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

## Question
Estimate the population slope β of the linear projection y = α + βx + ε in a synthetic firm panel, with standard errors that allow arbitrary correlation among observations of the same firm (independence across firms).

## Data and exclusions
`data/panel.csv` (SHA-256 968410b9…ce5a8) has 2000 rows (`n_input_rows`) and 3 columns; there are no missing values and no duplicate rows. The panel is balanced, with 50 firms (`n_clusters`) of 40 rows each (`obs_per_cluster_min`, `obs_per_cluster_max`). I applied no exclusions (see `out/exclusions.csv`, where every step removed 0 rows), so 2000 observations were analysed (`n_analysed`). No data problems were found (`data_issues` is empty).

## Method
I ran OLS of y on x with an intercept. The conventional homoskedastic standard error is the naïve baseline (`se_ols`). The main standard error is cluster-robust by `firm_id` (`se_cluster`). It uses the sandwich (X'X)⁻¹ [Σ_g X_g'e_g e_g'X_g] (X'X)⁻¹ with the CR1 small-sample factor G/(G−1)·(N−1)/(N−K) (`cluster_se_correction`). The 95% CI is beta_hat ± t₀.₉₇₅,G−1 · se_cluster, with G−1 = 49 df (`ci_critical_value_t_df`, `ci_critical_value`). The method involves no randomness (`random_seed` = null). Estimates match statsmodels exactly (see `out/state/checks.md`).

## Results
- Slope: beta_hat = 0.8051 (`beta_hat`). Intercept: alpha_hat = 1.4440 (`alpha_hat`).
- Firm-clustered SE = 0.0731 (`se_cluster`). Conventional OLS SE = 0.0241 (`se_ols`).
- 95% CI for β using the clustered SE: [0.6581, 0.9521] (`beta_ci_low`, `beta_ci_high`). The CI excludes 0.

## Uncertainty
The clustered SE is about 3.03 times the naïve SE (`se_ratio_cluster_to_ols`). This fits the strong within-firm dependence found in the audit (`out/state/data_audit.md`, not a results key): firm means account for about 45% of the variance of x and about 53% of the variance of the OLS residuals. Treating the 2000 observations as independent would therefore greatly overstate precision. With 50 balanced clusters, cluster-robust inference based on t(49) is standard and should be reasonably reliable.

## Limitations and open issues
- β is a linear-projection (association) coefficient, not a causal effect. Because x is correlated within firms, β pools within-firm and between-firm variation, and I did not estimate firm fixed effects (the task did not ask for them).
- Cluster-robust inference is asymptotic in the number of clusters. 50 clusters is adequate, but the CI is approximate. I did not run a wild cluster bootstrap.
- The data are synthetic, as stated in TASK.md.