# 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.