and Liquidity: Evidence and Interpretation,” IMF Working Paper 01/2 (Washington: International Monetary Fund).
- Eaton, Jonathan, and Raquel Fernandez, 1995, “Sovereign Debt,” NBER Working Paper 5131, Prepared for Handbook of International Economics.
- Frankel, Jeffrey, and Shang-Jin Wei, 2004, “Managing Macroeconomic Crises: Policy Lessons,” NBER Working Paper 10907, November. Available via the Internet http://ksghome.harvard.edu/~jfrankel/Managing%20Macroeconomic%20Crises%20Policy%20Lessons.pdf
- Ghosh, Swati, and Atish Ghosh, 2002, “Structural Vulnerabilities and Currency Crises,” IMF Working Paper 02/9 (Washington: International Monetary Fund).
- Haque, Nadeem U., Mark Nelson, and Donald J. Mathieson, 1998, “The Relative Importance of Political and Economic Variables in Creditworthiness Ratings,” IMF Working Paper 98/46 (Washington: International Monetary Fund).
- Hemming, Richard, and Nigel Chalk, 2000, “Assessing Fiscal Sustainability in Theory and Practice,” IMF Working Paper 00/81 (Washington: International Monetary Fund).
- Hemming, Richard, and Murray Petrie, 2002, “A Framework for Assessing Fiscal Vulnerability,” IMF Working Paper 00/52 (Washington: International Monetary Fund).
- Henisz, W. J., (2002), “The Institutional Environment for Infrastructure Investment,” Industrial and Corporate Change, Vol.11, Number 2, pp. 355–89.
- Jeanne, Olivier, 2000, “Debt Maturity and the Global Financial Architecture,” CEPR Discussion Paper 2520 (London: Centre for Economic Policy Research).
- Lane, Timothy, and Steven Phillips, 2001, “Does IMF Financing Result in Moral Hazard?” IMF Working Paper 00/168 (Washington: International Monetary Fund).
- Larrain, Guillermo, Helmut Reisen, and Julia von Maltzan, 1997, “Emerging Market Risk and Sovereign Credit Ratings,” OECD Development Centre Technical Paper 124 (Paris: Organization for Economic Co-operation and Development).
- Lee, Suk Hun, 1993, “Are the credit ratings assigned by bankers based on the willingness of LDC borrowers to repay?,” Journal of Development Economics, Vol. 40, pp. 349–59.
- Manasse, Paolo, Nouriel Roubini, and Axel Schimmelpfennig, 2003, “Predicting Sovereign Debt Crises,” IMF Working Paper 03/221 (Washington: International Monetary Fund).
- Mody, Ashoka, and Diego Saravia, 2003, “Catalyzing Private Capital Flows: Do IMF-Supported Programs Work as Commitment Devices?” (unpublished; Washington: International Monetary Fund).
- Obstfeld, Maurice, and Kenneth Rogoff, 1996, “Foundations of International Macroeconomics,” MIT Press (Cambridge, Massachusetts).
- Reinhart, Carmen M., 2002, “Default, Currency Crises and Sovereign Credit Ratings,” NBER Working Paper 8738. Also in The World Bank Economic Review, Vol. 16, No. 2, pp. 151–70.
- Reinhart, Carmen M., Kenneth S. Rogoff, and Miguel A. Savastano, 2003, “Debt Intolerance,” NBER Working Paper 9908 (Cambridge, Massachusetts: National Bureau of Economic Research).
- Rojas-Suarez, L., 2001, Rating Banks in Emerging Markets, (Washington: Institute for International Economics).
- Roubini, Nouriel, 2001, “Debt Sustainability: How to Assess Whether a Country is Insolvent,” unpublished (New York: New York University) (December).
- Roubini, Nouriel, and Brad Setser, 2004, Bailouts or Bail-ins? Responding to Financial Crises in Emerging Economies, (Washington: Institute for International Economics).

