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---

### Orbis balance sheet data: coverage, preprocessing, and sample composition
- Data coverage:
  - Firm-level balance sheet data are annual and sourced from the historical financial dataset of the Orbis database, which links several data vintages.
  - For advanced economies, data are used from 1995 onwards.
  - For emerging markets, Orbis data start at later dates, ranging from 1997 to 2005.
- Sample filtering and cleaning:
  - Firms with total assets below USD 1 million are dropped.
  - Balance-sheet cleaning procedure (following Kalemli-Ozcan et al. (2015)):
    1. Exclude firms if any condition is satisfied:
       - recorded sales, total assets, or total fixed assets are negative;
       - any of these items are missing: total assets, shareholders’ funds, current liabilities, noncurrent liabilities, operating revenue, net income;
       - observation is an outlier outside the 99.9% percentile of sales-to-asset ratio, employee-to-asset ratio, or employee-to-sales ratio.
    2. Drop duplicate records by eliminating unconsolidated accounts when consolidated account is available for that year.
    3. For each BvD ID select only the last observation in each report year (report year = fiscal year when closing date is after June; otherwise preceding fiscal year).
  - Financial ratios grouped into liquidity, profitability, solvency; predictors also include number of employees, total assets in constant price (2015 USD), and industry sector (NACE Rev. 2 main section).
  - To reduce outliers, only firm observations where the denominator in a financial ratio exceeds 100 USD are kept; otherwise the ratio is set to missing.
- Sample composition and representativeness:
  - After cleaning and filtering, the sample includes more than 4 million firms, of which around 35 thousand are publicly traded.
  - Coverage of privately held companies is visibly low in Brazil, Hong Kong, Netherlands, and the US, with medians of total assets exceeding 20 million USD in these countries.
  - Privately held firms typically have smaller scale and often have median solvency ratios that are lower; no consistent cross-country patterns emerge for return on assets and liquidity ratios relative to publicly traded firms.

### NUS-CRI 1-year Probability of Default dataset: model, inputs, and usage
- Model and inputs:
  - NUS-CRI calculates PDs using the forward-intensity model of Duan et al. (2012), modeling two independent doubly stochastic Poisson processes (default delisting and other delisting).
  - Economy-wide inputs: stock index return, short-term risk-free rate, median distance-to-default (DD) of listed firms.
  - Firm-level inputs: firm distance-to-default, idiosyncratic volatility, cash to total assets, current assets to current liabilities, net income to total assets, relative size and market-to-book ratio versus economy medians.
  - Calibration requires credit default events data from sources like Thomson Reuters Datastream and Bloomberg Backoffice License.
- Coverage and grouping:
  - Public companies worldwide are categorized into six calibration groups: North America, Europe, Asia-developed economies, Emerging Markets, China, and India.
  - The database provides 1-year PDs for more than 85,000 publicly traded firms, live and delisted, in 134 countries.
  - Sectoral predictors used by SKTL: sectoral 20, 50 (median), and 80 percentiles of the cross-sectional distribution of the Logit PD; sectoral PDs computed only for sectors with more than ten firms.

### Orbis bankruptcy data: construction, limitations, and suitability
- Construction and definition:
  - Orbis status categories include: Active, Bankruptcy, Dissolved, In liquidation, Inactive and Unknown, with some detailed labels (e.g., Dissolved (bankruptcy)).
  - Bankruptcy in this study is defined by status labels: “Bankruptcy”, “Dissolved (bankruptcy)”, “Active (insolvency proceedings)”, or “Active (default of payment)”.
  - Orbis does not include dates of bankruptcy or default; the study assumes a firm becomes bankrupt/default during the year following its last observation in Orbis; last observations of firms with “Active” status are discarded to reduce bias.
- Key limitations making Orbis bankruptcy data unsuitable for direct model estimation:
  - Severe under-recording and omissions (examples: Brazil: out of 15,000 firms only one bankruptcy recorded; China: 880,000 firms with 38 recorded bankruptcies).
  - Orbis generally documents fewer bankruptcies than NUS-CRI credit events among listed firms.
  - Difficulty distinguishing genuine market exits from dataset dropouts; many firms that stop being updated remain labeled “Active”.
  - Historical truncation: bankruptcy information deleted after five years of inactivity, producing a truncated sample that misses clustered defaults and sectoral/cyclical factors.
  - Conclusion: Orbis bankruptcy data are adequate for validation of models calibrated on other sources but not for direct estimation of default predictive models.

### Macroeconomic variables and publication lags
- Included macro variables:
  - Quarterly GDP growth (sourced from World Economic Outlook database).
  - Monthly interbank interest rates (sourced from the Federal Reserve Bank of Saint Louis FRED database).
- Real-time assumptions for prediction:
  - Assume quarterly GDP growth is published one month after the quarter ends.
  - Interbank interest rates are available at the end of each month.

### From public firms’ PDs to private firms’ financial conditions: framework and empirical findings
- Intuition and challenge:
  - Market-based PDs of publicly traded firms are high-frequency signals that may inform PDs of privately held firms via sectoral shocks.
  - Challenge: publicly traded firms differ systematically (easier equity financing, larger size), and Orbis bankruptcy data prevent direct regression of private credit incidents on listed firms’ PDs.
  - The study uses an indirect approach based on panel regressions of changes in private firms’ financial ratios on sectoral PDs of listed firms.
- Predictive framework and notation:
  - Focus financial conditions: solvency (shareholders’ fund-to-total assets), return on assets (net income/total assets), liquidity (current assets minus inventory to current liabilities).
  - Panel regression setup for each quarter k ∈ {1,2,3,4}:
    - ∆y_cs,t+1 = α_csk + β_k ∆PD_cs,t+0.25k + γ_k · controls + ε_cs,t+1|t+0.25k
    - ∆y_cs,t = median change in privately held firms’ financial ratio of yeart relative to previous year within sector s of economy c.
    - ∆PD_cs,t+0.25k = median change in publicly traded firms’ PD observed at end of quarter k of yeart+1 relative to end of yeart; scaled to unit variance within each economy-sector-quarter.
    - Controls: country real GDP growth rate, average intra-bank interest rates in year t and t+1, end-of-year sectoral PD and sectoral median of the financial ratio in yeart.
- Empirical findings:
  - A rise in sectoral PD of publicly traded firms predicts decreased solvency ratios, ROA, and liquidity ratios for all firms.
  - Informational value increases with later-quarter PD information: β_k increases with k.
    - Example: a one-standard-deviation shock to end-of-year sectoral PD (k = 3,4) predicts a lower expectation of current-year nonlisted-firm ROA by about 0.3 percent; same shock for earlier quarters (k = 1,2) lowers it by 0.2 percent.
  - Coefficient magnitudes generally higher among privately held firms than publicly traded firms, consistent with smaller firms’ greater susceptibility to macro shocks.
  - Conclusion: Sectoral PDs of listed firms are informative for privately held firms; including high-frequency market-based PDs could significantly improve predictive models for private firms’ financial conditions.

