## _wp0444

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---

### III. I. INTRODUCTION & DEFINITIONAL FRAME
- Question posed: What is a debt crisis?
- Proposed alternative definition (PesSy):
  - A debt crisis occurs when either:
    - there is a sovereign default (rating-agency definition, S&P); or
    - secondary market sovereign bond spreads exceed a critical threshold τ.
- Market practice and motivation:
  - Market participants often view sovereign bond spreads above the 1,000 basis points (10 percentage points) mark as signaling a significant probability of default.
  - EVT and kernel density estimation find that the 1,000 basis points threshold corresponds to significant tail events.
- Latent-variable formalization:
  - Latent variable y*: y* = x'β + u (equation (1)).
  - Default indicator I: I = 1 if y* > 0 (equation (2)).
  - Spreads-based indicator S: S indicates when s exceeds τ via h(.) with h(0)=0 (equation (3)).
  - Combined observable y% = 1 if (I,S) ≠ (0,0), else 0 (equation (4.b)).
- Empirical-sample notes:
  - Sample period: 1975–2002; spreads data start from 1994.
  - Sample summary (Table 1, authors’ calculations):
    - Total obs. (1975–2002): 886
      - Defaults: Percentage 24, Number of Crises 214
      - PesSy: Percentage 27, Number of Crises 238
    - (1994–2002): 287 obs.
      - Defaults: Percentage 16, Number of Crises 46
      - PesSy: Percentage 24, Number of Crises 70
    - (1975–1993): 599 obs.
      - Defaults: Percentage 28, Number of Crises 168

### III. THRESHOLD ESTIMATION: EXTREME VALUE THEORY (EVT)
- Rationale:
  - Sovereign spreads exhibit fat tails and volatility clustering; EVT appropriate for extreme-event thresholds.
- Method summary:
  - Use Hill estimator for tail parameter α (γ = 1/α) on pooled data; choose m (number of extreme order statistics).
  - For yearly data, plot ˆγ against m; recursive least squares forecasts used to identify stable regions (Appendix Figures A.1–A.3).
- EVT empirical results (Table A.2 / A1.2):
  - Yearly Data: m ∈ [42, 50], τ ∈ [969, 1072]
  - Monthly Data: m = 400, τ = 1117
  - Daily Data: m = 9000, τ = 1084
- Yearly-sample numeric conclusions:
  - m = 42 → threshold 1,073 bps
  - m = 50 → threshold 969 bps
  - Conclude threshold reasonably lies between 696 bps and 1,073 bps (authors’ stated interval).

### III. KERNEL DENSITY ESTIMATION & PSYCHOLOGICAL THRESHOLD
- Rationale:
  - If a psychological threshold τ exists, distribution of bond spreads should show a mode near τ.
- Approach and distributions fitted (Table 3):
  - Yearly and daily data analyzed; for yearly data fit Gamma and Weibul (Weibull) distributions and assess 90th percentile.
- Estimated 90th Percentile – 95% Confidence Interval (Table 3):
  - Gamma distribution: 1,036.52 [980.55, 1,093.85]
  - Weibul distribution: 978.82 [795.10, 1,248.05]
  - Percentile corresponding to 1,000 bps:
    - Gamma distribution: 0.88
    - Weibul distribution: 0.91
- Authors’ conclusion:
  - The 1,000 basis points threshold lies inside the 95 percent confidence bands for the 90th percentile of fitted distributions; kernel density estimation confirms a mode around 1,000 bps for both daily and yearly data.

### III. REGRESSION RESULTS: DEFAULTS VS PES SY (1975–2002)
- Model type:
  - GEE Logit population-averaged model, correlation exchangeable, Huber-White estimator.
  - Number of observations: 567 (full-sample regressions reported).
- Key coefficients (Default Definition, 567 Obs.):
  - Openness: Coeff. -0.05, Z -3.14, P>|z| 0.00, 95% CI -0.08, -0.02
  - Overvaluation: Coeff. 0.01, Z 3.40, P>|z| 0.00, 95% CI 0.01, 0.02
  - Total debt over GDP: Coeff. 0.06, Z 3.57, P>|z| 0.00, 95% CI 0.03, 0.09
  - Short-term debt over reserves: Coeff. 0.19, Z 2.07, P>|z| 0.04, 95% CI 0.01, 0.36
  - Real growth rate: Coeff. -0.08, Z -2.49, P>|z| 0.01, 95% CI -0.15, -0.02
  - Inflation (when included): Coeff. 0.00, Z 2.67, P>|z| 0.01, 95% CI 0.00, 0.00
  - Constant: -1.94 (without inflation) and -2.09 (with inflation)
  - Wald χ2: (5) 44.0; (6) 46.1
- Key coefficients (PesSy Indicator = Defaults + 1,000 bps, 567 Obs.):
  - Openness: Coeff. -0.03, z -2.06, P>|z| 0.04, 95% CI -0.07, 0.00
  - Overvaluation: Coeff. 0.01, z 2.75, P>|z| 0.01, 95% CI 0.00, 0.02
  - Total debt over GDP: Coeff. 0.06, z 4.26, P>|z| 0.00, 95% CI 0.03, 0.09
  - Short-term debt over reserves: Coeff. 0.30, z 2.50, P>|z| 0.01, 95% CI 0.06, 0.53
  - Real growth rate: Coeff. -0.09, z -2.61, P>|z| 0.01, 95% CI -0.17, -0.02
  - Inflation (when included): Coeff. 0.00, z 1.25, P>|z| 0.21
  - Constant: -2.80 (without inflation) and -3.08 (with inflation)
  - Wald χ2: (5) 48.0; (6) 45.4
- Authors’ summary:
  - Solvency and liquidity measures and macroeconomic controls statistically significant at 5 percent in both specifications except inflation is not significant in the PesSy specification.
  - Augmented model (PesSy) shows a higher Wald χ2 in the full sample.

