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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.\n\n## Inputs\n- `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.\n\n## Information-availability rules\n- 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.\n\n## Required outputs\n- `out/results.json` with the keys:\n\n| key | type | unit | meaning |\n|---|---|---|---|\n| `task_id` | string | – | \"F2-conf-03\" |\n| `coverage_conventional` | number | proportion | share of the 2000 replications in which the conventional interval contains β = 0.5 — the naïve baseline |\n| `coverage_hc1` | number | proportion | share of the 2000 replications in which the HC1 interval contains β = 0.5 |\n| `sd_beta_exact` | number | – | exact standard deviation of β̂ under the design (analytical, no simulation) |\n| `sd_beta_mc` | number | – | sample standard deviation (divisor R − 1) of the 2000 simulated β̂ |\n| `mean_se_conventional` | number | – | average of se_conv over the 2000 replications |\n| `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| `n_replications` | integer | – | number of replications (2000) |\n| `n_obs_per_replication` | integer | – | sample size per replication (30) |\n| `feasibility` | string | – | \"complete\", \"partial\" or \"infeasible\" |\n| `feasibility_reason` | string or null | – | explanation if not complete |\n| `data_issues` | array of {\"issue\",\"handling\"} | – | data problems detected and how handled (empty list if none) |\n| `n_input_rows` | object | – | {\"design.csv\": rows read} |\n| `random_seed` | integer | – | the seed used for the simulation (required; the simulation must be reproducible) |\n\n- `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.\n- `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).\n- `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.\n\n## Constraints\n- No network access. Python: `~/wp29/venv/bin/python` (numpy, pandas, scipy, statsmodels available). Do not modify `data/`.\n- This task is designed to be completed in well under 15 minutes of work; the simulation itself should take seconds.\ndesign.csv","is_error":false}]},"uuid":"5d16c6b3-b400-4887-8e91-a511a896af29","timestamp":"2026-10-30T04:16:28.590Z","toolUseResult":{"stdout":"# F2-conf-03 — Monte Carlo coverage of OLS confidence intervals in a small heteroskedastic sample\n\n## Scientific question\nHow 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). 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Do not redraw x or σ.\n\n## Required outputs\n- `out/results.json` with the keys:\n\n| key | type | unit | meaning |\n|---|---|---|---|\n| `task_id` | string | – | \"F2-conf-03\" |\n| `coverage_conventional` | number | proportion | share of the 2000 replications in which the conventional interval contains β = 0.5 — the naïve baseline |\n| `coverage_hc1` | number | proportion | share of the 2000 replications in which the HC1 interval contains β = 0.5 |\n| `sd_beta_exact` | number | – | exact standard deviation of β̂ under the design (analytical, no simulation) |\n| `sd_beta_mc` | number | – | sample standard deviation (divisor R − 1) of the 2000 simulated β̂ |\n| `mean_se_conventional` | number | – | average of se_conv over the 2000 replications |\n| `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| `n_replications` | integer | – | number of replications (2000) |\n| `n_obs_per_replication` | integer | – | sample size per replication (30) |\n| `feasibility` | string | – | \"complete\", \"partial\" or \"infeasible\" |\n| `feasibility_reason` | string or null | – | explanation if not complete |\n| `data_issues` | array of {\"issue\",\"handling\"} | – | data problems detected and how handled (empty list if none) |\n| `n_input_rows` | object | – | {\"design.csv\": rows read} |\n| `random_seed` | integer | – | the seed used for the simulation (required; the simulation must be reproducible) |\n\n- `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.\n- `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).\n- `out/claims.json` — a list of the central claims of the report: [{\"id\": \"C1\", \"text\": \"...\", \"results_keys\": [\"...\"], \"status\": \"supported\"|\"unsupported\"|\"not_evaluated\", \"caveat\": \"...\"}]. 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data. Write the analysis."}],"container":null,"stop_reason":"tool_use","stop_sequence":null,"stop_details":null,"usage":{"input_tokens":2,"cache_creation_input_tokens":458,"cache_read_input_tokens":3905,"output_tokens":1725,"output_tokens_details":{"thinking_tokens":0},"server_tool_use":{"web_search_requests":0,"web_fetch_requests":0},"service_tier":"standard","cache_creation":{"ephemeral_1h_input_tokens":0,"ephemeral_5m_input_tokens":458},"inference_geo":"global","iterations":[],"speed":"standard","fallback_credit":null},"input_transformations":[],"diagnostics":null,"context_management":null},"apiBlockIndex":0,"requestId":"req_011CgXmu8Zwvb8Be6TBDgUuG","requestRef":"7c13bc04-bef4-4eef-aca4-4c992930d459","requestedModel":"claude-opus-5-5","type":"assistant","uuid":"72a998e7-8a22-4d9f-a1c0-01bf3e0ff7ce","timestamp":"2026-10-30T04:16:30.628Z","effort":"medium","perTurnEffort":"medium","userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0056","sessionId":"fe3d3c52-0b32-42a3-a868-c59731699afe","version":"2.1.314","gitBranch":"HEAD"}21{"parentUuid":"72a998e7-8a22-4d9f-a1c0-01bf3e0ff7ce","isSidechain":false,"message":{"model":"claude-opus-5-5","id":"msg_011CgXmu8mbzgA78NGMJsbWD","type":"message","role":"assistant","content":[{"type":"tool_use","id":"toolu_01X2FR6LBLMc5yRBERNBEDde","name":"Bash","input":{"command":"mkdir -p out; cat > out/analysis.py <<'EOF'\n\"\"\"F2-conf-03: Monte Carlo coverage of OLS CIs under heteroskedasticity.\nRun from workspace root: ~/wp29/venv/bin/python out/analysis.py\"\"\"\nimport json, os\nimport numpy as np, pandas as pd\nfrom scipy import stats\n\nSEED = 20261030\nALPHA, BETA, N, R = 1.0, 0.5, 30, 2000\n\nd = pd.read_csv(\"data/design.csv\")\nissues = []\nn_rows = len(d)\nif d[[\"x\", \"sigma\"]].isna().any().any():\n issues.append({\"issue\": \"missing values\", \"handling\": \"none dropped; would fail\"})\nif (d[\"sigma\"] <= 0).any():\n issues.append({\"issue\": \"non-positive sigma\", \"handling\": \"flagged\"})\nif n_rows != N:\n issues.append({\"issue\": f\"expected {N} rows, got {n_rows}\", \"handling\": \"flagged\"})\nx = d[\"x\"].to_numpy(float); sig = d[\"sigma\"].to_numpy(float)\nn = len(x)\n\nX = np.column_stack([np.ones(n), x])\nXtXi = np.linalg.inv(X.T @ X)\nA = XtXi @ X.T # beta_hat = A y\na = A[1] # weights for slope\nsd_exact = float(np.sqrt(np.sum(a**2 * sig**2))) # Var = sum w_i^2 sigma_i^2\nsd_naive_formula = float(np.sqrt(np.mean(sig**2) * XtXi[1, 1])) # for reference only\n\ntcrit = stats.t.ppf(0.975, n - 2)\nrng = np.random.default_rng(SEED)\neps = rng.standard_normal((R, n))\nY = ALPHA + BETA * x + sig * eps # R x n\nB = Y @ A.T # R x 2\nresid = Y - B @ X.T\ns2 = (resid**2).sum(1) / (n - 2)\nse_conv = np.sqrt(s2 * XtXi[1, 1])\nhc0 = (resid**2) @ (a**2) # slope element of sandwich\nse_hc1 = np.sqrt(hc0 * n / (n - 2))\nbh = B[:, 1]\ncov_conv = float(np.mean(np.abs(bh - BETA) <= tcrit * se_conv))\ncov_hc1 = float(np.mean(np.abs(bh - BETA) <= tcrit * se_hc1))\nmcse_conv = float(np.sqrt(cov_conv * (1 - cov_conv) / R))\nmcse_hc1 = float(np.sqrt(cov_hc1 * (1 - cov_hc1) / R))\nmcse_nominal = float(np.sqrt(0.95 * 0.05 / R))\n# Decision rule: binomial MC SE at R (evaluated at the estimated coverage)\nundercovers = bool(cov_conv < 0.95 - 3 * mcse_conv)\n\nres = {\n \"task_id\": \"F2-conf-03\",\n \"coverage_conventional\": cov_conv,\n \"coverage_hc1\": cov_hc1,\n \"sd_beta_exact\": sd_exact,\n \"sd_beta_mc\": float(np.std(bh, ddof=1)),\n \"mean_se_conventional\": float(se_conv.mean()),\n \"conventional_ci_undercovers\": undercovers,\n \"n_replications\": R,\n \"n_obs_per_replication\": n,\n \"feasibility\": \"complete\",\n \"feasibility_reason\": None,\n \"data_issues\": issues,\n \"n_input_rows\": {\"design.csv\": n_rows},\n \"random_seed\": SEED,\n}\nos.makedirs(\"out\", exist_ok=True)\nwith open(\"out/results.json\", \"w\") as f:\n json.dump(res, f, indent=2)\nextra = {\"mcse_coverage_conventional\": mcse_conv, \"mcse_coverage_hc1\": mcse_hc1,\n \"mcse_at_nominal_0.95\": mcse_nominal,\n \"undercovers_using_nominal_mcse\": bool(cov_conv < 0.95 - 3 * mcse_nominal),\n \"t_crit\": float(tcrit), \"mean_se_hc1\": float(se_hc1.mean()),\n \"homoskedastic_formula_sd_with_mean_sigma2\": sd_naive_formula,\n \"mean_beta_hat\": float(bh.mean())}\nwith open(\"out/diagnostics.json\", \"w\") as f:\n json.dump(extra, f, indent=2)\nprint(json.dumps(res, indent=2)); print(json.dumps(extra, indent=2))\nEOF\n~/wp29/venv/bin/python out/analysis.py","description":"Write and run analysis script"},"caller":{"type":"direct"}}],"container":null,"stop_reason":"tool_use","stop_sequence":null,"stop_details":null,"usage":{"input_tokens":2,"cache_creation_input_tokens":458,"cache_read_input_tokens":3905,"output_tokens":1725,"output_tokens_details":{"thinking_tokens":0},"server_tool_use":{"web_search_requests":0,"web_fetch_requests":0},"service_tier":"standard","cache_creation":{"ephemeral_1h_input_tokens":0,"ephemeral_5m_input_tokens":458},"inference_geo":"global","iterations":[],"speed":"standard","fallback_credit":null},"input_transformations":[],"diagnostics":null,"context_management":null},"apiBlockIndex":1,"requestId":"req_011CgXmu8Zwvb8Be6TBDgUuG","requestRef":"7c13bc04-bef4-4eef-aca4-4c992930d459","requestedModel":"claude-opus-5-5","type":"assistant","uuid":"3f91ef14-0222-4bae-96c3-a9f01bc52081","timestamp":"2026-10-30T04:16:43.148Z","effort":"medium","perTurnEffort":"medium","userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0056","sessionId":"fe3d3c52-0b32-42a3-a868-c59731699afe","version":"2.1.314","gitBranch":"HEAD"}22{"type":"last-prompt","lastPrompt":"Begin","leafUuid":"3f91ef14-0222-4bae-96c3-a9f01bc52081","sessionId":"fe3d3c52-0b32-42a3-a868-c59731699afe"}23{"type":"atis-latch","atis":"","sessionId":"fe3d3c52-0b32-42a3-a868-c59731699afe"}24{"parentUuid":"3f91ef14-0222-4bae-96c3-a9f01bc52081","isSidechain":false,"promptId":"47fca238-6847-43c8-83fa-e6e3ccb609ae","type":"user","message":{"role":"user","content":[{"tool_use_id":"toolu_01X2FR6LBLMc5yRBERNBEDde","type":"tool_result","content":"{\n \"task_id\": \"F2-conf-03\",\n \"coverage_conventional\": 0.844,\n \"coverage_hc1\": 0.902,\n \"sd_beta_exact\": 0.2679805928302833,\n \"sd_beta_mc\": 0.2696473708309126,\n \"mean_se_conventional\": 0.1902623101215266,\n \"conventional_ci_undercovers\": true,\n \"n_replications\": 2000,\n \"n_obs_per_replication\": 30,\n \"feasibility\": \"complete\",\n \"feasibility_reason\": null,\n \"data_issues\": [],\n \"n_input_rows\": {\n \"design.csv\": 30\n },\n \"random_seed\": 20261030\n}\n{\n \"mcse_coverage_conventional\": 0.00811369213120636,\n \"mcse_coverage_hc1\": 0.006648157639526908,\n \"mcse_at_nominal_0.95\": 0.004873397172404482,\n \"undercovers_using_nominal_mcse\": true,\n \"t_crit\": 2.048407141795244,\n \"mean_se_hc1\": 0.24080977065103962,\n \"homoskedastic_formula_sd_with_mean_sigma2\": 0.19623471734704753,\n \"mean_beta_hat\": 0.4961965063446237\n}","is_error":false}]},"uuid":"4dd372ae-1409-4d8a-aab7-59e66f42ef97","timestamp":"2026-10-30T04:16:43.656Z","toolUseResult":{"stdout":"{\n \"task_id\": \"F2-conf-03\",\n \"coverage_conventional\": 0.844,\n \"coverage_hc1\": 0.902,\n \"sd_beta_exact\": 0.2679805928302833,\n \"sd_beta_mc\": 0.2696473708309126,\n \"mean_se_conventional\": 0.1902623101215266,\n \"conventional_ci_undercovers\": true,\n \"n_replications\": 2000,\n \"n_obs_per_replication\": 30,\n \"feasibility\": \"complete\",\n \"feasibility_reason\": null,\n \"data_issues\": [],\n \"n_input_rows\": {\n \"design.csv\": 30\n },\n \"random_seed\": 20261030\n}\n{\n \"mcse_coverage_conventional\": 0.00811369213120636,\n \"mcse_coverage_hc1\": 0.006648157639526908,\n \"mcse_at_nominal_0.95\": 0.004873397172404482,\n \"undercovers_using_nominal_mcse\": true,\n \"t_crit\": 2.048407141795244,\n \"mean_se_hc1\": 0.24080977065103962,\n \"homoskedastic_formula_sd_with_mean_sigma2\": 0.19623471734704753,\n \"mean_beta_hat\": 0.4961965063446237\n}","stderr":"","interrupted":false,"isImage":false,"noOutputExpected":false},"sandboxed":false,"sourceToolAssistantUUID":"3f91ef14-0222-4bae-96c3-a9f01bc52081","permissionDecision":{"decision":"accept","source":"config","reasonType":"rule","ruleSource":"cliArg"},"userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0056","sessionId":"fe3d3c52-0b32-42a3-a868-c59731699afe","version":"2.1.314","gitBranch":"HEAD"}25{"type":"api-request-blob","sessionId":"fe3d3c52-0b32-42a3-a868-c59731699afe","hash":"0be30bcdab4ee7ac7ee095db80e53c9e3af0104abb304bd0d908e37da7fd88ac","message":{"role":"assistant","content":[{"type":"text","text":"Clean data. Write the analysis."},{"type":"tool_use","id":"toolu_01X2FR6LBLMc5yRBERNBEDde","name":"Bash","input":{"command":"mkdir -p out; cat > out/analysis.py <<'EOF'\n\"\"\"F2-conf-03: Monte Carlo coverage of OLS CIs under heteroskedasticity.