1[2 {"id":"C1","text":"The OLS slope of y on x is about 0.805 with intercept about 1.444, using all 2000 observations from 50 firms.","results_keys":["beta_hat","alpha_hat","n_analysed","n_clusters"],"status":"supported","caveat":"Linear-projection slope on synthetic data; not causal."},3 {"id":"C2","text":"The firm-clustered SE (0.0731) is about 3 times the naive OLS SE (0.0241).","results_keys":["se_cluster","se_ols","se_ratio_cluster_over_ols"],"status":"supported","caveat":"CR1 small-sample correction."},4 {"id":"C3","text":"The 95% clustered CI for beta is [0.658, 0.952] and excludes zero.","results_keys":["beta_ci_low","beta_ci_high","ci_t_critical_df_Gminus1"],"status":"supported","caveat":"t(G-1) critical value; asymptotic in number of clusters (G=50)."},5 {"id":"C4","text":"The naive OLS CI [0.758, 0.852] understates uncertainty relative to the clustered CI.","results_keys":["baseline_ci_ols_low","baseline_ci_ols_high","beta_ci_low","beta_ci_high"],"status":"supported","caveat":"Relies on firms being independent of each other."}6]7