### III. SUBSAMPLE ANALYSIS: 1994–2002
- General finding:
  - PesSy (defaults + bond-spread threshold) performs better than defaults-only in the 1994–2002 subsample.
- Wald χ2 comparison (1994–2002, 207 Obs. unless otherwise stated):
  - Without inflation: PesSy 29.4 vs Defaults 8.4
  - With inflation: PesSy 59.5 vs Defaults 41.6
- Selected coefficients (PesSy, subsample 1994–2002):
  - Total debt over GDP: Coeff. 0.06, z 3.31, P>|z| 0.00, 95% CI 0.02, 0.09
  - Short-term debt over reserves: Coeff. 0.34, z 1.89, P>|z| 0.06, 95% CI -0.01, 0.70 (alternate spec: Coeff. 0.39, z 2.00, P>|z| 0.05, 95% CI 0.01, 0.76)
  - Inflation: significant (Z 3.87, P>|z| 0.00) when included
- Interpretation:
  - Liquidity indicators (short-term debt over reserves) are significant when crises incorporate bond-market turbulence but not when crises are defined as defaults only.
  - Real growth and openness lose significance in the subsample regressions as previously observed; inflation becomes more important in 1994–2002.

### III. OUT-OF-SAMPLE FORECAST EVALUATION (ESTIMATE TO 1993, PREDICT 1994–2002)
- Estimation sample: 1975–1993 (Defaults model reported, 360 Obs.)
  - Example coefficients (Defaults, 360 Obs.):
    - Openness: Coeff. -0.04, Z -2.92, P>|z| 0.00, 95% CI -0.07, -0.01
    - Total debt over GDP: Coeff. 0.07, Z 4.05, P>|z| 0.00, 95% CI 0.03, 0.10
    - Short-term debt over reserves: Coeff. 0.23, Z 1.97, P>|z| 0.05, 95% CI 0.00, 0.46
    - Real growth rate: Coeff. -0.09, Z -1.93, P>|z| 0.05, 95% CI -0.17, 0.00
    - Wald χ2 (6): 39.20
- Forecast evaluation constructs:
  - A (Matched Crises), B (False Alarms), C (Missing Crises), D (Matched Tranquil periods). Optimal threshold T* minimizes L(T)=B(T)/A(T)+C(T)/D(T).
  - T* estimated via bootstrapping.
- Matched Crises over Total Crises and False Alarms over Tranquil Periods (90 percent confidence intervals, out-of-sample):
  - Defaults: Matched Crises 0.43 [0.33, 0.50]; False Alarms over Tranquil Periods [0.12, 0.13]
  - PesSy: Matched Crises 0.61 [0.55, 0.67]; False Alarms over Tranquil Periods [0.19, 0.20]
  - In-sample (point estimate): Matched Crises 0.86; False Alarms 0.14
  - In-sample (bootstrap intervals): Matched Crises 0.69 [0.52, 0.91]; False Alarms [0.02, 0.20]
- Loss-function bootstrap results (reported 10th percentile, mean, 90th percentile as CI):
  - Default definition Loss (Out Sample): 1.50 [1.13 – 2.14]
  - Associated Optimal Threshold (Defaults): .34 [0.23 and 0.28]
  - PesSy definition Loss (Out Sample): .89 [0.75, 1.07]
  - Associated Optimal Threshold (PesSy): 0.32 [0.38 and 0.36]
  - In Sample Result Loss: 0.39
  - In Sample Result Optimal Threshold: 0.57
- Statistical comparison (Table 11: two-sample t-test):
  - Significance = 0.000
  - Lower bound = 0551
  - Upper bound = 067
- Authors’ out-of-sample conclusions:
  - PesSy matches predicted crises better (higher Matched Crises share) but yields higher False Alarms.
  - PesSy yields lower out-of-sample loss (.89 vs 1.50) and associated optimal threshold near 0.32.
  - Two-sample t-test rejects the null: PesSy performs better than defaults at any confidence level.

