# F2-smoke-01 — Slope estimate with firm-clustered uncertainty

## Question
Estimate the population slope β in the linear projection y = α + βx + ε for a synthetic firm panel. The standard error must allow for errors that are correlated within a firm and independent across firms.

## Data and exclusions
- Input: `data/panel.csv` (synthetic). 2000 rows read; 50 firms, each with exactly 40 observations.
- Checks: no missing values, no non-finite values, no exact duplicate rows. Nothing was excluded, so all 2000 observations were analysed. `data_issues` is empty.

## Method
- Pooled OLS of y on x with an intercept (statsmodels).
- Naïve baseline: conventional homoskedastic OLS standard error.
- Main inference: cluster-robust (Liang–Zeger / CR1) variance clustered by `firm_id`, using the standard small-sample correction G/(G−1)·(N−1)/(N−K).
- 95% CI: β̂ ± t(0.975, G−1)·SE_cluster with G−1 = 49 degrees of freedom (t = 2.0096). This is a common conservative choice when there are few clusters.
- The procedure is deterministic, so no random seed is needed. `out/analysis.py` regenerates `out/results.json`.

## Results
| quantity | value |
|---|---|
| β̂ (slope) | 0.8051 |
| α̂ (intercept) | 1.4440 |
| SE, OLS (naïve) | 0.0241 |
| SE, clustered by firm | 0.0731 |
| 95% CI (clustered) | [0.6581, 0.9521] |
| N / clusters | 2000 / 50 |

## Uncertainty
The firm-clustered SE is about 3.03 times the naïve OLS SE. This means x and the residuals are both strongly correlated within firms. The naïve SE therefore greatly overstates precision: its implied interval would be roughly a third as wide. Even with the clustered SE, the interval excludes 0 by a wide margin.

## Limitations and open issues
- With 50 equal-sized clusters, CR1 with t(G−1) critical values should be reasonably reliable. Still, cluster-robust inference is asymptotic in G. A wild cluster bootstrap was not run.
- β is a linear-projection (association) coefficient, not a causal effect. Pooled OLS mixes within-firm and between-firm variation in x. A firm fixed-effects slope could differ but was not the estimand asked for.
- The data are synthetic, so the conclusions describe only this dataset.