# 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
| quantity | value |
|---|---|
| beta_leverage | 0.2638 |
| clustered SE | 0.0408 |
| 95% CI | [0.184, 0.344] |
| p-value | 9.7e-11 |
| cash_ratio coef (SE) | −0.186 (0.059) |
| log_sales coef (SE) | −0.0073 (0.0076) |
| R² | 0.0349 |
| baseline default rate | 0.1211 |

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.