### SKTL: Signal-Knowledge Transfer Learning model — motivation and implementation
- Motivation:
  - Absence of reliable, publicly available default data for privately held firms motivates SKTL, which leverages PD information from publicly traded firms.
- Implementation steps:
  1. Choose functional form (Gradient-Boosting Tree) for mapping predictors to Logit(PD).
  2. Train model to predict PDs of publicly traded firms using predictors available for private firms: balance sheet variables, market-based sectoral PDs, macro indicators.
  3. Evaluate performance conditional on systemic credit cycle and balance sheet staleness.
  4. Extend model to privately held firms and validate using Orbis bankruptcy data.

### Mixed-frequency model setup (Section 5.1)
- Mixed-frequency predictors and publication lags:
  - Balance sheet data: annual (assumed released three months after fiscal year closing; alternate analysis assumes 15-month lag).
  - Sectoral PD and interest rates: available monthly at end-of-month.
  - GDP: quarterly with one-month publication lag.
- Notation and modeling choices:
  - i: firm index; s(i): sector of firm i; t: end of month observed.
  - l: months since release of most recent annual balance sheet, l ∈ {0,1,2,...,11}.
  - Firm_i,t−l, Macro_t, Sectoral_s(i),t denote most recent firm features, macro variables, and sectoral PD predictors available at t.
  - Models F_cl estimated for each staleness l and calibration group cl.
- Forecasting objective and formulation:
  - Generate end-of-month PD using available predictors at that time:
    - Logit(PD_it) = F_cl[Macro_t, Sectoral_s(i),t, Firm_i,t−l] + ε_i,t|t−l
  - Practical notes:
    - F_cl varies with l to account for balance sheet staleness.
    - Truncate one-year PDs by assigning 10^{-5} to all values below that threshold to avoid logit explosion.
    - Duan and Li (2021) suggest AAA rating corresponds to PD below 3.5·10^{-5}.
    - Model calibration groups follow NUS-CRI groupings.

### Gradient-boosting Tree specification and hyper-parameters (Section 5.2)
- Functional form and justification:
  - Gradient-Boosting Tree chosen for F_cl; tree-based ensemble methods are strong performers on tabular data.
  - LightGBM used for efficient implementation.
- Mathematical training objective (key equations summarized in source):
  - Ensemble prediction ˆy_i = Σ_{k=1}^K ρ_k f_k.
  - Sequential updates φ_k = φ_{k−1} + ρ_k f_k; ideal base learner approximates negative gradient of loss.
- LightGBM engineering highlights:
  - Trains on subset of dataset by keeping observations with larger gradients.
  - Bundles features to reduce dimensionality.
  - Offers significant speed and memory improvements with minor performance compromises.
- Overfitting guardrails and hyper-parameters tuned in SKTL:
  - Tuned hyper-parameters:
    - Number of boosting rounds (K).
    - Maximum depth of regression tree.
    - Learning rate (shrinkage parameter) for base learners.
    - Path smoothing (controls smoothing of leaf outputs).
  - Hyperparameter search:
    - Grid search over combinations to minimize cross-validation MSE using data up to 2015.
    - Training and validation set design described in Section 5.3 of source.

### Predictive performance highlights and validation metrics
- Discrimination and linkage:
  - NUS-CRI / CRI-PD AUROC examples for CRI credit events or Orbis exits (CRI-PD AUROC column preserved):
    - BR: 0.87; CN: 0.83; DE: 0.87; ES: 0.76; FR: 0.87; GB: 0.89; HK: 0.82; IN: 0.86; IT: 0.89; JP: 0.93; KR: 0.90; MY: 0.86; NL: 0.88; PH: 0.84; PL: 0.88; RU: 0.66; SG: 0.87; TH: 0.91; US: 0.93; VN: (entry blank).
- R^2 performance (selected entries preserved exactly):
  - Table 5 (balance sheets published 3 months after closing dates) — Median (Cross-validation Year ≤ 2015): Main 0.55, PM 0.37, Benchmark 0.30; Out of sample Year > 2015: Main 0.57, PM 0.45, Benchmark 0.18.
  - Country Main (Cross-validation / Out of sample) examples:
    - BR: 0.60 / 0.63; CN: 0.79 / 0.64; DE: 0.46 / 0.46; JP: 0.66 / 0.73; US: 0.55 / 0.51; RU: 0.43 / 0.48.
  - Table A.1 (balance sheets published 15 months after closing dates) — Median (Cross-validation Year ≤ 2015): Main 0.51, PM 0.35, Benchmark 0.29; Out of sample Year > 2015: Main 0.51, PM 0.41, Benchmark 0.15.
    - Country Main examples (Cross-validation / Out of sample): BR: 0.53 / 0.58; CN: 0.72 / 0.51; DE: 0.41 / 0.37; JP: 0.66 / 0.67; US: 0.48 / 0.41; RU: 0.42 / 0.41.
- Loss decomposition of forecasting Logit PD (selected median and examples preserved):
  - Table 6 (3-month publication lag) — Median squared bias and idiosyncratic loss differentials by PD regime:
    - Left panel (̄PD_t ≤ qt25): −0.25, −0.03.
    - Middle (qt25 ≤ ̄PD_t ≤ qt75): −0.13, −0.04.
    - Right (̄PD_t ≥ qt75): −0.33, −0.04.
  - Table A.2 (15-month publication lag) — Median:
    - Left: −0.26, −0.04; Middle: −0.14, −0.03; Right: −0.322, −0.04.
- Feature importance and interpretability:
  - Main influential predictors (feature names preserved): Profit margin (earnings before tax based); Current ratio; Debt-to-suppliers to operating revenue ratio; Gross profit margin; ROE using net income; Liquidity ratio; ROA (EBIT based); Solvency ratio (asset based); Acid test; Gearing ratio; Earnings to debt ratio; Tangible fixed assets to debt ratio; Sectoral 20/80 percentiles and median of logit PD; Total assets in constant price; Number of employee; Inter-bank overnight rates; Interest coverage ratio; Stock turnover; Industry sector.
  - Country-specific top features preserved (examples):
    - BR: Profit margin (earnings before tax based); Current ratio; Debt-to-suppliers to operating revenue ratio.
    - CN: Sectoral median of logit PD; Gearing ratio; Sectoral 20 percentile of logit PD.
    - DE: Liquidity ratio; Acid test; Gearing ratio.
    - US: Industry sector; Tangible fixed assets to debt ratio; Sectoral 80 percentile of logit PD; Acid test; ROA (EBIT based).

