# F2-smoke-01 — Slope of y on x with firm-clustered uncertainty

## Question
What is the population linear-projection slope β in y = α + βx + ε for a synthetic panel of firms? The standard error has to allow for a shared firm-level shock, meaning observations are independent across firms but not within a firm. β is a descriptive projection coefficient, not a causal effect.

## Data and exclusions
- Input: `data/panel.csv` (simulated). Rows read: 2000 (`n_input_rows`). SHA-256 `968410b9…5ce5a8` (`input_sha256`).
- No values are missing, no values are non-finite and no rows are exact duplicates. There are 50 firms with exactly 40 observations each (`n_clusters`, `obs_per_cluster_min`, `obs_per_cluster_max`).
- No rows were excluded (`out/exclusions.csv`). Observations analysed: 2000 (`n_analysed`). No data issues were found (`data_issues` = []).

## Method
- OLS of y on a constant and x (statsmodels).
- Baseline: the conventional homoskedastic OLS standard error (`se_ols`) and its 95% t-interval (`baseline_ci_ols_low`, `baseline_ci_ols_high`).
- Main uncertainty measure: a cluster-robust (CR1) sandwich standard error clustered by `firm_id`, using the small-sample factor G/(G−1)·(N−1)/(N−K) (`se_cluster`). The script cross-checks it against a manual sandwich computation.
- 95% CI = beta_hat ± t₀.₉₇₅(G−1)·se_cluster, with G−1 = 49 degrees of freedom (`cluster_df`) and critical value `t_crit_cluster_df` = 2.0096.
- The analysis is deterministic and uses no randomness (`random_seed` = null).

## Results
- Slope `beta_hat` = 0.8051; intercept `alpha_hat` = 1.4440.
- 95% cluster-robust CI for β: [0.6581, 0.9521] (`beta_ci_low`, `beta_ci_high`).
- Status: `feasibility` = "complete".

## Uncertainty
- Naive OLS SE `se_ols` = 0.0241. Firm-clustered SE `se_cluster` = 0.0731.
- The ratio `se_ratio_cluster_to_ols` is 3.03, so the naive SE understates sampling uncertainty by a factor of about 3. As a result, the naive interval [0.7578, 0.8524] (`baseline_ci_ols_low`, `baseline_ci_ols_high`) is far too narrow. This pattern fits a model in which both x and the error share firm-level components.
- Both intervals exclude zero.

## Limitations and open issues
- Cluster-robust inference is asymptotic in the number of clusters. With G = 50 balanced clusters the CR1 + t(G−1) approach is usually adequate, but its coverage is not guaranteed. A wild cluster bootstrap was not run because the task does not require it.
- β is a linear projection coefficient. It carries no causal interpretation, and firm fixed effects were deliberately not included, because they would change the estimand to a within-firm slope.
- The data are synthetic. The conclusions describe this sample only.