### III. APPENDIX METHODS & DIAGNOSTICS
- Ljung-Box Q-test (yearly-data) diagnostics (Table A1.1 highlights):
  - Number of rejections of the null hypothesis of no correlation: 4 (mainly for short sample countries).
  - Sample country entries (as reported exactly):
    - Algeria: Null 0, p-value 0.11, Q-Statistic 7.47, Critical Value 9.49
    - Argentina: Null 0, p-value 0.27, Q-Statistic 11.07, Critical Value 16.92
    - China: Null 1, p-value 0.02, Q-Statistic 12.93, Critical Value 11.07
    - Malaysia: Null 1, p-value 0.01, Q-Statistic 22.60, Critical Value 16.92
    - South Africa: Null 1, p-value 0.04, Q-Statistic 11.71, Critical Value 11.07
- EVT vs fixed 1,000 bps comparison (Table A1.3 summary for 1994–2002):
  - Yearly:
    - Total number: 216
    - Matching EVT: [211, 213]
    - Adding EVT: [5, 0]
    - Crossing out: [0, 3]
    - As percentage: Matching EVT [0.98, 0.99]; Adding EVT [0, 0.023]; Crossing out [0, 0.014]; Total 1
  - Monthly:
    - Total number: 2,344 (authors’ tabulation: 2,252; 92; 0; Total 2,344)
    - As percentage: 96; 39; 0; 1
  - Daily:
    - Total number: 50,329 (authors’ tabulation: 48,863; 1,466; 0; Total 50,329)
    - As percentage: 97; 029; 0; 1
  - Interpretation: EVT-derived thresholds are broadly consistent with the 1,000 bps mark; using 1,000 bps does not significantly change crisis classification.
- Kernel-density robustness:
  - Pooled and daily spreads kernel density plots using multiple kernels (Epanechnikov, Normal, Box, Triangle) show density features around threshold regions.
- Metropolis-Hastings (M-H) implementation for sampling:
  - Acceptance probability α(x,y) = min{1, [f(y) q(y,x)] / [f(x) q(x,y)] } (equation (0.2)).
  - Authors discard the first 1,000 observations as burn-in and display sample convergence (Figure A3.1).

### III. KEY EMPIRICAL FINDINGS & POLICY-ORIENTED IMPLICATIONS
- Empirical findings:
  - Defining debt crises as defaults only performs poorly post-1994; defaults rare despite multiple sovereign credit events.
  - Using PesSy (defaults + 1,000 bps spread threshold) yields:
    - Higher matched-crisis share out-of-sample (PesSy 0.61 [0.55, 0.67] vs Defaults 0.43 [0.33, 0.50]).
    - Lower out-of-sample loss (.89 [0.75, 1.07] vs 1.50 [1.13 – 2.14]).
    - Liquidity indicators become significant under PesSy but not under defaults-only for 1994–2002.
  - EVT and kernel density estimation both support the 1,000 basis points threshold as statistically consistent with extreme-event behavior and distributional modes (threshold lies inside 95% CIs for 90th percentiles).
- Policy implication:
  - Incorporating secondary-market bond-spread thresholds (empirically at the 1,000 basis points mark) into debt-crisis definitions provides a more comprehensive indicator of sovereign debt distress in an environment with significant bond-market access.
  - This augmented definition captures liquidity-driven tensions and near-default episodes that rating-agency default indicators alone miss.

*Source: _wp0444 — IMF Working Paper content as supplied (authors’ calculations, tables, and appendix).*

### 1. Number of Crises, by Definition......................................................................................

### 1. Number of Crises, by Definition..........................................................................................12

### Main Sections
- 1. Number of Crises, by Definition..........................................................................................12
- 2. Extreme Value Theory Versus 1,000 bps thresholds (Sample: 1994–2002).......................15
- 3. Thresholds for Bond Spreads from Kernel Density Estimations.........................................16
- 4. Regression Results Using Default Definition: 1975–2002..................................................18
- 5. Regression Results Using PesSy Indicator: 1975–2002......................................................18
- 6. Regression Results Using Default Definition: Subsample for 1994–2002..........................19
- 7. Regression Results Using PesSy Indicator: Sub-Sample for 1994–2002............................19
- 8. Regression Results Using Default Definition: Sample for 1975–1993...............................20
- 9. “Matched Crises over Total Crises” and “False Alarms over Tranquil Periods”................21
- 10. Bootstrapped Results for Standard and PesSy Distress Definition Loss Function ........... 21
- 11. Comparing Standard and PesSy Distress Definitions: Loss Functions.............................21

### Appendix Tables
- A1.1. Ljung-Box Q-test Results...................................................................... ........................23
- A1.2. Extreme Value Theory: Estimated m and Implied Threshold τ.....................................24
- A1.3. Comparison: EVT and 1,000 bps Thresholds, Sample for 1994–2002.........................24

### Figures
- 1. Comparison Between PesSy and Standard Defaults............................................................11

### Appendix Figures
- A1.1. Estimated Alphas w.r.t m and ILS for the Forecast, Yearly Data..................................24
- A1.2. Estimated Alphas w.r.t m and ILS for the Forecast, Daily Data....................................25
- A2.1. Pooled Spreads, Kernel Density.....................................................................................26
- A2.2. Daily Spreads, Kernel Density.......................................................................................27
- A3.1. Markov Chain Monte Carlo Convergence Checked: Sample Path................................29

