1# F2-conf-03 — Coverage of OLS slope CIs in a small heteroskedastic sample23## Question4For the fixed design in `data/design.csv` (n = 30, y = 1 + 0.5x + σ_i ε), does the nominal 95% t-interval for the slope reach 95% coverage? We compare the conventional SE with HC1, compute the exact SD of β̂, and compute the mean conventional SE.56## Data and exclusions7I used one input, `design.csv`: 30 rows (`n_input_rows`), with no missing values and no duplicates. All σ are positive. No rows were excluded (`out/exclusions.csv`), and `data_issues` is empty. σ_i grows with |x_i|, so the high-leverage observations are also the noisiest.89## Method10Each replication draws ε ~ N(0, I₃₀) and computes OLS β̂, se_conv (s² with divisor 28) and se_HC1 (HC0 × 30/28). An interval counts as covering when |β̂ − 0.5| ≤ t₀.₉₇₅(28)·se, with t = 2.0484 (`t_critical_28`). There are R = 2000 replications (`n_replications`) and the numpy default_rng seed is 20261030 (`random_seed`). The exact SD is sqrt(Σ w_i² σ_i²), where w is the slope row of (X'X)⁻¹X'. The undercoverage rule is coverage < 0.95 − 3·sqrt(0.95·0.05/2000) = 0.9354 (`undercover_threshold`). The binomial SE is evaluated at the nominal level.1112## Results13- Conventional-interval coverage: **0.844** (`coverage_conventional`). HC1-interval coverage: **0.902** (`coverage_hc1`).14- Exact SD(β̂) = 0.2680 (`sd_beta_exact`). The MC SD is 0.2696 (`sd_beta_mc`) and the MC mean of β̂ is 0.4962 (`mean_beta_mc`).15- Mean se_conv = 0.1903 (`mean_se_conventional`), about 29% below the true SD. Mean se_HC1 = 0.2408 (`mean_se_hc1`), closer to the true SD but still too small.16- **The conventional interval does not attain nominal coverage** (`conventional_ci_undercovers` = true). HC1 improves coverage but also stays below 95%. Its gap of 0.048 is about 7 MCSE (`coverage_hc1`, `mcse_coverage_hc1`); the report applies no formal test to it.1718## Uncertainty19These are binomial Monte Carlo standard errors at R = 2000:20- Conventional: 0.0081 using the estimated p (`mcse_coverage_conventional`).21- HC1: 0.0066 (`mcse_coverage_hc1`).22- At p = 0.95: 0.0049 (`mcse_at_nominal_0.95`).2324The conventional shortfall (0.106) is about 13 MCSE under either SE definition, so the undercoverage conclusion does not depend on which one is used (`undercovers_using_estimated_p_se` = true). The sampling uncertainty in sd_beta_mc relative to the exact value is about 1.6%; the observed difference is 0.6%.2526## Limitations and open issues27- All results are conditional on this one synthetic design and normal errors. Other designs or error distributions will give different coverage.28- The task does not say whether the binomial SE in the undercoverage rule should use p = 0.95 or the estimated p. I used p = 0.95 and checked that the alternative gives the same conclusion.29- Monte Carlo estimates depend on the seed. Different seeds will move the coverages by roughly ±2 MCSE.30- Small-sample corrections stronger than HC1, such as HC2/HC3 or a Bell–McCaffrey degrees-of-freedom correction, were not evaluated because the task does not require them.31