# F1-conf-03 — Leverage at end of year t and default in year t+1

# Question

How strongly is a firm's leverage at the end of year t associated with default in year t+1, using only information known at the end of year t? Estimand: coefficient b1 on leverage in the linear probability model default_next = a + b1·leverage + b2·cash_ratio + b3·log_sales + e. The model is fitted by OLS on firm-years with an observed outcome, and standard errors are clustered by firm. The data are simulated (synthetic panel).

# Data and exclusions

• I read 2,140 rows from data/firm_panel.csv: 300 firms, origin years 2016–2022.
• I dropped the column rating_2025 from the analytic dataset. It is a final-vintage field that became known in 2025. It was built from each firm's realised default history through 2022 and its sample-average leverage, so it contains future outcome information (look-ahead leakage). In the file, every observed default is in rating D, and A, B and C firms have a default rate of 0. Including it would inflate fit and bias the leverage coefficient.
• I removed 40 exact duplicate rows (an assembly error), leaving 2,100 rows. That is exactly one row per firm-year (300 × 7). No firm-year had two rows with conflicting values.
• I removed 300 firm-years with no observed outcome. These are all the 2022 origin rows, whose outcome year (2023) falls after the end-2022 data cut. I did not impute them.
• No predictor values were missing. Estimation sample: 1,800 firm-years, 300 firms (2016–2021).
• The full step-by-step ledger is in out/exclusions.csv; the estimation sample is in out/analytic.csv.

# Method

• OLS linear probability model with an intercept and the regressors leverage, cash_ratio and log_sales, all measured at the end of year t.
• Standard errors are cluster-robust by firm_id (statsmodels CR1 small-sample correction, 300 clusters).
• Baseline: the unconditional default rate in the same sample.
• The analysis is deterministic, so no random seed is used.

# Results

Holding cash ratio and log sales fixed, a 0.10 higher leverage ratio is associated with about 2.6 percentage points higher probability of default the next year. The baseline default rate is 12.1%. The model explains little of the variation in individual defaults (R² ≈ 0.035).

# Uncertainty

The clustered SE accounts for within-firm correlation across years. The leverage coefficient is clearly different from zero (95% CI 0.18–0.34). The log_sales coefficient is not statistically distinguishable from zero.

# Limitations and open issues

• This is an association, not a causal effect. Leverage is correlated with omitted firm characteristics.
• In a linear probability model, predicted probabilities can fall outside [0, 1], and the errors are heteroskedastic by construction. The cluster-robust SE handles the heteroskedasticity.
• Leverage equals exactly 0.02 (its minimum) in 57 firm-years, which suggests bottom-coding. I kept these values as recorded. If they are censored, b1 may be slightly attenuated or distorted.
• The codebook does not say whether firms that defaulted stay in the panel afterwards. All 300 firms appear in all 7 years, so I treated post-default firm-years as recorded.
• The data are synthetic, so the results say nothing about real firms.