### Appendix Sections
- I. Extreme-Value-Theory Approach........................................................................................23
- II. Kernel Density Approach....................................................................................................25
- III. The Metropolis-Hasting Algorithm...................................................................................28

*Source: _wp0444 - 1. Number of Crises, by Definition (PDF chapter/section).*

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

### _wp0444 - References

### I. INTRODUCTION
- Question posed: What is a debt crisis? Literature has focused on sovereign defaults as the primary credit event for foreign debt contracts.
- Observation: According to Moody’s (2003) only seven rated sovereign bond issuers have defaulted on their foreign-currency denominated bonds since 1985 and all of those defaults happened between 1998 and 2002.
- Argument: Defining debt crises solely as sovereign defaults overlooks developments in international capital markets, notably the advent of the bond market for emerging market sovereign issuers.
- Proposed alternative definition: A debt crisis is an event when either there is a sovereign default or secondary market bond spreads are higher than a critical threshold.
- Market practice cited: Market participants often view sovereign bond spreads above the 1,000 basis points (10 percentage points) mark as signaling a significant probability of default.
- Statistical approach mentioned: Extreme value theory and kernel density estimation find that the 1,000 basis points threshold corresponds to significant tail events.
- Latent variable framework: Foreign-debt-servicing difficulties represented by an unobservable latent variable; indicators (defaults or market-based measure) infer seriousness of difficulties.
- Empirical finding preview: Solvency and liquidity measures and macroeconomic controls explain the proposed definition of debt crises better than definitions based solely on defaults. Liquidity indicators are significant under the proposed definition but do not play a role in explaining defaults after 1994.
- Paper organization: Literature review (Section II), alternative model and statistical motivation (Section III), estimation and out-of-sample comparison (Section IV), conclusion (Section V).

### II. REVIEW OF LITERATURE
- Common practice: Most studies assume the relevant sovereign credit event is a sovereign default; many empirical works focus on determinants of default risk.
- Theoretical definitions: In foreign debt crisis theory, a credit event often defined as non-repayment of pre-agreed debt service (Sachs, 1984); surprise inflation can be a domestic sovereign credit event (Calvo, 1988; Alesina, Prati, and Tabellini, 1990).
- Empirical proxies for default probability:
  - Spreads between country interest rates and a benchmark (Edwards, 1984).
  - Backing out default probability from spreads, Duffie and Singleton (2003), credit default swaps (Chan-Lau, 2003).
  - Credit ratings or rating transitions (Schmukler, 2001; Juttner and McCarthy, 1998; Hu, Kiesel and Perraudin, 2001).
- Broader financial crises literature links sovereign debt variables to capital flow reversals, currency crises, and interactions between default and currency crises (Radelet and Sachs, 1998; Rodrick and Velasco, 1999; Frankel and Rose, 1996; Milesi-Ferretti and Razin, 1998; Berg and Pattillo, 1999; Bussière and Mulder, 1999; Reinhart, 2002).

A. What Is a Debt Crisis?
- Debt Crises as Sovereign Defaults:
  - Moody’s (2003) default conditions:
    - Missed or delayed disbursement of interest and/or principal, even if within a grace period.
    - Distressed exchange where the issuer offers securities that amount to diminished financial obligations (lower coupon or par value) or the exchange had the apparent purpose of avoiding a stronger event of default.
  - Standard and Poor’s (Chambers and Alexeeva, 2002) default definition:
    - Failure to meet a principal or interest payment on due date (or within grace period).
    - For bonds/notes/bills: scheduled debt service not paid on due date or exchange offers with less favorable terms.
    - For bank loans: scheduled debt service not paid or rescheduling at less-favorable terms than the original loan.
    - Central bank currency: default occurs when notes are converted into new currency of less-than-equivalent face value.
  - Beim and Calomiris (2001) classification:
    - Periods (six months or more) where all or part of interest and/or principal payments due were reduced or rescheduled; exclusions: intergovernmental loans, voluntary refinancing.

- Debt Crises as Large Arrears (Detragiache and Spilimbergo, 2001):
  - Crisis if either:
    - Arrears of principal or interest on external obligations to commercial creditors > 5 percent of total commercial debt outstanding; or
    - Rescheduling or debt restructuring with commercial creditors as listed in the World Bank’s Global Development Finance.
  - Sensitivity tests: threshold set at 15 percent of commercial debt service due.
  - Excludes arrears/rescheduling of official debt and observations where commercial debt is zero.
  - Episode termination: arrears fall below 5 percent threshold; crises beginning within four years treated as continuation.
  - Baseline sample identifies 54 debt crises; only 4 crises in the 1994-1998 period.

- Debt Crises as Large IMF Loans (Manasse, Roubini, and Schimmelpfennig, 2003; MRS):
  - Types of sovereign-debt servicing difficulties: outright default, semi-coercive restructuring, rollover/liquidity crises.
  - MRS crisis definition: country experiences a debt crisis if:
    - It is classified as in default by Standard and Poor’s; or
    - It receives a large non-concessional IMF loan defined as access in excess of 100 percent of quota.
  - Rationale: include near-defaults avoided by large-scale official financing.