### Case study — Germany utility sector during 2022 energy crisis (Section 6.1)
- Background:
  - Russia-Ukraine conflict in 2022 led to depletion of Russian gas supplies to Europe; wholesale prices for electricity and gas surged by as much as 15-fold above their early 2021 levels.
  - Numerous utility companies faced restrictions on passing additional costs to clients; governments intervened to provide liquidity to energy firms.
- Data and modeling approach:
  - As of July 2023, sample contains 821 privately held firms and 13 publicly traded firms.
  - SKTL used for real-time analysis accounting for a 15-month lag in Orbis publication of balance sheet data for most firms.
  - Benchmark model omits real-time sectoral PD information.
- PD distribution evolution and tail risk:
  - SKTL privately held PD distributions rose steadily since early 2021 and spiked in September 2022 (Nord Stream 2 event), remaining elevated afterward.
  - Without real-time sectoral PD (benchmark), privately held PDs would have remained stagnant until July 2022 then risen gradually.
  - Privately held PD distribution shows greater dispersion than publicly traded distribution, particularly in the right tail.
  - Top-decile statistic: the top decile of the privately held PD distribution averages around 45 basis points (interpretation: ~one out of ten privately held firms faced risk of losing its BBB- rating and potentially being downgraded to junk).
  - Affected firm types include municipal utilities and renewable energy firms; publicly traded firms remained investment-grade.
- Drivers of increased default risk (Shapley value analysis):
  - Predictor groups: liquidity, profitability, solvency, macro, sectoral PD.
  - Worsening of liquidity, profitability, and solvency are primary drivers behind increased PDs.
  - Sectoral PD variables contribute 88 percent to the rise in PDs, overshadowing other variable groups.
- Implication:
  - Including real-time sectoral PD information substantially alters assessment of default risk during episodic shocks; SKTL expands coverage of tail risks and responds quickly to macroeconomic events (example: immediate PD jump at Nord Stream 2 event and persistent elevation afterward).

_Italic: Source: wpiea2024115-print-pdf_

### 3.1  Orbis Balance Sheet Data

### 3.1  Orbis Balance Sheet Data

### Data coverage and preprocessing
- Firm-level balance sheet data are annual and sourced from the historical financial dataset of the Orbis database, which links several data vintages.
- For advanced economies, data are used from 1995 onwards to match CRI-PD dataset availability.
- For emerging markets, Orbis data start at later dates, ranging from 1997 to 2005.
- Firms with total assets below USD 1 million are dropped for three reasons:
  - Bajgar et al. (2020) shows Orbis’ smaller firms coverage is not representative of the actual small firms’ distribution;
  - Altman et al. (2017) shows financial ratios of small firms tend to be unstable and not suitable for financial distress prediction;
  - The SKTL model’s calibration is based on data from publicly-traded firms, which are, on average, significantly larger than privately-held firms.
- Balance-sheet cleaning procedure follows Kalemli-Ozcan et al. (2015):
  1. Exclude firms if any condition is satisfied:
     - any of recorded sales, total assets, or total fixed assets are negative;
     - any of these balance sheet items are missing: total assets, shareholders’ funds, current liabilities, noncurrent liabilities, operating revenue, and net income;
     - the observation is an outlier outside the 99.9% percentile of sales-to-asset ratio, number of employee-to-asset ratio, and number of employee-to-sales ratio.
  2. Drop duplicate records by eliminating observations from the unconsolidated account when the consolidated account is available for that year.
  3. For each unique Orbis identifier (BvD ID) select only the last observation in each report year. The report year is the fiscal year when the closing date is after June; otherwise it is the preceding fiscal year.
- Financial ratios are grouped into liquidity, profitability, and solvency. Predictors also include number of employees, total assets in constant price (2015 USD), and industry sector (NACE Rev. 2 main section).
- To reduce outliers, only firm observations for which the denominator in their financial ratio exceeds 100 USD are kept; if violated the ratio is set to missing. This also removes negative denominators.

### Sample composition and representativeness
- After cleaning and filtering, the sample includes more than 4 million firms, of which around 35 thousand are publicly traded.
- Representativeness varies across countries due to differences in reporting requirements and firm coverage (e.g., mandatory reporting in many European countries versus no mandatory reporting in the US).
- Coverage of privately held companies is visibly low in Brazil, Hong Kong, Netherlands, and the US, with medians of total assets exceeding 20 million USD in these countries.
- In general, privately held firms typically have smaller scale and often have median solvency ratios that are lower (less equity financing and reduced capital buffers). No consistent cross-country patterns emerge for return on assets and liquidity ratios relative to publicly traded firms.

