# F2-conf-03 — Monte Carlo coverage of OLS confidence intervals in a small heteroskedastic sample

# Scientific question

How well do nominal 95% confidence intervals for an OLS slope perform in a small sample with heteroskedastic errors? Consider the fixed-design linear model y_i = α + β x_i + σ_i ε_i, i = 1, …, 30, with α = 1, β = 0.5, ε_i independent standard normal, and x_i and σ_i fixed at the values given in data/design.csv (the same 30 values in every replication; only ε is redrawn). The estimands are (a) the true coverage probability of the interval β̂ ± t_{0.975}(28) · se_conv, where se_conv is the conventional OLS standard error (residual variance with divisor n − 2); (b) the true coverage probability of the interval β̂ ± t_{0.975}(28) · se_HC1, where se_HC1 is the heteroskedasticity-robust HC1 standard error (White's HC0 sandwich multiplied by n/(n − 2)); (c) the exact finite-sample standard deviation of β̂ under this design; (d) the expected value of se_conv. Estimate (a), (b) and (d) by Monte Carlo with exactly R = 2000 replications and a fixed seed, compute (c) analytically, and conclude whether the conventional interval attains its nominal coverage.

# Inputs

data/design.csv — the fixed design (a synthetic design generated for this exercise; not observed data). Columns: i (observation index 1–30), x (regressor value, dimensionless), sigma (standard deviation of the error for observation i, dimensionless). 30 rows.

# Information-availability rules

Use only data/design.csv; the model constants (α = 1, β = 0.5, n = 30, R = 2000, nominal level 95%, t(28) critical value) are given above. Do not redraw x or σ.

# Required outputs

out/results.json with the keys:

key	type	unit	meaning
`task_id`	string	–	"F2-conf-03"
`coverage_conventional`	number	proportion	share of the 2000 replications in which the conventional interval contains β = 0.5 — the naïve baseline
`coverage_hc1`	number	proportion	share of the 2000 replications in which the HC1 interval contains β = 0.5
`sd_beta_exact`	number	–	exact standard deviation of β̂ under the design (analytical, no simulation)
`sd_beta_mc`	number	–	sample standard deviation (divisor R − 1) of the 2000 simulated β̂
`mean_se_conventional`	number	–	average of se_conv over the 2000 replications
`conventional_ci_undercovers`	boolean	–	true if and only if `coverage_conventional` is below 0.95 by more than three Monte Carlo standard errors (binomial standard error at R = 2000)
`n_replications`	integer	–	number of replications (2000)
`n_obs_per_replication`	integer	–	sample size per replication (30)
`feasibility`	string	–	"complete", "partial" or "infeasible"
`feasibility_reason`	string or null	–	explanation if not complete
`data_issues`	array of {"issue","handling"}	–	data problems detected and how handled (empty list if none)
`n_input_rows`	object	–	{"design.csv": rows read}
`random_seed`	integer	–	the seed used for the simulation (required; the simulation must be reproducible)

out/report.md with the sections: Question; Data and exclusions; Method; Results; Uncertainty (include the Monte Carlo standard error of each coverage estimate); Limitations and open issues.
out/analysis.py — running ~/wp29/venv/bin/python out/analysis.py from the workspace root must regenerate out/results.json from data/ only, deterministically (same seed, same numbers).
out/claims.json — a list of the central claims of the report: [{"id": "C1", "text": "...", "results_keys": ["..."], "status": "supported"|"unsupported"|"not_evaluated", "caveat": "..."}]. Include one claim about whether the conventional interval attains its nominal coverage.

# Constraints

No network access. Python: ~/wp29/venv/bin/python (numpy, pandas, scipy, statsmodels available). Do not modify data/.
This task is designed to be completed in well under 15 minutes of work; the simulation itself should take seconds.

spb/wp29_uqo