- Debt Crises as Distress (Sy, 2003):
  - Defines sovereign bonds as distressed when bond spreads are trading 1,000 basis points or more above U.S. Treasuries.
  - Using this definition, Sy finds 140 distressed debt events (about 14 percent of observations) from 1994 to 2002, associated with reduced access to sovereign bond markets.

### III. AN ALTERNATIVE MODEL OF SOVEREIGN DEBT CRISIS
- Problem identified: Defaults were good proxies for debt-servicing difficulties in the 1980s but perform poorly post-1994; there is a dearth of defaults post-1994 despite multiple sovereign credit events.
- IMF bailout evidence: Percentage of countries receiving IMF assistance of more-than-100 percent of quota increased marginally to 4.8 percent post-1994 from 4.5 percent previously; percentage of countries receiving IMF bail-outs without defaulting rose to 36 percent post-1994 from 32 percent previously (for a sample of 76 countries mostly non-OECD).
- Implication: Complementing defaults with IMF bailouts increases total crisis events but does not change the relative number of crisis events between periods; puzzle of reduced defaults post-1994 remains.
- Two considerations motivating a bond-market-based measure:
  - Increased access to sovereign bond markets reduced the share of bank loans in total debt; debt-servicing difficulties may manifest in bonds, banks, or both.
  - Bond contracts are harder to renegotiate; usual default definitions miss renegotiation-related debt-servicing difficulties in bond markets.
- Proposed indicator (PesSy): A debt crisis occurs when either:
  - There is a default as defined by rating agencies (S&P); or
  - Secondary market sovereign bond spreads exceed a critical threshold τ.
- Data summary:
  - Sample period 1975 to 2002; spreads data start from 1994.
  - Figure 1 (authors’ calculations) shows number of crises signaled by defaults (S&P) and by PesSy.
  - Table 1 (authors’ calculations) key figures:
    - Total obs. (1975–2002): 886
      - Defaults: Percentage 24, Number of Crises 214
      - PesSy: Percentage 27, Number of Crises 238
    - (1994–2002): 287 obs.
      - Defaults: Percentage 16, Number of Crises 46
      - PesSy: Percentage 24, Number of Crises 70
    - (1975–1993): 599 obs.
      - Defaults: Percentage 28, Number of Crises 168
- Latent-variable formalization:
  - Latent variable y* represents foreign debt-servicing difficulties; regression y* = x'β + u (equation (1)).
  - Default indicator I: I = 1 if y* > 0, else 0 (equation (2)).
  - Spreads-based indicator S: define s and threshold τ; spreads in excess of threshold relate to latent variable via h(.) with h(0)=0; construct S indicating when s exceeds τ (equation (3)).
  - Combined observable: (I, S) as bivariate indicator (equation (4.a)), reduced to univariate indicator y% = 1 if (I,S) ≠ (0,0), else 0 (equation (4.b)).
  - Misspecification risk: Using defaults-only (model (2)) or spreads-only (model (3)) may treat conditional probabilities as unconditional and miss information from bond markets as emerging markets gain bond access.
  - Interpretation: No historical structural break necessarily between latent variable and covariates; rather, difficulty lies in choosing proper observable indicator.
- Threshold estimation strategy: Use anecdotal evidence and statistical methods to estimate τ.

A. Psychological/Market Threshold
- Market anecdote: 1,000 basis points often considered a psychological threshold; price quotes shift to recovery value focus when spreads cross 1,000 bps.
- Altman (1998) reference: Distressed securities include those yielding a minimum of 1,000 basis points (10 percent) over comparable U.S. Treasuries; some market participants consider loss of one-third of value as distressed.

B. Estimating the Threshold for Bond Spreads Using Extreme Value Theory
- Characteristics of sovereign spreads: fat tails and volatility clustering; extreme value approaches appropriate for determining extreme event thresholds.
- Methodology:
  - Use extremal analysis (Pozo-Amuedo (2003), Koedijk (1992), Hols and De Vries (1991)).
  - Focus on tail parameter α or γ = 1/α using the Hill estimator; requires stationary and serially uncorrelated data.
  - Spreads series stationary at any frequency but serially correlated at high frequency; focus on annual data (see Table A.1).
  - Pool data, rank-order observations S1 ... Sn, compute estimator:
    - ˆγ = (1/m) * Σ_{i=n-m+1}^{n} ln(S_i / S_{n-m})  (presentation simplified to mirror text—estimator described in source)
  - Critical choice: variable m. Plot ˆγ against m and use recursive least squares to regress ˆγ on a time trend and constant, adding observations and obtaining one-step-ahead forecast with 20 percent confidence interval (Appendix Figures A.1 to A.3).
- Empirical conclusion on threshold:
  - For yearly sample, m between 42 and 50 makes ˆ1/α relatively stable.
  - Using m = 42 leads to a value of 1,073 bps and m = 50 to 969 bps.
  - Relationship between m and extreme value threshold monotonic; conclude threshold reasonably lies between 696 bps and 1,073 bps.