---

### 3.2  NUS-CRI Probability of Default dataset

### Model and inputs
- NUS-CRI calculates PDs using the forward-intensity model of Duan et al. (2012), which models two independent doubly stochastic Poisson processes: one for delisting due to default and one for delisting due to other reasons.
- Economy-wide inputs: stock index return, short-term risk-free rate, median distance-to-default (DD) of listed firms in the economy.
- Firm-level inputs: firm distance-to-default, idiosyncratic volatility, cash to total assets, current assets to current liabilities, net income to total assets, relative size and market-to-book ratio versus economy medians.
- Calibration requires credit default events data gathered from sources like Thomson Reuters Datastream and Bloomberg Backoffice License.

### Coverage, grouping, and usage
- Calibrating models by individual economy can be infeasible where credit default events are sparse. Public companies worldwide are therefore categorized into six calibration groups: North America, Europe, Asia-developed economies, Emerging Markets, China, and India.
- The study uses 1-year PDs from the NUS-CRI PD database, which provides time series for more than 85,000 publicly traded firms, both live and dead (delisted), in 134 countries.
- The 1-year NUS-CRI PDs are appropriate for SKTL calibration; prediction accuracy measured by AUROC is comparable to other commercial PD providers.
- SKTL uses sectoral predictors: the 20, 50 (median), and 80 percentile of the cross-sectional distribution of the Logit PD in each sector. Sectoral PDs are computed only for sectors with more than ten firms to reduce noise.

---

### 3.3  Orbis Bankruptcy Data

### Construction and limitations
- Orbis status categories include: Active, Bankruptcy, Dissolved, In liquidation, Inactive and Unknown. Some statuses have detailed labels, e.g., Active (default of payment), Dissolved (bankruptcy).
- Bankruptcy in this study is defined by status labels: “Bankruptcy”, “Dissolved (bankruptcy)”, “Active (insolvency proceedings)”, or “Active (default of payment)”.
- Orbis does not include dates of bankruptcy or default. For validation, the study assumes a firm becomes bankrupt/default during the year following its last observation in Orbis. To reduce bias, last observations of firms with “Active” status are discarded.

### Reasons Orbis bankruptcy data are unsuitable for direct model estimation
- Data scarcity and omissions: Example counts show severe under-recording (e.g., Brazil: out of 15,000 firms only one bankruptcy recorded; China: 880,000 firms with 38 recorded bankruptcies).
- Comparison with NUS-CRI credit events among listed firms indicates Orbis documents fewer bankruptcies in most countries, suggesting substantial omissions.
- Difficulty distinguishing market exits from dataset dropouts: many firms that stop being updated remain labeled “Active”.
- Historical truncation: bankruptcy information is deleted after five years of inactivity, producing a truncated sample that misses clustered defaults and sectoral/cyclical factors (Das et al., 2007).
- Conclusion: Orbis bankruptcy data are adequate for validation of models calibrated on other data sources but not for direct estimation of default predictive models.

---

### 3.4  Macroeconomic Variables
- Included macro variables: quarterly GDP growth and monthly interbank interest rates.
- Quarterly GDP growth sourced from the World Economic Outlook database; interbank interest rates from the Federal Reserve Bank of Saint Louis FRED database.
- For real-time PD prediction, assume quarterly GDP growth is published one month after the quarter ends and interbank interest rates are available at the end of each month.

---

### 4  From Public Firms’ PDs to Private Firms’ Financial Conditions

### Intuition and challenge
- Market-based PDs of publicly traded firms are high-frequency signals (daily/weekly/monthly) that may inform PDs of privately held firms through sectoral shocks.
- Challenge: publicly traded firms differ (easier equity financing, larger size); representativeness for private firms is not guaranteed.
- Direct logit regression of private firms’ credit incidents on listed firms’ PDs is infeasible due to Orbis bankruptcy data deficiencies; an indirect approach is used.

### Predictive framework and variables
- Focused private-firm financial conditions: solvency (shareholders’ fund-to-total assets ratio), return on assets (net income-to-total assets ratio), and liquidity (current assets minus inventory to current liability ratio, i.e., current ratio).
- Panel regressions with economy-sector fixed effects estimate whether intra-year changes in sectoral PDs of publicly traded firms forecast changes in privately held firms’ end-of-year financial ratios.
- Notation and setup:
  - ∆y_cs,t = median change in privately held firms’ financial ratio of yeart relative to previous year within sector s of economy c.
  - ∆PD_cs,t+0.25k = median change in publicly traded firms’ PD observed at end of quarter k of yeart+1 relative to end of yeart. ∆PD_cs,t+0.25k is scaled to unit variance within each economy-sector-quarter.
  - For each quarter k ∈ {1,2,3,4}, regression:
    ∆y_cs,t+1 = α_csk + β_k ∆PD_cs,t+0.25k + γ_k · controls + ε_cs,t+1|t+0.25k
  - Controls include country real GDP growth rate, average intra-bank interest rates in year t and t+1, and end-of-year sectoral PD and sectoral median of the financial ratio in yeart.

### Empirical findings
- A rise in sectoral PD of publicly traded firms predicts decreased solvency ratios, ROA, and liquidity ratios for all firms, regardless of listing status.
- The informational value increases as later-quarter PD information is used: β_k increases with k.
  - Example: a one-standard-deviation shock to end-of-year sectoral PD (k = 3,4) predicts a lower expectation of current-year nonlisted-firm ROA by about 0.3 percent, whereas the same shock for earlier quarters (k = 1,2) lowers it by 0.2 percent.
- Coefficient magnitudes are generally higher among privately held firms than publicly traded firms, consistent with smaller firms’ greater susceptibility to macro shocks (Fama and French (1993)).
- Conclusion: Sectoral PDs of listed firms are informative for privately held firms; including high-frequency market-based PDs could significantly improve predictive models for private firms’ financial conditions.

