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

## Question
Model: y_i = 1 + 0.5 x_i + σ_i ε_i, with i = 1…30, ε_i iid N(0,1), and x_i, σ_i held fixed at the values in `data/design.csv`. Do nominal 95% intervals β̂ ± t_{0.975}(28)·se reach 95% coverage when se is (a) the conventional OLS standard error, or (b) the HC1 robust standard error? The task also asks for (c) the exact SD of β̂ and (d) E[se_conv].

## Data and exclusions
`data/design.csv` has 30 rows (`n_input_rows`) and SHA-256 b1a17a52…74f0. It has no missing values, no duplicates, and every σ is positive. No rows were excluded (`out/exclusions.csv`: 30 − 0 = 30). No data issues were found, so `data_issues` = []. In this design σ_i is smallest near x = 0 and largest at the extreme x values (largest of all at x = 1.99). That means the error variance is highest at the high-leverage points. See `out/state/data_audit.md`.

## Method
- **Exact SD.** 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`). No simulation is involved.
- **Monte Carlo.** R = 2000 replications (`n_replications`), using numpy `default_rng` with seed 20261030 (`random_seed`). Only ε is redrawn in each replication; x and σ stay fixed. Each replication fits OLS and computes:
  - se_conv = sqrt(s²/S_xx), where s² = RSS/(n − 2);
  - se_HC1 = sqrt(n/(n−2) · Σ(x_i − x̄)² e_i² / S_xx²).
- **Coverage** is the share of replications with |β̂ − 0.5| ≤ t_{0.975}(28)·se, where t = 2.0484 (`t_critical`).
- **Checks.** For replication 1, both SEs match statsmodels to within 1e-10 (`statsmodels_crosscheck_passed`).
- **Decision rule.** `conventional_ci_undercovers` is true if coverage_conventional < 0.95 − 3·sqrt(0.95·0.05/2000). The binomial standard error here is evaluated at the nominal level, which is the null hypothesis being tested. I also checked the conclusion using the SE evaluated at the estimated coverage.

## Results
- **Conventional interval (baseline).** Coverage is **0.844** (`coverage_conventional`), well short of the nominal 0.95.
- **HC1 interval.** Coverage is **0.902** (`coverage_hc1`). This is better than the conventional interval but still below 0.95 (`hc1_ci_undercovers` = true).
- **SD of β̂.** The exact value is 0.2680 (`sd_beta_exact`). The Monte Carlo SD is 0.2696 (`sd_beta_mc`), which agrees with it. The mean of β̂ across replications is 0.4962 (`mean_beta_hat`), consistent with β̂ being unbiased.
- **Conventional SE.** Its average is 0.1903 (`mean_se_conventional`), only 0.71 of the true SD (`ratio_mean_se_conv_to_sd_exact`). The exact value of sqrt(E[s²]/S_xx) is 0.1932 (`sqrt_expected_conv_variance_exact`). So the conventional SE is biased downward by design, not by sampling noise: it ignores the fact that the error variance is largest at high-leverage x values. The mean HC1 SE is 0.2408 (`mean_se_hc1`), closer to the true value but still too small.
- **Conclusion.** The conventional interval does **not** attain its nominal coverage (`conventional_ci_undercovers` = true). The threshold is 0.9354 (`undercover_threshold`).

## Uncertainty
- **Coverage estimates (binomial Monte Carlo standard errors):**
  - conventional: 0.844 ± 0.0081 (`mcse_coverage_conventional`);
  - HC1: 0.902 ± 0.0066 (`mcse_coverage_hc1`);
  - at the nominal level: 0.0049 (`mcse_at_nominal`).
- **Size of the shortfall.** The conventional coverage falls about 22 nominal-level MCSEs below 0.95. The conclusion is the same if the MCSE at the estimated coverage is used instead (`conventional_ci_undercovers_using_estimated_mcse` = true).
- **SD estimate.** The Monte Carlo standard error of `sd_beta_mc` is about 0.0043 (`mcse_sd_beta_mc_approx`). The gap from the exact value (0.0017) is within that noise.
- **What the numbers depend on.** All values are for this seed and this R. A different seed would shift the coverages by roughly ±1–2 MCSE.

## Limitations and open issues
- The results apply only to this fixed design, this σ pattern, n = 30 and Gaussian errors. They should not be generalised to other designs.
- E[se_conv] is estimated by simulation, as the task specifies. Only E[s²] is available in closed form, and it is reported as a diagnostic.
- The undercoverage flag depends on how "binomial standard error" is defined, so I checked two definitions: SE at p = 0.95 and SE at the estimated p. Both give the same conclusion.
- HC1 also undercovers in this small, high-leverage sample. Alternatives such as HC2/HC3 or wild bootstrap were not evaluated because the task does not ask for them.
- Feasibility: complete.