*Source: _wp0444 - References, authors’ calculations.*

### Appendix Table A.2).

### _wp0444 - Appendix Table A.2)

### Extreme Value Theory Versus 1,000 bps Threshold
- Using the 1,000 basis points mark versus an EVT-derived critical threshold does not significantly change crisis classification.
- Table 2 (Sample: 1994–2002) summary:
  - Total number: 216
  - Matching EVT: [211, 213]
  - Adding EVT: [5, 0]
  - Crossing out: [0, 3]
  - As percentage: Matching EVT [0.98, 0.99]; Adding EVT [0, 0.023]; Crossing out [0, 0.014]; Total 1
- Authors’ conclusion: EVT estimates are consistent with anecdotal evidence; the 1,000 basis points mark lies between the lower and upper EVT estimates and does not significantly affect the binary crisis dependent variable.

*Source: Authors’ calculations.*

### Kernel Density Estimation for the 1,000 bps Threshold
- Rationale:
  - Crossing a rounded number (1,000 bps) assumed to be a psychological threshold for market participants.
  - If psychological threshold exists, distribution of bond spreads should show a mode near that value; mode may lie slightly left because the body of the distribution lies to the left of the threshold.
- Empirical approach:
  - Analyzed both yearly and daily data; strong autocorrelation in daily data yields multimodality but does not spoil detection of modes.
  - For yearly data (less correlated), fit Gamma and Extreme Value Distributions and assess the 90th percentile.
  - For daily data, multimodality prevents good single-distribution fits; authors focus on extremes relative to tranquil periods by sampling mainly from the highest mode using an MCMC sampling method.
- Results (Table 3: Estimated 90th Percentile – 95% Confidence Interval):
  - Gamma distribution: 1,036.52 [980.55, 1,093.85]
  - Weibul distribution: 978.82 [795.10, 1,248.05]
  - Percentile corresponding to 1,000 bps:
    - Gamma distribution: 0.88
    - Weibul distribution: 0.91
- Authors’ conclusion: The 1,000 basis points threshold lies inside the 95 percent confidence bands for the 90th percentile of fitted distributions; kernel density estimation confirms a mode around 1,000 bps for both daily and yearly data.

*Source: Authors’ calculations.*

### Regression Results: Defaults Definition (1975–2002)
- Model: GEE Logit population-averaged model, correlation exchangeable, Huber-White estimator. 567 Obs.
- Key coefficients and significance (Default Definition):
  - Openness: Coeff. -0.05, Z -3.14, P>|z| 0.00, 95% CI -0.08, -0.02
  - Overvaluation: Coeff. 0.01, Z 3.40, P>|z| 0.00, 95% CI 0.01, 0.02
  - Total debt over GDP: Coeff. 0.06, Z 3.57, P>|z| 0.00, 95% CI 0.03, 0.09
  - Short-term debt over reserves: Coeff. 0.19, Z 2.07, P>|z| 0.04, 95% CI 0.01, 0.36
  - Real growth rate: Coeff. -0.08, Z -2.49, P>|z| 0.01, 95% CI -0.15, -0.02
  - Inflation (when included): Coeff. 0.00, Z 2.67, P>|z| 0.01, 95% CI 0.00, 0.00
  - Constant: -1.94 (without inflation) and -2.09 (with inflation)
  - Wald χ2: (5) 44.0; (6) 46.1
- Authors’ summary: Solvency and liquidity measures and macroeconomic controls statistically significant at 5 percent in both specifications (with or without inflation).

*Source: Authors’ calculations.*

### Regression Results: PesSy Indicator (Defaults + 1,000 bps) (1975–2002)
- Model: GEE Logit population-averaged model, correlation exchangeable, Huber-White estimation. 567 Obs.
- Key coefficients and significance (PesSy indicator):
  - Openness: Coeff. -0.03, z -2.06, P>|z| 0.04, 95% CI -0.07, 0.00
  - Overvaluation: Coeff. 0.01, z 2.75, P>|z| 0.01, 95% CI 0.00, 0.02
  - Total debt over GDP: Coeff. 0.06, z 4.26, P>|z| 0.00, 95% CI 0.03, 0.09
  - Short-term debt over reserves: Coeff. 0.30, z 2.50, P>|z| 0.01, 95% CI 0.06, 0.53
  - Real growth rate: Coeff. -0.09, z -2.61, P>|z| 0.01, 95% CI -0.17, -0.02
  - Inflation (when included): Coeff. 0.00, z 1.25, P>|z| 0.21
  - Constant: -2.80 (without inflation) and -3.08 (with inflation)
  - Wald χ2: (5) 48.0; (6) 45.4
- Authors’ summary: All regressors significant at 5 percent in both specifications except inflation; augmented default model shows a higher Wald statistic though interpretation is not clear-cut.

*Source: Authors’ calculations.*

### Subsample Comparisons: 1994–2002
- General finding: Model using bond markets to complement defaults (PesSy) performs better than default-based model in 1994–2002 subsample.
- Wald statistics comparison (1994–2002):
  - Without inflation: PesSy 29.4 vs Defaults 8.4
  - With inflation: PesSy 59.5 vs Defaults 41.6
- Additional observations:
  - Real growth and openness not significant in subsample as before; inflation is strongly significant in both setups for 1994–2002.
  - Short-term debt variable: not significant when crises are defined as defaults; significant and with expected sign when crises include bond-market turbulence.
- Selected coefficient highlights (PesSy, subsample 1994–2002, 207 Obs.):
  - Total debt over GDP: Coeff. 0.06, z 3.31, P>|z| 0.00, 95% CI 0.02, 0.09
  - Short-term debt over reserves: Coeff. 0.34, z 1.89, P>|z| 0.06, 95% CI -0.01, 0.70 (becomes significant at P>|z| 0.05 in alternate specification: Coeff. 0.39, z 2.00, P>|z| 0.05, 95% CI 0.01, 0.76)
  - Inflation: significant (Z 3.87, P>|z| 0.00) when included

*Source: Authors’ calculations.*

### Out-of-Sample Comparisons (Estimate to 1993, Predict 1994–2002)
- Estimation for 1975–1993 (Defaults model, 360 Obs.) results:
  - Openness: Coeff. -0.04, Z -2.92, P>|z| 0.00, 95% CI -0.07, -0.01
  - Overvaluation: Coeff. 0.01, Z 2.27, P>|z| 0.02, 95% CI 0.00, 0.02
  - Total debt over GDP: Coeff. 0.07, Z 4.05, P>|z| 0.00, 95% CI 0.03, 0.10
  - Short-term debt over reserves: Coeff. 0.23, Z 1.97, P>|z| 0.05, 95% CI 0.00, 0.46
  - Real growth rate: Coeff. -0.09, Z -1.93, P>|z| 0.05, 95% CI -0.17, 0.00
  - Inflation: Coeff. 0.00, Z 0.88, P>|z| 0.38
  - Constant: -2.37
  - Wald χ2 (6): 39.20
- Forecast evaluation framework:
  - Construct A (Matched Crises), B (False Alarms), C (Missing Crises), D (Matched Tranquil periods) as functions of threshold T; optimal threshold T* minimizes L(T)=B(T)/A(T)+C(T)/D(T); T* estimated via bootstrapping.
- Matched Crises over Total Crises and False Alarms over Tranquil Periods (90 percent confidence intervals):
  - Defaults: Matched Crises 0.43 [0.33, 0.50]; False Alarms over Tranquil Periods [0.12, 0.13]
  - PesSy: Matched Crises 0.61 [0.55, 0.67]; False Alarms over Tranquil Periods [0.19, 0.20]
  - In-sample (point estimate): Matched Crises 0.86; False Alarms 0.14
  - In-sample (bootstrap intervals): Matched Crises 0.69 [0.52, 0.91]; False Alarms [0.02, 0.20]
- Loss-function bootstrap results (10th percentile, mean value, 90th percentile reported as CI):
  - Default definition Loss (Out Sample): 1.50 [1.13 – 2.14]
  - Associated Optimal Threshold (Defaults): .34 [0.23 and 0.28]
  - PesSy definition Loss (Out Sample): .89 [0.75, 1.07]
  - Associated Optimal Threshold (PesSy): 0.32 [0.38 and 0.36]
  - In Sample Result Loss: 0.39
  - In Sample Result Optimal Threshold: 0.57
- Statistical comparison:
  - Table 11: Comparing Standard and PesSy Distress Definitions: Loss Functions (t-test 2 samples for different means)
    - Significance = 0.000
    - Lower bound = 0551
    - Upper bound = 067
- Authors’ conclusions from out-of-sample analysis:
  - PesSy indicator matches predicted crises better (higher Matched Crises share) though with higher False Alarms.
  - PesSy yields lower out-of-sample loss (.89 vs 1.50) and associated optimal threshold near 0.32.
  - Two-sample t-test rejects the null; PesSy performs better than defaults at any confidence level.
  - Defining debt crises as defaults is too strict to capture debt-servicing difficulties; incorporating bond-market information improves predictive alignment.

*Source: Authors’ calculations.*

### Overall Conclusion
- Proposal: Define debt crises as events when either a default occurs or secondary-market bond spreads exceed a critical threshold (empirically supported at the 1,000 basis points mark).
- Evidence:
  - EVT and kernel density estimation support the 1,000 basis points threshold as statistically consistent with extreme-event behavior and distributional modes.
  - Econometric models using the PesSy indicator (defaults + 1,000 bps) perform better in capturing debt-servicing difficulties, especially post-1994 when bond markets grew in importance.
  - PesSy improves matched-crisis rates and reduces out-of-sample loss compared with the defaults-only definition.
- Policy implication: Incorporating secondary-market bond-spread thresholds into crisis definitions provides a more comprehensive indicator of sovereign debt distress in an environment where bond markets matter.