---

### 5  SKTL: A Mixed-Frequency Gradient-Boosting Tree PD Model

### Motivation and overall approach
- The absence of reliable, publicly available default and bankruptcy data for privately held firms motivates the Signal-Knowledge Transfer Learning (SKTL) model, which leverages PD information from publicly traded firms.
- Implementation steps:
  1. Choose a functional form (Gradient-Boosting Tree) for mapping predictors to Logit(PD).
  2. Train the model to predict PDs of publicly traded firms using predictors available for private firms: balance sheet variables, market-based sectoral PDs, and macroeconomic indicators.
  3. Evaluate model performance conditional on systemic credit cycle and balance sheet data staleness.
  4. Extend the model to privately held firms and validate using Orbis bankruptcy data.

### 5.1  Mixed-frequency model setup
- Mixed-frequency predictors are essential:
  - Balance sheet data: annual (assumed released three months after fiscal year closing).
  - Sectoral PD and interest rates: available monthly at end-of-month.
  - GDP: quarterly with one-month publication lag.
- Notation and modeling choices:
  - i: firm index; s(i): sector of firm i; t: end of month observed.
  - l: number of months since release of most recent annual balance sheet, l ∈ {0,1,2,...,11}.
  - Firm_i,t−l, Macro_t, Sectoral_s(i),t denote most recent firm features, macro variables, and sectoral PD predictors available at t.
  - Models F_cl are estimated for each staleness l and for each calibration group cl.
- Forecasting objective:
  - Generate end-of-month PD using available predictors at that time.
  - Formulation:
    Logit(PD_it) = F_cl[Macro_t, Sectoral_s(i),t, Firm_i,t−l] + ε_i,t|t−l
- Practical modeling notes:
  - F_cl accounts for balance sheet staleness by varying with l; GDP staleness is mitigated by its three-month lag aligning with PD data.
  - Using Logit(PD) ensures implied PD ∈ [0,1] and magnifies sensitivity of squared loss for small PD values.
  - To avoid logit explosion for extremely small PDs, truncate one-year PDs by assigning 10^{-5} to all values below that threshold.
  - Duan and Li (2021) suggest AAA rating corresponds to PD below 3.5·10^{-5}.
- The model adopts a modular approach consistent with NUS-CRI groupings for calibration.

*Source: wpiea2024115-print-pdf - 3.1  Orbis Balance Sheet Data*

### 5.2  Gradient-boosting Tree and its Hyper-parameters

### 5.2  Gradient-boosting Tree and its Hyper-parameters

### Overview of method
- Functional form: gradient-boosting tree chosen for F_cl.  
- Justification: tree-based ensemble methods (regression/classification trees) are among the best algorithms for regression and classification on tabular data (Grinsztajn et al., 2022). Gradient Boosting (Friedman, 2001) trains base learners sequentially, with each learner attempting to rectify mistakes of preceding learners.

### Mathematical formulation and training objective
- Dataset notation: nsamples and mfeatures D = {(x_i, y_i)} (|D| = n, x_i ∈ R^m, y_i ∈ R). Mapping function y_i = φ(x_i).
- Ensemble approximation: ˆy_i = φ_K(x_i) = Σ_{k=1}^K ρ_k f_k, with f_k = f(x_i, θ_k). (Equation (3))
- Loss minimization: construct φ_K to minimize expected value of loss L(y_i, ˆy_i) via K steps.
- Initial constant approximation: φ*_0 = arg min_α Σ_{i=1}^n L(y_i, α). (Equation (4))
- Sequential update: φ_k = φ_{k−1} + ρ_k f_k, for k = 1,...,K. (Equation (5))
- Each step chooses {ρ_k, f_k} = arg min_{ρ,f} Σ_{i=1}^n L(y_i, φ_{k−1} + ρ f(x_i, θ)). (Equation (6))
- Ideal step (negative gradient): f_k(x_i) = −g_k(x_i) = −[∂L(y_i, φ(x_i))/∂φ(x_i)]_{φ(x_i)=φ_{m−1}(x_i)}. (Equation (7))
- Line search for ρ_k: ρ_k = arg min_ρ Σ_{i=1}^n L(y_i, φ_{k−1}(x_i) + ρ f_k(x_i)). (Equation (8))
- Practical approximation: choose f_k(x_i) = f(x_i, θ_k) closest to g_k across data distribution by solving θ_k = arg min_{β,θ} Σ_{i=1}^n [−g_k(x_i) − ρ_k f(x_i, θ)]^2. (Equation (9))

### Trees as base learners
- Weak learners: regression trees. Tree structure: root → internal nodes (decision criteria) → leaves (scores/outputs).
- Prediction mechanism: follow decision rules from root to leaf; leaf outputs are combined into final prediction.
- Optimization challenge: finding ideal tree structure and leaf weights is computationally intensive; conventional "greedy" method evaluates every possible split point for each feature.

### Efficient implementations: LightGBM
- Engineering advances: algorithms and engineering solutions proposed to train gradient boosting more efficiently (Tyree et al. (2011); Chen and Guestrin (2016); Ke et al. (2017b)).
- LightGBM highlights (Ke et al., 2017b):
  - Trains on a subset of dataset by keeping observations associated with larger gradients and discarding the rest.
  - Bundles features to reduce data dimensionality.
- Performance trade-off: LightGBM significantly outperforms other gradient-boosting implementations in computational speed and memory consumption with minor compromises on predictive performance (Bentéjac et al., 2021).

### Overfitting and hyper-parameters tuned in SKTL
- Tree-based methods tend to overfit; LightGBM hyperparameters guard against overfitting.
- In the SKTL model only a subset of hyperparameters are tuned; others set to default.
- Hyperparameters tuned to improve model performance:
  - The number of boosting rounds.
    - Purpose: find optimal number of base learners, K (as in equation (3)). Setting a lower K would reduce overfitting.
  - The maximum depth of regression tree.
    - Purpose: restrict the maximum depth of base learner f(x, θ). A lower maximum depth reduces base learner complexity and overfitting.
  - The learning rate of base learners.
    - Purpose: shrink each base learner ρ_k f_k in equation (3) towards zero by applying a shrinkage parameter. A lower learning rate, coupled with more boosting rounds, leads to a better fit to the sample.
  - Path smoothing.
    - Purpose: governs how each base learner tree generates outputs to reduce overfitting in leaves with fewer samples.
    - Behavior: under default value of zero, output determined solely by samples from each leaf. A positive path-smoothing parameter makes tree outputs the weighted average of the current leaf output and outputs of past nodes that lead to the leaf. Leaf/node output defined as average sample dependent variables in the leaf or node.