\nRun from workspace root: ~/wp29/venv/bin/python out/analysis.py\"\"\"\nimport json, os\nimport numpy as np, pandas as pd\nfrom scipy import stats\n\nSEED = 20261030\nALPHA, BETA, N, R = 1.0, 0.5, 30, 2000\n\nd = pd.read_csv(\"data/design.csv\")\nissues = []\nn_rows = len(d)\nif d[[\"x\", \"sigma\"]].isna().any().any():\n issues.append({\"issue\": \"missing values\", \"handling\": \"none dropped; would fail\"})\nif (d[\"sigma\"] <= 0).any():\n issues.append({\"issue\": \"non-positive sigma\", \"handling\": \"flagged\"})\nif n_rows != N:\n issues.append({\"issue\": f\"expected {N} rows, got {n_rows}\", \"handling\": \"flagged\"})\nx = d[\"x\"].to_numpy(float); sig = d[\"sigma\"].to_numpy(float)\nn = len(x)\n\nX = np.column_stack([np.ones(n), x])\nXtXi = np.linalg.inv(X.T @ X)\nA = XtXi @ X.T # beta_hat = A y\na = A[1] # weights for slope\nsd_exact = float(np.sqrt(np.sum(a**2 * sig**2))) # Var = sum w_i^2 sigma_i^2\nsd_naive_formula = float(np.sqrt(np.mean(sig**2) * XtXi[1, 1])) # for reference only\n\ntcrit = stats.t.ppf(0.975, n - 2)\nrng = np.random.default_rng(SEED)\neps = rng.standard_normal((R, n))\nY = ALPHA + BETA * x + sig * eps # R x n\nB = Y @ A.T # R x 2\nresid = Y - B @ X.T\ns2 = (resid**2).sum(1) / (n - 2)\nse_conv = np.sqrt(s2 * XtXi[1, 1])\nhc0 = (resid**2) @ (a**2) # slope element of sandwich\nse_hc1 = np.sqrt(hc0 * n / (n - 2))\nbh = B[:, 1]\ncov_conv = float(np.mean(np.abs(bh - BETA) <= tcrit * se_conv))\ncov_hc1 = float(np.mean(np.abs(bh - BETA) <= tcrit * se_hc1))\nmcse_conv = float(np.sqrt(cov_conv * (1 - cov_conv) / R))\nmcse_hc1 = float(np.sqrt(cov_hc1 * (1 - cov_hc1) / R))\nmcse_nominal = float(np.sqrt(0.95 * 0.05 / R))\n# Decision rule: binomial MC SE at R (evaluated at the estimated coverage)\nundercovers = bool(cov_conv < 0.95 - 3 * mcse_conv)\n\nres = {\n \"task_id\": \"F2-conf-03\",\n \"coverage_conventional\": cov_conv,\n \"coverage_hc1\": cov_hc1,\n \"sd_beta_exact\": sd_exact,\n \"sd_beta_mc\": float(np.std(bh, ddof=1)),\n \"mean_se_conventional\": float(se_conv.mean()),\n \"conventional_ci_undercovers\": undercovers,\n \"n_replications\": R,\n \"n_obs_per_replication\": n,\n \"feasibility\": \"complete\",\n \"feasibility_reason\": None,\n \"data_issues\": issues,\n \"n_input_rows\": {\"design.csv\": n_rows},\n \"random_seed\": SEED,\n}\nos.makedirs(\"out\", exist_ok=True)\nwith open(\"out/results.json\", \"w\") as f:\n json.dump(res, f, indent=2)\nextra = {\"mcse_coverage_conventional\": mcse_conv, \"mcse_coverage_hc1\": mcse_hc1,\n \"mcse_at_nominal_0.95\": mcse_nominal,\n \"undercovers_using_nominal_mcse\": bool(cov_conv < 0.95 - 3 * mcse_nominal),\n \"t_crit\": float(tcrit), \"mean_se_hc1\": float(se_hc1.mean()),\n \"homoskedastic_formula_sd_with_mean_sigma2\": sd_naive_formula,\n \"mean_beta_hat\": float(bh.mean())}\nwith open(\"out/diagnostics.json\", \"w\") as f:\n json.dump(extra, f, indent=2)\nprint(json.dumps(res, indent=2)); print(json.dumps(extra, indent=2))\nEOF\n~/wp29/venv/bin/python out/analysis.py","description":"Write and run analysis 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match statsmodels. Now report and claims."}],"container":null,"stop_reason":"tool_use","stop_sequence":null,"stop_details":null,"usage":{"input_tokens":2,"cache_creation_input_tokens":418,"cache_read_input_tokens":6523,"output_tokens":2334,"output_tokens_details":{"thinking_tokens":0},"server_tool_use":{"web_search_requests":0,"web_fetch_requests":0},"service_tier":"standard","cache_creation":{"ephemeral_1h_input_tokens":0,"ephemeral_5m_input_tokens":418},"inference_geo":"global","iterations":[],"speed":"standard","fallback_credit":null},"input_transformations":[],"diagnostics":null,"context_management":null},"apiBlockIndex":0,"requestId":"req_011CgXmvTJ6wxNdNqcLGnFca","requestRef":"cf84d246-55f8-4a29-bdc9-b4e2052d94fe","requestedModel":"claude-opus-5-5","type":"assistant","uuid":"b7aae674-0c43-447b-8bf0-7dd485509eb3","timestamp":"2026-10-30T04:16:48.554Z","effort":"medium","perTurnEffort":"medium","userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0056","sessionId":"fe3d3c52-0b32-42a3-a868-c59731699afe","version":"2.1.314","gitBranch":"HEAD"}33{"parentUuid":"b7aae674-0c43-447b-8bf0-7dd485509eb3","isSidechain":false,"message":{"model":"claude-opus-5-5","id":"msg_011CgXmvTaE5cbd7SqxZyHiL","type":"message","role":"assistant","content":[{"type":"tool_use","id":"toolu_01XWUxz7oxJMbPLh1iSzb7h5","name":"Bash","input":{"command":"cat > out/report.md <<'EOF'\n# F2-conf-03 — Coverage of OLS confidence intervals, n = 30, heteroskedastic errors\n\n## Question\nIn the fixed-design model y_i = 1 + 0.5 x_i + σ_i ε_i (n = 30, ε_i iid N(0,1), x and σ fixed from `data/design.csv`), what is the true coverage of nominal 95% intervals β̂ ± t_{0.975}(28)·se for (a) the conventional OLS SE and (b) the HC1 robust SE? Also: (c) the exact SD of β̂, (d) E[se_conv]. Does the conventional interval reach 95%?\n\n## Data and exclusions\n`data/design.csv`: 30 rows read and 30 used. There were no missing values, σ was always positive, and nothing was excluded. In this design σ_i grows with |x_i|: it ranges from 0.60 near x = 0 to 1.79 at x = 1.99. This is the kind of heteroskedasticity that makes the conventional SE too small.\n\n## Method\n- **Exact SD (c):** β̂ = Σ w_i y_i, where w is the slope row of (X'X)⁻¹X'. So Var(β̂) = Σ w_i² σ_i². This is analytic, with no simulation.\n- **Monte Carlo (a, b, d):** R = 2000 replications, using `numpy.random.default_rng(20261030)`. Only ε is redrawn in each replication.\n - se_conv² = s²·[(X'X)⁻¹]₂₂, where s² = RSS/(n−2).\n - se_HC1² = (n/(n−2))·Σ w_i² e_i², i.e. the HC0 sandwich scaled by n/(n−2).\n - Critical value: t_{0.975}(28) = 2.0484.\n - Coverage is the share of replications with |β̂ − 0.5| ≤ t·se.\n- **Check against statsmodels:** for one sample, the SE formulas matched statsmodels' OLS SE and HC1 SE to machine precision.\n- **Decision rule:** the conventional interval \"undercovers\" if its coverage is below 0.95 − 3·MCSE, where MCSE = √(p̂(1−p̂)/R) is evaluated at the estimated coverage. Evaluating it at p = 0.95 instead gives MCSE 0.0049, and the conclusion stays the same.\n\n## Results\n| Quantity | Value |\n|---|---|\n| Coverage, conventional CI | **0.844** |\n| Coverage, HC1 CI | **0.902** |\n| Exact SD of β̂ | 0.2680 |\n| MC SD of β̂ (R−1 divisor) | 0.2696 |\n| Mean se_conv | 0.1903 |\n| Mean se_HC1 (diagnostic) | 0.2408 |\n| Mean β̂ (diagnostic) | 0.4962 |\n\nThe conventional SE is badly biased downward. On average it is about 0.190, against a true SD of 0.268, so it understates the SD by roughly 29%. As a result, the conventional interval covers only about 84% of the time. Its shortfall from 95% is 10.6 percentage points, about 13 MCSEs, so `conventional_ci_undercovers = true`: **the conventional interval does not attain its nominal 95% coverage.**\n\nHC1 does much better, at about 90%. It still undercovers, though. At n = 30 the robust SE is itself biased downward (mean 0.241 vs 0.268) and noisy, especially here, where the observations with the largest variance also have high leverage. The MC SD of β̂ agrees with the exact value to within simulation error, which supports the correctness of the simulation.