1# F2-conf-03 — Monte Carlo coverage of OLS confidence intervals (n = 30, heteroskedastic)23## Question4In the fixed-design model y_i = 1 + 0.5 x_i + σ_i ε_i (i = 1..30, ε_i iid N(0,1), x_i and σ_i fixed from `data/design.csv`), do nominal 95% t(28) intervals for the slope based on (a) the conventional OLS SE and (b) the HC1 robust SE reach 95% coverage? We also report (c) the exact SD of β̂ and (d) E[se_conv].56## Data and exclusions7`data/design.csv` (SHA-256 b1a17a52…3e74f0): 30 rows, columns i, x, sigma; no missing values, no duplicates, all σ_i > 0 (`n_input_rows`, `data_issues` = []). No rows excluded (`out/exclusions.csv`: 30 → 30). Design feature: σ_i² is strongly correlated with x_i² (r = 0.987, computed in the audit), so error variance is largest at high-leverage points.89## Method10Analytic: β̂ − β = Σ w_i σ_i ε_i with w_i = (x_i − x̄)/S_xx, so SD(β̂) = sqrt(Σ w_i² σ_i²). Monte Carlo: R = 2000 replications (`n_replications`), n = 30 (`n_obs_per_replication`), numpy `default_rng` seed 20261030 (`random_seed`); only ε redrawn. In each replication OLS is fitted; se_conv = sqrt(s²/S_xx), s² = RSS/(n−2); se_HC1 = sqrt(n/(n−2)) · sqrt(Σ (x_i−x̄)² e_i²)/S_xx. Intervals β̂ ± t_{0.975}(28)·se with t = 2.0484 (`t_critical`). Both SE formulas were cross-checked against statsmodels for one replication. Undercoverage rule: coverage_conventional < 0.95 − 3·sqrt(0.95·0.05/2000) = 0.9354 (`undercoverage_threshold`). The conventional interval's coverage is the naïve baseline (`baseline_coverage_conventional`).1112## Results13- Conventional interval coverage: **0.844** (`coverage_conventional`), far below 95%.14- HC1 interval coverage: **0.902** (`coverage_hc1`) — better, but still below 95%.15- Exact SD of β̂: **0.2680** (`sd_beta_exact`); Monte Carlo SD: 0.2696 (`sd_beta_mc`), agreeing within MC error.16- Mean conventional SE: **0.1903** (`mean_se_conventional`), about 29% below the true SD of 0.2680; mean HC1 SE is 0.2408 (`mean_se_hc1`), also below the true SD. Analytic check: sqrt(E[s²]/S_xx) = 0.1932 (`sqrt_E_s2_over_Sxx_analytic`), consistent with the MC mean SE (Jensen).17- Conclusion: the conventional interval does **not** attain its nominal coverage (`conventional_ci_undercovers` = true).1819## Uncertainty20Binomial Monte Carlo standard errors: conventional coverage 0.844 ± 0.0081 (`mcse_coverage_conventional`); HC1 coverage 0.902 ± 0.0067 (`mcse_coverage_hc1`); MC SE at the nominal 0.95 is 0.0049 (`mcse_at_nominal`). The conventional shortfall (0.106) is about 22 nominal-level MC SEs; the HC1 shortfall (0.048) is about 10. The sd_beta_exact value is analytic (no MC error); sd_beta_mc differs from it by 0.6%, below its approximate MC relative SE of 1.6%.2122## Limitations and open issues23- Results are conditional on this one synthetic fixed design and normal errors; they do not generalise to other designs.24- Coverage estimates depend on the seed at the level of the reported MC SEs; R = 2000 is fixed by the task.25- HC1 also undercovers in this n = 30 high-leverage design; small-sample refinements (HC2/HC3, wild bootstrap) were not evaluated because the task does not require them.26- No data problems were found; feasibility is "complete".27