### Hyperparameter search and validation
- A grid search over combinations of the hyper-parameters listed above is used to find values that minimize cross-validation MSE using data up to 2015.
- The design of training and validation sets for cross-validation and out-of-sample evaluation is described subsequently in Section 5.3.

*Source: wpiea2024115-print-pdf - 5.2  Gradient-boosting Tree and its Hyper-parameters*

### 6.1  Case 1: Germany Utility Sector During 2022 Energy Crisis

### 6.1 Case 1: Germany Utility Sector During 2022 Energy Crisis

### Background: 2022 energy shock and government response
- The Russia-Ukraine conflict in 2022 led to depletion of Russian gas supplies to Europe.
- Wholesale prices for electricity and gas surged by as much as 15-fold above their early 2021 levels.
- Numerous utility companies faced restrictions on passing additional costs to clients.
- Governments intervened quickly to provide liquidity to energy firms.

### Data coverage and modeling approach
- As of July 2023, there were 821 privately held firms and 13 publicly traded firms in the sample.
- The SKTL model is used for real-time analysis while accounting for a 15-month lag in Orbis’s publication of balance sheet data for most firms.
- Comparisons are made to a benchmark model that omits real-time sectoral PD information.

### Evolution of probability of default (PD) distributions
- PD distributions for privately held firms (SKTL) rose steadily since early 2021 and spiked in September 2022 (coinciding with the Nord Stream 2 gas pipeline event), remaining elevated afterward.
- Without real-time sectoral PD (benchmark model), PDs for privately held utilities would have remained stagnant until July 2022 and then risen gradually.
- The privately held PD distribution shows greater dispersion than the publicly traded distribution, particularly in the right tail (tail risk).
- Top-decile statistic:
  - The top decile of the privately held PD distribution averages around 45 basis points.
  - Interpretation: approximately one out of ten privately held firms faced the risk of losing its BBB- rating and potentially being downgraded to junk.
- Affected firm types: municipal utilities and renewable energy firms were among the most affected.
- Publicly traded firms: all remained comfortably within the investment-grade rating.

### Drivers of increased default risk (Shapley value analysis)
- Shapley values compare median contributions of predictor groups between July 2023 and July 2019.
- Predictor groups: liquidity, profitability, solvency, macroeconomic conditions (macro), and sectoral PD.
- Key findings:
  - Worsening of liquidity, profitability, and solvency are primary drivers behind increased PDs.
  - Sectoral PD variables, which embed forward-looking information, contribute 88 percent to the rise in PDs, overshadowing other variable groups.

### Implications from the case
- Including real-time sectoral PD information substantially alters the assessment of default risk during episodic shocks.
- SKTL expands coverage of tail risks and responds quickly to macroeconomic events (example: immediate PD jump at Nord Stream 2 event and persistent elevation afterward).

_Italic: Source: wpiea2024115-print-pdf - 6.1 Case 1: Germany Utility Sector During 2022 Energy Crisis_

### References

### References and empirical materials

### Key methodological and empirical sources
- Foundational models and methods cited:
  - Altman, E. I. (1968); Altman (2013); Altman et al. (2017) — Z-score and financial distress prediction.
  - Beaver (1966); Zmijewski (1984); Shumway (2001) — financial ratios and hazard models for failure prediction.
  - Black and Scholes (1973); Merton (1974); Duffie and Singleton (1999) — structural and reduced-form approaches to credit risk and defaultable securities.
  - Breiman et al. (1984); Friedman (2001); Chen and Guestrin (2016); Ke et al. (2017a, 2017b); Lundberg and Lee (2017) — tree-based methods, gradient boosting, XGBoost, LightGBM, and SHAP interpretation.
  - Duan (1994); Duan et al. (2012, 2018, 2021) — distance-to-default, multiperiod default prediction, PD-implied ratings.
  - Duffie, Saita, and Wang (2007); Giesecke et al. (2014); Koopman et al. (2012) — multi-period default prediction, macroeconomic effects, dynamic factor models.
  - KMV and industry models: Crosbie and Bohn (2002); Falkenstein et al. (2000); Cangemi et al. (2003).
- Machine learning, transfer learning, and tabular-data references:
  - Barboza et al. (2017); Bentéjac et al. (2021); Grinsztajn et al. (2022); Weiss et al. (2016); Ben-David et al. (2010); Storkey (2009).
- Data coverage and firm-level representativeness:
  - Bajgar et al. (2020); Kalemli-Ozcan et al. (2015); Kalemli-Ozcan et al. (2020).

### Empirical applications and context-specific studies cited
- Corporate debt and crises: Adasoro, Hardy, and Tarashev (2021); Das et al. (2007); Giesecke et al. (2014).
- Private firm default and PD estimation: Duan et al. (2018); Zhou et al. (2006); Bhimani et al. (2010); Dierkes et al. (2013).
- Sectoral studies and case-specific analyses: Fioretti et al. (2023); Eckert and Buli (2022); Miao et al. (2018).

