# F2-conf-03 — Coverage of OLS slope confidence intervals under heteroskedasticity

# Question

In the fixed-design model y_i = 1 + 0.5 x_i + σ_i ε_i (n = 30, with x_i and σ_i from data/design.csv and ε_i iid N(0,1)), do nominal 95% intervals β̂ ± t_0.975(28)·se cover β = 0.5 with probability 0.95? The question is asked for (i) the conventional OLS standard error and (ii) the HC1 robust standard error.

# Data and exclusions

The only input is data/design.csv: 30 rows, columns i, x, sigma. Its SHA-256 matches the bundle manifest. It has no missing values and no duplicate rows, and every σ is positive (n_input_rows). No rows were excluded (out/exclusions.csv: 30 → 30), and no data issues were found (data_issues = []). σ_i is largest at the extreme x values, so the heteroskedasticity sits on the high-leverage points. Full audit: out/state/data_audit.md.

# Method

• Exact SD of β̂ (computed analytically): sqrt(Σ w_i² σ_i²), with w_i = (x_i − x̄)/S_xx. I confirmed it against the sandwich matrix formula.
• 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; x and σ are held fixed.
• Standard errors in each replication:
  • se_conv = sqrt(s²/S_xx), where s² uses divisor n − 2.
  • se_HC1 = sqrt(n/(n−2) · Σ(x_i−x̄)² e_i² / S_xx²).
• Critical value: t_0.975(28) = 2.0484 (t_crit_28).
• Coverage: the share of replications with |β̂ − 0.5| ≤ t·se.
• Undercoverage decision: the conventional interval is declared to undercover if 0.95 − coverage > 3 × sqrt(0.95·0.05/2000). This uses the binomial MC standard error at the nominal level (mcse_at_nominal = 0.00487), which gives a threshold of 0.9354 (undercoverage_threshold).
• Baseline: the conventional interval is the naïve baseline.
• Cross-check: statsmodels OLS/HC1 on the same draws reproduced every number exactly.

# Results

The conventional interval does not reach its nominal 95% coverage.

• Its coverage of 0.844 is about 10.6 points below 0.95, far beyond the three-MC-SE threshold of 0.9354.
• On average, se_conv (0.190) understates the true sampling SD (0.268) by about 29%. Its mean is close to the homoskedastic-formula SD (0.196, sd_beta_homoskedastic_formula), which ignores the fact that error variance is concentrated at high-leverage x.

HC1 helps but also undercovers in this small sample.

• HC1 coverage is 0.902 and its mean SE is 0.241. Both are better than the conventional interval but still below nominal.
• No formal three-SE test was specified for HC1. Even so, 0.902 is about 7 binomial MC SEs at the estimate (≈ 0.0066) below 0.95.

The MC SD of β̂ agrees with the exact SD (z ≈ 0.39), and mean β̂ is consistent with unbiasedness (out/state/checks.md).

# Uncertainty

• Conventional coverage: MC standard error at the estimate is sqrt(p(1−p)/R) = 0.0081 (mcse_coverage_conventional), giving an approximate 95% MC interval of 0.828–0.860.
• HC1 coverage: MC standard error is 0.0066 (mcse_coverage_hc1), giving an approximate interval of 0.889–0.915.
• At the nominal level: the MC SE is 0.0049 (mcse_at_nominal).
• SD estimate: sd_beta_mc has a sampling SE of about sd/sqrt(2(R−1)) ≈ 0.0042.
• Other sources: the results depend on the single seed 20261030. Under the stated model the analytic SD has no uncertainty.

# Limitations and open issues

• The results apply only to this specific synthetic design (n = 30, these x and σ) with normal errors. They should not be generalised to other designs or error distributions.
• I used one seed and R = 2000, as the task specifies. Coverage figures are therefore accurate only to about ±0.016 (2 MC SEs).
• The undercoverage rule uses the binomial SE at the nominal 0.95. Using the SE at the estimate (0.0081) gives the same conclusion.
• Small-sample HC variants (HC2, HC3, wild bootstrap) were outside scope and were not evaluated.