# Contract (P0)

Question: Do nominal 95% t-intervals for the OLS slope achieve 95% coverage in a fixed design (n=30) with heteroskedastic errors y_i = 1 + 0.5 x_i + sigma_i eps_i, eps_i iid N(0,1)?

Estimands (conditional on the fixed x, sigma in data/design.csv; only eps redrawn):

• (a) P(beta in beta_hat ± t_.975(28) se_conv), se_conv = conventional OLS SE with s^2 = RSS/(n-2). MC, R=2000.
• (b) same with se_HC1 = sqrt(n/(n-2)) * HC0 sandwich SE. MC, R=2000.
• (c) exact SD of beta_hat = sqrt(sum w_i^2 sigma_i^2), w_i = (x_i - xbar)/Sxx. Analytic.
• (d) E[se_conv], MC average. Meaningful under model as stated (normal errors, known fixed design).

Outputs: out/results.json (keys: task_id, coverage_conventional, coverage_hc1, sd_beta_exact, sd_beta_mc, mean_se_conventional, conventional_ci_undercovers, n_replications, n_obs_per_replication, feasibility, feasibility_reason, data_issues, n_input_rows, random_seed), out/report.md (Question; Data and exclusions; Method; Results; Uncertainty; Limitations and open issues), out/analysis.py, out/claims.json. Undercovers flag: coverage_conventional < 0.95 - 3sqrt(0.950.05/2000).

Information rules: only data/design.csv plus given constants; do not redraw x or sigma.

Invalidity: missing/malformed design (not 30 rows, non-positive sigma), wrong R, non-reproducible seed.