---

### Predictors used (Table 1)
- Panel A: Liquidity
  - Acid test: Cash and cash equivalent / Current liabilities
  - Cash to operating revenue ratio: Cash flow / Operating revenue turnover
  - Current ratio: Current assets / Current liabilities
  - Debt-to-suppliers to operating revenue ratio: Debt to suppliers / Operating revenue
  - Interest coverage ratio: Operating profit / Interest paid
  - Liquidity ratio: (Current assets - inventory) / Current Liabilities
  - Tangible fixed assets to debt ratio: Tangible fixed assets / (Current liability + Noncurrent liability)
  - Trade receivables to operating revenue ratio: Trade receivables / Operating revenue
  - Working capital: (Current assets - Current liabilities) / Total assets
- Panel B: Profitability and activity
  - Asset turnover: Operating revenue / Total assets
  - Gross profit margin: (Operating revenue - Costs of goods sold) / Operating revenue
  - Profit margin (earnings before tax based): Earnings before tax / Operating revenue
  - Profit margin (net income based): Net income / Operating revenue
  - ROA (earnings after tax based): Earnings after tax / Total assets
  - ROA (earnings before tax based): Earnings before tax / Total assets
  - ROA (EBIT based): EBIT / Total assets
  - ROA (EBITDA based): EBITDA / Total assets
  - ROA (net income based): Net income / Total assets
  - ROCE using net income: (Net income + Interest paid) / (Shareholders’ funds + Non current liabilities)
  - ROCE using profit before tax: (Profit before tax + Interest paid) / (Shareholders’ funds + Non current liabilities)
  - ROE using net income: Net income / Shareholders’ funds
  - Stock turnover: Operating revenue / Inventory
- Panel C: Solvency
  - Earnings to debt ratio: EBIT / (Current liabilities + Noncurrent liabilities)
  - Gearing ratio: (Noncurrent liabilities+Loans) / Shareholders’ funds
  - Shareholders liquidity ratio: Shareholders’ funds / Noncurrent liabilities
  - Solvency ratio (asset based): Shreholders’ funds / Total assets
  - Solvency ratio (liability based): Shreholders’ funds / (Non current liabilities+Current liabilities)
- Panel D: Other firm characteristics
  - Industry sector: NACE Rev. 2 main section
  - Number of employee
  - Total assets in constant price: In 2015 USD
- Panel E: PD and macro economic variables
  - Inter-bank overnight rates: End-of-month values
  - Real quarterly GDP growth
  - Sectoral 20 percentile of logit PD: End-of-month values
  - Sectoral 80 percentile of logit PD: End-of-month values
  - Sectoral median of logit PD: End-of-month values

---

### Sample and summary statistics (Table 2)
- Sample selection:
  - Focus: observations with total assets above 1 million USD and with data available on total assets, sharholders’ funds, current liabilities, noncurrent liabilities, operating revenue and net income.
- Table entries (selected country-level medians and aggregates; figures preserved exactly as in source):
  - For each country (BR, CN, DE, ES, FR, GB, HK, IN, IT, JP, KR, MY, NL, PH, PL, RU, SG, TH, US, VN) the table reports:
    - Start and End years (mostly 1995 to 2023; VN: 2001 to 2023)
    - Aggregate total asset (fiscal year 2021) in billion of USD (e.g., BR 404, CN 2,576, DE 1,155, ... , US 8,819)
    - Median total asset in million of USD (e.g., BR 651.6, CN 9,251, DE 120.56, ... , US 243.66)
    - Median shareholders’ fund to total assets ratio (Solvency Ratio) (e.g., BR 23.37, CN 4.5, DE 12.79, ... , US 26.1)
    - Median net income to total assets ratio (%) (Return on Assets) (e.g., BR 0.39, CN 0.53, DE 0.41, ... , US 0.48)
    - Median current asset minus inventory to current liability ratio (Liquidity Ratio) (e.g., BR 3.18, CN 3.87, DE 2.25, ... , US 1.53)
- Notes (preserved):
  - 1 Aggregate total asset of fiscal year 2021 in billion of USD.
  - 2 Median total asset in million of USD.
  - 3 Median shareholders’ fund to total assets ratio.
  - 4 Median net income to total assets ratio (%).
  - 5 Median current asset minus inventory to current liability ratio.

---

### Default events linkage and discrimination (Table 3)
- Comparison of Orbis exits and NUS-CRI credit events; CRI-PD AUROC when applied to predict CRI credit events or Orbis firms’ exits in the next twelve months:
  - Selected AUROCs (CRI-PD AUROC column preserved as in table):
    - BR: 0.87
    - CN: 0.83
    - DE: 0.87
    - ES: 0.76
    - FR: 0.87
    - GB: 0.89
    - HK: 0.82
    - IN: 0.86
    - IT: 0.89
    - JP: 0.93
    - KR: 0.90
    - MY: 0.86
    - NL: 0.88
    - PH: 0.84
    - PL: 0.88
    - RU: 0.66
    - SG: 0.87
    - TH: 0.91
    - US: 0.93
    - VN: (entry blank in source)

---

### Regression links between sectoral PD and sectoral financial ratios (Table 4)
- Intra-year prediction regression of sectoral financial ratios against sectoral PD; coefficients (selected) and significance indicators preserved:
  - Shareholders’ funds to total assets ratio (Solvency ratio, %), Return on assets (Net income to total assets, %), Liquidity ratio (Current asset minus inventory to current liability ratio, %)
  - Quarterly months reported: March, June, September, December.
  - Coefficient examples (Listed / Non-listed where shown) with significance markers:
    - March: Solvency listed −0.02, non-listed −0.07; Return on assets listed −0.11∗∗, non-listed −0.17; Liquidity listed 0.11, non-listed 0.40∗∗∗. (Standard errors in parentheses preserved in table.)
    - September: Solvency listed −0.11∗∗∗, non-listed −0.29∗∗∗; Return on assets listed −0.24∗∗, non-listed −0.37∗∗∗; Liquidity listed −0.39∗∗∗, non-listed −0.55∗∗∗. (Standard errors shown in table.)
  - Each regression includes 5044 country-sector-year observations; standard errors derived from clustered covariance matrix.
  - Significance legend preserved:
    - ∗ indicates pvalues between 5 percent and 10 percent;
    - ∗∗ indicates pvalues between 1 percent and 5 percent;
    - ∗∗∗ indicates pvalues below 1 percent.