*Source: Authors’ calculations.*

### 1994. More precisely, we find that when our definition is used, the typical determinants of bank

### _wp0444 - 1994. More precisely, we find that when our definition is used, the typical determinants of bank

### Key empirical findings on debt-servicing difficulties and defaults
- Using the authors' definition, the typical determinants of bank loan defaults in the 1980s retain predictive power for debt-servicing difficulties—on both bank loans and bonds—in the period from 1994 to 2002.
- In contrast, the standard definition of default implies a much worse out-of-sample performance for the period from 1994 to 2002.
- Liquidity indicators:
  - Are significant in explaining the authors' definition of debt crises.
  - Do not play any role in explaining defaults in the period from 1994 to 2002, under the standard default definition.
- Bond-market measurement note:
  - The analysis uses the level of spreads to capture problems in the bond markets.
  - The approach can be applied with secondary-market bond prices because of the inverse relationship between bond prices and spreads; debt-servicing problems also arise when bond prices or returns fall below a critical threshold.

### Data and serial-correlation diagnostics
- Sample: data for 31 countries ranging from 1975 to 2002.
- Ljung-Box Q-test yearly-data summary:
  - Number of rejections of the null hypothesis of no correlation: 4 (mainly for short sample countries).
  - Selected p-values, Q-Statistics, and Critical Values (as reported in Table A1.1 for individual countries; examples preserved exactly as in the source):
    - Algeria: Null 0, p-value 0.11, Q-Statistic 7.47, Critical Value 9.49
    - Argentina: Null 0, p-value 0.27, Q-Statistic 11.07, Critical Value 16.92
    - China: Null 1, p-value 0.02, Q-Statistic 12.93, Critical Value 11.07
    - Malaysia: Null 1, p-value 0.01, Q-Statistic 22.60, Critical Value 16.92
    - South Africa: Null 1, p-value 0.04, Q-Statistic 11.71, Critical Value 11.07
  - (Full country-level Ljung-Box Q-test entries are presented in Table A1.1 of the source.)

### Extreme-Value-Theory (EVT) approach and implied thresholds
- EVT applied to yearly, monthly, and daily spreads as a robustness check.
- Serial autocorrelation complicates interpretation for monthly and daily data.
- Estimated m and implied threshold τ (Table A1.2):
  - Yearly Data: m ∈ [42, 50], τ ∈ [969,1072]
  - Monthly Data: m =400, τ =1117
  - Daily Data: m =9000, τ =1084
- EVT vs. fixed 1,000 bps threshold, sample 1994–2002 (Table A1.3):
  - Yearly:
    - Matching EVT: Total number [211, 213]
    - Adding EVT: [5, 0]
    - Crossing out: [0, 3]
    - Total: 216
    - As percentage: [0.98, 0.99], [0,  0.023], [0, 0.014], 1
  - Monthly:
    - Total number: 2,252; 92; 0; Total 2,344
    - As percentage: 96; 39; 0; 1
  - Daily:
    - Total number: 48,863; 1,466; 0; Total 50,329
    - As percentage: 97; 029; 0; 1
- Interpretation highlights:
  - For monthly data, stabilization around m =400 (i.e., 1,118 bps) seems plausible.
  - For daily data, the critical threshold indicated is m = 9,000 (i.e., a bond spreads value of 1,084 bps).

### Kernel density estimation approach: reasoning and robustness
- Conceptual example explaining mode formation around a psychological threshold τ:
  - Stochastic process {y_t} constructed so that when an “inner” random walk crosses τ it does so with probability (1-p); with probability p the value is drawn between previous state y_{t-1} and τ via a generic distribution D with support [y_{t-1}, τ].
  - With errors uniformly distributed between –a and a, and for δ < a, sampling from this model produces a mode around the threshold τ because recurrence implies hitting τ with probability one.
- Kernel density empirical diagnostics:
  - Pooled spreads kernel density plots and daily spreads kernel density plots are presented using multiple kernels: Epanechnikov, Normal, Box, Triangle.
  - Closer views and robustness checks to other kernels show the density features around the threshold regions (figures report axis scales in scientific notation as in the source).

### Metropolis-Hastings algorithm implementation details
- Purpose: sample from an analytically unknown density f(x).
- Reversibility condition required in principle:
  - (0.1) f(x) q(x,y) = f(y) q(y,x)
- Practical construction via acceptance probability α:
  - Define α(x,y) = min{1, [f(y) q(y,x)] / [f(x) q(x,y)] }  (equation (0.2) in the source with the same functional form)
  - Then q_mh(x,y) = q(x,y) α(x,y) satisfies reversibility (equation (0.3) and (0.4) in the source).
- Algorithm steps as presented:
  - (1) Choose an initial value x_0; let i = 0.
  - (2) Generate x_i from q_{·}(·) for i = 1, 2, ...
  - For j = 1 to N:
    - draw x* from q(x* | x(j))
    - draw u ~ U(0,1)
    - if u < min{1, [q(x(j) | x*) f(x*)] / [q(x* | x(j)) f(x(j))]} then x(j+1) = x* else x(j+1) = x(j)
  - return {x(1), ..., x(N)}
- Practical notes from the implementation:
  - The authors show a Markov chain sample path (Figure A3.1).
  - Because the mode is known, the initial burn-in is not strictly necessary; authors throw out the first 1,000 observations and consider the displayed sample convergence achieved.

*Source: Authors’ calculations, as presented in the appendix of the supplied IMF Working Paper content.*

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