# F2-conf-03 — Coverage of OLS slope CIs in a small heteroskedastic fixed design

## Question
In y_i = 1 + 0.5 x_i + σ_i ε_i (n = 30; x_i, σ_i fixed from `data/design.csv`; ε_i iid N(0,1)), does the nominal 95% interval β̂ ± t₀.₉₇₅(28)·se reach 95% coverage, using (a) the conventional OLS SE and (b) the HC1 robust SE? Also: (c) the exact SD of β̂ and (d) E[se_conv].

## Data and exclusions
One input, `data/design.csv` (30 rows, columns i, x, sigma; `n_input_rows`). The audit (`out/state/data_audit.md`) found no missing values, duplicates or non-positive σ; `data_issues` is empty and no rows were excluded (`out/exclusions.csv`: 30 → 30). Note on the design: σ_i increases with |x_i − x̄| (correlation 0.84, computed during the audit), so the error variance is largest at the high-leverage points.

## Method
- Exact SD (no simulation): 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) (`sd_beta_exact`).
- Monte Carlo: R = 2000 replications (`n_replications`), n = 30 (`n_obs_per_replication`), numpy `default_rng` with seed 20261030 (`random_seed`). Only ε is redrawn. For each replication: OLS fit, se_conv = sqrt(s²·[(X'X)⁻¹]₂₂) with s² = RSS/(n−2), se_HC1 = sqrt(n/(n−2)·Σ a_i² e_i²) (HC0 sandwich × n/(n−2)), critical value t₀.₉₇₅(28) = 2.0484 (`t_critical_28`). Coverage = share of replications whose interval contains 0.5.
- The baseline is the conventional interval coverage (`coverage_conventional`), which the task names as the naïve baseline.
- Decision rule: the conventional CI undercovers iff coverage_conventional < 0.95 − 3·MCSE, with MCSE = sqrt(0.95·0.05/2000) = 0.00487 (`mcse_coverage_at_nominal`), giving a threshold of 0.9354 (`undercoverage_threshold`). I evaluated the binomial SE at the nominal 0.95 because the rule tests against that value.
- Validation: results matched statsmodels (OLS nonrobust and HC1) for the first 5 replications to machine precision. A clean re-run reproduced the results exactly (`out/state/checks.md`).

## Results
| quantity | value | key |
|---|---|---|
| Coverage, conventional CI | 0.844 | `coverage_conventional` |
| Coverage, HC1 CI | 0.902 | `coverage_hc1` |
| Exact SD of β̂ | 0.2680 | `sd_beta_exact` |
| MC SD of β̂ | 0.2696 | `sd_beta_mc` |
| Mean se_conv | 0.1903 | `mean_se_conventional` |
| Mean se_HC1 | 0.2408 | `mean_se_hc1` |
| Mean β̂ | 0.4962 | `mean_beta_hat` |
| Conventional CI undercovers | true | `conventional_ci_undercovers` |

The conventional interval does **not** reach its nominal 95% coverage. Its coverage of 0.844 (`coverage_conventional`) is far below the threshold of 0.9354 (`undercoverage_threshold`). The cause is that se_conv is too small on average: 0.190 (`mean_se_conventional`), compared with the true SD of 0.268 (`sd_beta_exact`), about 29% too small. The homoskedastic formula ignores that the noisiest observations are also the high-leverage ones. The HC1 interval does better at 0.902 (`coverage_hc1`) but still falls short of 95%, because its mean SE of 0.241 (`mean_se_hc1`) is still below 0.268 at n = 30. The MC SD of β̂ agrees with the exact value (0.2696 vs 0.2680), and the mean of β̂ is consistent with 0.5. This is a check that the simulation is correct.

## Uncertainty
- Monte Carlo SE of coverage_conventional, from the estimated proportion: 0.0081 (`mcse_coverage_conventional`). Approximate 95% MC interval: 0.828–0.860.
- Monte Carlo SE of coverage_hc1: 0.0066 (`mcse_coverage_hc1`). Approximate 95% MC interval: 0.889–0.915.
- MCSE at the nominal level used in the decision rule: 0.00487 (`mcse_coverage_at_nominal`). The decision does not change if the estimate-based MCSE is used instead: 0.95 − 3·0.0081 = 0.926 > 0.844.
- MC SE of mean se_conv: 0.0007 (`mcse_mean_se_conventional`). MC SE of mean β̂: 0.0060 (`mcse_mean_beta_hat`).
- `sd_beta_exact` is analytic and has no MC error.

## Limitations and open issues
- The results apply only to this one fixed design, with normal errors and R = 2000. Coverage estimates carry the MC error stated above. Other seeds would move them by about ±0.016 (2 MCSE).
- The task does not fix how the "binomial standard error" in the undercoverage rule should be computed. I used the value at the nominal p = 0.95, and the conclusion is the same under the alternative.
- The HC1 shortfall below 95% is a descriptive finding. I did not test it formally against a threshold, although 0.902 is more than 7 MCSE below 0.95.