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### Predictive performance: R^2 of Logit PD and model comparisons (Tables 5 and A.1)
- Definition preserved: R^2 of Model A over Model B is defined as R^2_A/B = 1 − MSE_A / MSE_B. Model “PM” denotes prevailing mean model (forecasts are sample mean in training sets).
- Table 5 (assuming balance sheets published 3 months after closing dates) — selected R^2 values:
  - Median (Cross-validation Year ≤ 2015): Main 0.55, PM 0.37, Benchmark 0.30; (Out of sample Year > 2015): Main 0.57, PM 0.45, Benchmark 0.18.
  - Country examples (Cross-validation / Out of sample Main values):
    - BR: 0.60 / 0.63
    - CN: 0.79 / 0.64
    - DE: 0.46 / 0.46
    - JP: 0.66 / 0.73
    - US: 0.55 / 0.51
    - RU: 0.43 / 0.48
- Table A.1 (assuming balance sheets published 15 months after closing dates) — selected R^2 values:
  - Median (Cross-validation Year ≤ 2015): Main 0.51, PM 0.35, Benchmark 0.29; (Out of sample Year > 2015): Main 0.51, PM 0.41, Benchmark 0.15.
  - Country examples (Cross-validation / Out of sample Main values):
    - BR: 0.53 / 0.58
    - CN: 0.72 / 0.51
    - DE: 0.41 / 0.37
    - JP: 0.66 / 0.67
    - US: 0.48 / 0.41
    - RU: 0.42 / 0.41

---

### Loss decomposition of forecasting Logit PD (Tables 6 and A.2)
- Table 6 (3-month publication lag): Decomposition of loss differentials between main model and benchmark model across conditional levels of cross-sectional average PD (groups: ̄PD_t ≤ qt25; qt25 ≤ ̄PD_t ≤ qt75; ̄PD_t ≥ qt75). Entries are squared bias differentials and squared idiosyncratic loss differentials; sample of country-level rows with values preserved:
  - Median: Left panel −0.25, −0.03; Middle −0.13, −0.04; Right −0.33, −0.04.
  - Examples:
    - BR: −0.22, −0.04; −0.13, −0.06; −0.28, −0.06.
    - CN: −0.17, −0.16; −0.18, −0.13; −0.31, −0.18.
    - US: −0.39, −0.05; −0.11, −0.03; −0.30, −0.04.
- Table A.2 (15-month publication lag): Same decomposition with preserved values:
  - Median: Left −0.26, −0.04; Middle −0.14, −0.03; Right −0.322, −0.04 (note: Right panel entry preserved verbatim as in source).
  - Examples:
    - BR: −0.16, −0.08; −0.12, −0.07; −0.32, −0.10.
    - CN: −0.22, −0.16; −0.12, −0.11; −0.55, −0.12.
    - US: −0.33, −0.05; −0.13, −0.05; −0.33, −0.03.

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### Model timing sensitivity (Figures 1–2 and A.1)
- Cross-validation and out-of-sample training/testing split schematics:
  - Figure 1(a) and 1(b) depict training/validation/testing partitioning across firm groups and time blocks with threshold T (schematic).
- MSE behavior at different lags since publication of annual reports:
  - Figure 2: For each country, MSE at different lags are divided by the zero-lag MSE; figure reports median, 0.2 and 0.8 quantiles across countries.
  - Figure A.1: Same exercise for data released 15 months after closing dates (median, 0.2, 0.8 quantiles preserved in figure).

---

### Discrimination performance and bankruptcy prediction (Figures 3 and A.2)
- AUC of predicting non-listed firm bankruptcy shown by region and selected countries:
  - Figure 3 panels: (a) Advanced Asia, (b) Europe, (c) CN, (d) IN, (e) US.
  - Figure A.2 repeats AUC exercise with 15 month publication lag across same panels.

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### Feature importance and interpretability: Shapley value distributions (Figures 4–5)
- Main features with high Shapley value importance across countries (preserved feature names):
  - Common influential predictors: Profit margin (earnings before tax based), Current ratio, Debt-to-suppliers to operating revenue ratio, Gross profit margin, ROE using net income, Liquidity ratio, ROA (EBIT based), Solvency ratio (asset based), Acid test, Gearing ratio, Earnings to debt ratio, Tangible fixed assets to debt ratio, Sectoral 20/80 percentiles and median of logit PD, Total assets in constant price, Number of employee, Inter-bank overnight rates, Interest coverage ratio, Stock turnover, Industry sector.
- Country-specific highlights (selected from figure labels):
  - BR top features: Profit margin (earnings before tax based); Current ratio; Debt-to-suppliers to operating revenue ratio.
  - CN top features: Sectoral median of logit PD; Gearing ratio; Sectoral 20 percentile of logit PD.
  - DE top features: Liquidity ratio; Acid test; Gearing ratio.
  - US top features: Industry sector; Tangible fixed assets to debt ratio; Sectoral 80 percentile of logit PD; Acid test; ROA (EBIT based).

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### Case studies: sectoral PD dynamics and Shapley-based attribution (Figures 6–7)
- Case study 1 — German gas and electricity firms (Figure 6):
  - PD series timeline: 2019-07 to 2023-07 with markers for A-, BBB, BBB- upper thresholds, Nord Stream explosion.
  - Shapley value changes (July 2023 minus July 2019) aggregated by category (Liquidity, Profitability, Solvency, Other, Macro, Sectoral_PD). Values preserved as listed:
    - July 2023 category contributions: Liquidity 0.00; Profitability 0.50; Solvency 1.00; Other 0.50; Macro 0.05; Sectoral_PD 1.87 (note: the table shows two adjacent values 1.64 and 1.87 — preserved ordering from source).
- Case study 2 — UK commercial real estate firms (Figure 7):
  - Logit PD series: 2006 to 2024 showing PD levels (axis ticks 4.5 to 8.5 preserved).
  - Shapley value changes (July 2023 minus July 2007) by category with preserved values:
    - Liquidity −0.12; Profitability −0.21; Solvency −0.09; Other 0.02; Macro 0.10; Sectoral_PD −0.31.

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*This overlay summarizes the References section and associated tables and figures from the source PDF.*

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_Source: https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024115-print-pdf.pdf_