### Appendix I — An Application of BRT To Heart Attacks (empirical illustration)
- Dataset:
  - Total patients: 215
  - Early deaths (died not more than 30 days following the heart attack): 37 (17 percent)
  - Survivors: 178 (83 percent)
  - Number of measurements per patient: 19 (examples: minimum systolic blood pressure, history of heart attacks, presence of tachycardia, concentration of enzymes, sex, age)
- Tree splitting sequence and node outcomes:
  - Rule 1: “Is minimum systolic blood pressure below 91?”
    - Yes → Terminal Node 1:
      - Cases: 20
      - Survivors: 6 (30 percent)
      - Early deaths: 14 (70 percent)
      - Classified: High Risk
    - No → Right child (195 patients; survivors 172 (88 percent); early deaths 12 percent)
      - Rule 2: “Is age below 62.5 years?”
        - Yes → Terminal Node 2:
          - Cases: 104
          - Survivors: 102 (98 percent)
          - Early deaths: 2 (2 percent)
          - Classified: Low Risk
        - No → Further split:
          - Rule 3: “Was there sinus tachycardia present?”
            - Yes → Terminal Node 3:
              - Cases: 28
              - High Risks: 14 (50 percent)
              - Low Risks: 14 (50 percent)
              - Classified: High Risk
            - No → Terminal Node 4:
              - Cases: 63
              - High Risks: 7 (11 percent)
              - Low Risks: 56 (89 percent)
              - Classified: Low Risk
- Key diagnostic interpretation:
  - Patients in Terminal Node 1: conditional probability of early death is 70 percent.
  - Patients in Terminal Node 3 (age > 62.5 and tachycardia): conditional probability of early death is 50 percent.
- Definitions (as given in source):
  - Systolic blood pressure: the maximum blood pressure that occurs with each heart cycle during contraction of the left-sided pumping chamber.
  - Sinus tachycardia: present if the sinus node heart rate ever exceeded 100 beats per minute during the first 24 hours following admission to the hospital; the sinus node is the normal electrical pacemaker of the heart and is located in the right atrium.

### Some Basic Concepts of Binary Classification Trees (BRT)
- Notation:
  - N, N_j, N(t), N_j(t): number of observations in the sample, number of class-j observations in the sample, number of observations in node t, number of class-j observations in node t.
  - π_j , j=1...J: a priori distribution for class j.
  - p(t|j) = N_j(t)/N_j
  - p(j,t) = π_j N_j(t)/N_j
  - p(t) = ∑_j p(j,t)
  - p(j | t) = p(j,t)/p(t) = [π_j N_j(t)/N_j] / [ ∑_j π_j N_j(t)/N_j ] with ∑_j p(j | t)=1
  - If π_j = N_j/N then p(j | t)=N_j(t)/N(t)
- Impurity and splits:
  - Node impurity: i(t) = φ( p(1 | t), p(2 | t),..., p(J | t) )
  - φ(.) must: (i) achieve a maximum at (1/J,...,1/J); (ii) achieve its minimum only at unit vectors; (iii) be symmetric in p(j | t).
  - Goodness of split: ∆i(s,t)= i(t) - p_R i(t_R) - p_L i(t_L)
  - Gini index (a φ function): G(t) = ∑_{j≠i} p(j | t) p(i | t)
  - Alternate expression: G(t) = 1 - ∑_j p^2(j | t)
  - Examples: For J=4, G(t) minimum = 0 when node contains single class; maximum = 3/4 when classes are equally frequent.
- Classification and stopping rules:
  - Class assignment at node t: j*(t) = arg max_j p(j | t). If ties occur assign arbitrarily among maximizers.
  - Example for j=1,2 with uniform prior (1/2,1/2): classify t as class 1 if N_1(t)/N_2(t) > N_1/N_2.
  - Majority rule when prior = sample frequency: classify as class 1 if N_1(t) > N_2(t).
  - Early stopping rule: declare node t terminal if max_s ∆i(s,t) < β (threshold β).
  - Breiman et al. (1984) approach: grow a very large tree and then “prune” it back.
- Goodness-of-fit and misclassification:
  - Node misclassification probability: r(t) = 1 - max_i p(i | t)
  - Overall misclassification rate estimate: R(t) = ∑_{t∈T} p(t) r(t) where summation is over terminal nodes t.
  - Cost-sensitive Gini variant: G(t) = ∑_{j≠i} C(i│j) p(j | t) p(i | t) if misclassification costs C(i│j) differ.

*Italic: Source content from _wp0542 - References (PDF chapter/section).*

---


_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2005/_wp0542.pdf_