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

# Question

Estimate the population slope β in the linear projection y = α + βx + ε using a synthetic firm panel (data/panel.csv), with standard errors that allow arbitrary correlation within a firm (a shared firm-level shock) while treating firms as independent of one another.

# Data and exclusions

• Input: data/panel.csv, SHA-256 968410b9…5ce5a8, 2000 rows read (n_input_rows), columns firm_id, x, y.
• No missing values, no duplicate rows. The panel is balanced: 50 firms with 40 observations each (n_clusters = 50).
• No exclusions (out/exclusions.csv): n_analysed = 2000. No data issues were found (data_issues = []).
• Information rule respected: only data/panel.csv is read.

# Method

• OLS of y on x with an intercept, giving beta_hat and alpha_hat.
• Naïve baseline: the conventional homoskedastic OLS standard error (se_ols), which assumes all observations are independent.
• Main inference: a cluster-robust (CR1) sandwich variance clustered by firm_id. It uses the usual small-sample factor G/(G−1)·(N−1)/(N−K), the same one Stata and statsmodels use. This gives se_cluster. The 95% CI is beta_hat ± t₀.₉₇₅(G−1)·se_cluster, with critical value ci_t_critical_df_Gminus1 (G−1 = 49 df).
• The computation is deterministic and uses no randomness (random_seed = null). The script is out/analysis.py.

# Results

• beta_hat = 0.8051; alpha_hat = 1.4440.
• se_ols = 0.0241 (naïve); se_cluster = 0.0731.
• 95% clustered CI for β: [beta_ci_low, beta_ci_high] = [0.658, 0.952].
• feasibility = "complete".

# Uncertainty

• The clustered SE is about 3.0 times the naïve OLS SE (se_ratio_cluster_over_ols = 3.03). The naïve SE therefore seriously understates the sampling uncertainty when observations share a firm-level shock. For comparison, the naïve 95% CI would be [baseline_ci_ols_low, baseline_ci_ols_high] = [0.758, 0.852], which is much too narrow.
• Even with the clustered interval, β is clearly positive: the CI excludes 0.
• Cross-check: statsmodels' cluster-robust SE agrees with se_cluster to machine precision. Statsmodels' default interval uses a normal critical value ([0.662, 0.948]), so it is slightly narrower than the reported t(49) interval.

# Limitations and open issues

• β is a linear-projection (associational) slope, not a causal effect. The data are synthetic.
• Cluster-robust inference is asymptotic in the number of clusters. With 50 balanced clusters, CR1 with t(G−1) is generally adequate, but it is not exact. No wild-cluster bootstrap was run because the task does not require it.
• The CI's coverage depends on firms being independent of each other. The data cannot be used to test that assumption.