# F2-smoke-01 — OLS slope with firm-clustered uncertainty

## Question
Estimate the population linear-projection slope β in y = α + βx + ε for a synthetic panel of firms, with a standard error that allows observations from the same firm to share a common shock (independent across firms, dependent within firms). This is an association, not a causal effect.

## Data and exclusions
`data/panel.csv` (SHA-256 `968410b9…5ce5a8`, key `input_sha256`) has 2000 rows (`n_input_rows`). Columns are firm_id, x, y. Nothing is missing or non-finite, and no rows are duplicated. There are 50 firms (`n_clusters`), each with exactly 40 observations (`obs_per_cluster_min`, `obs_per_cluster_max`). The two exclusion rules (missing values; non-finite values) removed 0 rows (`out/exclusions.csv`), so `n_analysed` = 2000. `data_issues` is empty.

## Method
- Pooled OLS of y on x with an intercept.
- Baseline: the conventional homoskedastic, non-clustered SE (`se_ols`).
- Main inference: the cluster-robust sandwich SE by firm_id with the CR1 small-sample correction G/(G−1)·(n−1)/(n−k) (`se_cluster`). The 95% CI is β̂ ± t₀.₉₇₅,G−1·se_cluster, with G−1 = 49 degrees of freedom (`ci_df`, `t_crit_cluster` = 2.0096).
- The method uses no randomness (`random_seed` = null).
- `out/analysis.py` regenerates every number. I checked the clustered SE against an independent numpy sandwich computation, and the two agree to about 1e-17 (`out/state/checks.md`).

## Results
- Slope: `beta_hat` = 0.8051.
- Intercept: `alpha_hat` = 1.4440.
- 95% cluster-robust CI for β: [`beta_ci_low`, `beta_ci_high`] = [0.6581, 0.9521]. It excludes zero.

## Uncertainty
- Clustered SE: `se_cluster` = 0.0731.
- Naïve OLS SE: `se_ols` = 0.0241.
- The clustered SE is about 3.0 times the naïve one (`se_ratio_cluster_over_ols` = 3.03).
- The naïve 95% CI [`baseline_ci_low_ols`, `baseline_ci_high_ols`] = [0.7578, 0.8524] is therefore far too narrow.

This inflation is expected for two reasons found in the audit (`out/state/data_audit.md`): about 45% of the variance of x and 53% of the variance of the OLS residual lie between firms, and each firm has 40 observations.

## Limitations and open issues
- Cluster-robust inference here relies on asymptotics in the number of firms. With G = 50 balanced clusters the CR1 + t(G−1) approach is generally adequate, but it is not exact. I did not run a wild-cluster bootstrap, because the task does not require it.
- β is a linear-projection coefficient. It is not a causal effect, and it is not a within-firm (fixed-effects) slope. Because x has a large firm-level component, a within-firm slope could differ.
- The data are synthetic. Feasibility is "complete".