\n\n## Uncertainty\nBinomial Monte Carlo standard errors at R = 2000:\n- **Conventional coverage:** 0.844, MCSE 0.0081, approximate 95% MC interval [0.828, 0.860].\n- **HC1 coverage:** 0.902, MCSE 0.0067, approximate 95% MC interval [0.889, 0.915].\n- **MCSE at the nominal 0.95:** 0.0049.\n\nThe MC SD of β̂ has a relative standard error of about 1/√(2R) ≈ 1.6%. All of these values are reproducible with seed 20261030; a different seed would change them within these MC errors.\n\n## Limitations and open issues\n- The results hold for this single fixed design and for normal errors only. They should not be generalised to other x/σ configurations or to other sample sizes.\n- HC1 was not compared with HC2/HC3 or with wild-bootstrap intervals. Those might bring coverage closer to 95%, but they were outside the task.\n- R = 2000 limits coverage precision to about ±1.6 percentage points (2·MCSE).\n- Additional diagnostics are written to `out/diagnostics.json`. They are not part of the required results.\nEOF\ncat > out/claims.json <<'EOF'\n[\n {\"id\":\"C1\",\"text\":\"The conventional OLS 95% interval does not attain nominal coverage under this design: estimated coverage 0.844, more than three MC standard errors below 0.95.\",\"results_keys\":[\"coverage_conventional\",\"conventional_ci_undercovers\"],\"status\":\"supported\",\"caveat\":\"Specific to this fixed design, normal errors, n=30; MCSE ~0.008 at R=2000.\"},\n {\"id\":\"C2\",\"text\":\"The HC1 robust interval covers better (0.902) but still falls short of 95% at n=30.\",\"results_keys\":[\"coverage_hc1\"],\"status\":\"supported\",\"caveat\":\"MCSE ~0.0067; HC2/HC3 not evaluated.\"},\n {\"id\":\"C3\",\"text\":\"The conventional SE is biased downward: its mean (0.190) is well below the exact SD of the slope estimator (0.268).\",\"results_keys\":[\"mean_se_conventional\",\"sd_beta_exact\"],\"status\":\"supported\",\"caveat\":\"Mean se_conv is a Monte Carlo estimate.\"},\n {\"id\":\"C4\",\"text\":\"The simulated SD of the slope estimator (0.270) agrees with the analytic exact SD (0.268) within Monte Carlo error, validating the simulation.\",\"results_keys\":[\"sd_beta_mc\",\"sd_beta_exact\"],\"status\":\"supported\",\"caveat\":\"Relative MC error of the SD is about 1.6%.\"}\n]\nEOF\n~/wp29/venv/bin/python out/analysis.py >/dev/null && md5 out/results.json && ~/wp29/venv/bin/python out/analysis.py >/dev/null && md5 out/results.json","description":"Write report and claims, verify reproducibility"},"caller":{"type":"direct"}}],"container":null,"stop_reason":"tool_use","stop_sequence":null,"stop_details":null,"usage":{"input_tokens":2,"cache_creation_input_tokens":418,"cache_read_input_tokens":6523,"output_tokens":2334,"output_tokens_details":{"thinking_tokens":0},"server_tool_use":{"web_search_requests":0,"web_fetch_requests":0},"service_tier":"standard","cache_creation":{"ephemeral_1h_input_tokens":0,"ephemeral_5m_input_tokens":418},"inference_geo":"global","iterations":[],"speed":"standard","fallback_credit":null},"input_transformations":[],"diagnostics":null,"context_management":null},"apiBlockIndex":1,"requestId":"req_011CgXmvTJ6wxNdNqcLGnFca","requestRef":"cf84d246-55f8-4a29-bdc9-b4e2052d94fe","requestedModel":"claude-opus-5-5","type":"assistant","uuid":"61b696fe-ebb5-41a5-81f0-62c7376757d2","timestamp":"2026-10-30T04:17:07.729Z","effort":"medium","perTurnEffort":"medium","userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0056","sessionId":"fe3d3c52-0b32-42a3-a868-c59731699afe","version":"2.1.314","gitBranch":"HEAD"}34{"parentUuid":"61b696fe-ebb5-41a5-81f0-62c7376757d2","isSidechain":false,"promptId":"47fca238-6847-43c8-83fa-e6e3ccb609ae","type":"user","message":{"role":"user","content":[{"tool_use_id":"toolu_01XWUxz7oxJMbPLh1iSzb7h5","type":"tool_result","content":"MD5 (out/results.json) = 67879644da732090cb4236ecef056496\nMD5 (out/results.json) = 67879644da732090cb4236ecef056496","is_error":false}]},"uuid":"40c10b86-fb5e-4216-8040-783d6879214f","timestamp":"2026-10-30T04:17:08.690Z","toolUseResult":{"stdout":"MD5 (out/results.json) = 67879644da732090cb4236ecef056496\nMD5 (out/results.json) = 67879644da732090cb4236ecef056496","stderr":"","interrupted":false,"isImage":false,"noOutputExpected":false},"sandboxed":false,"sourceToolAssistantUUID":"61b696fe-ebb5-41a5-81f0-62c7376757d2","permissionDecision":{"decision":"accept","source":"config","reasonType":"rule","ruleSource":"cliArg"},"userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0056","sessionId":"fe3d3c52-0b32-42a3-a868-c59731699afe","version":"2.1.314","gitBranch":"HEAD"}35{"type":"api-request-blob","sessionId":"fe3d3c52-0b32-42a3-a868-c59731699afe","hash":"62eda9fbbf767fe7c4c043716250f43ec33c4c887671d5377c4e52bab3b0587c","message":{"role":"assistant","content":[{"type":"text","text":"Formulas match statsmodels. Now report and claims."},{"type":"tool_use","id":"toolu_01XWUxz7oxJMbPLh1iSzb7h5","name":"Bash","input":{"command":"cat > out/report.md <<'EOF'\n# F2-conf-03 — Coverage of OLS confidence intervals, n = 30, heteroskedastic errors\n\n## Question\nIn the fixed-design model y_i = 1 + 0.5 x_i + σ_i ε_i (n = 30, ε_i iid N(0,1), x and σ fixed from `data/design.csv`), what is the true coverage of nominal 95% intervals β̂ ± t_{0.975}(28)·se for (a) the conventional OLS SE and (b) the HC1 robust SE? Also: (c) the exact SD of β̂, (d) E[se_conv]. Does the conventional interval reach 95%?\n\n## Data and exclusions\n`data/design.csv`: 30 rows read and 30 used. There were no missing values, σ was always positive, and nothing was excluded. In this design σ_i grows with |x_i|: it ranges from 0.60 near x = 0 to 1.79 at x = 1.99. This is the kind of heteroskedasticity that makes the conventional SE too small.\n\n## Method\n- **Exact SD (c):** β̂ = Σ w_i y_i, where w is the slope row of (X'X)⁻¹X'. So Var(β̂) = Σ w_i² σ_i². This is analytic, with no simulation.\n- **Monte Carlo (a, b, d):** R = 2000 replications, using `numpy.random.default_rng(20261030)`. Only ε is redrawn in each replication.\n - se_conv² = s²·[(X'X)⁻¹]₂₂, where s² = RSS/(n−2).\n - se_HC1² = (n/(n−2))·Σ w_i² e_i², i.e. the HC0 sandwich scaled by n/(n−2).\n - Critical value: t_{0.975}(28) = 2.0484.\n - Coverage is the share of replications with |β̂ − 0.5| ≤ t·se.\n- **Check against statsmodels:** for one sample, the SE formulas matched statsmodels' OLS SE and HC1 SE to machine precision.\n- **Decision rule:** the conventional interval \"undercovers\" if its coverage is below 0.95 − 3·MCSE, where MCSE = √(p̂(1−p̂)/R) is evaluated at the estimated coverage. Evaluating it at p = 0.95 instead gives MCSE 0.0049, and the conclusion stays the same.\n\n## Results\n| Quantity | Value |\n|---|---|\n| Coverage, conventional CI | **0.844** |\n| Coverage, HC1 CI | **0.902** |\n| Exact SD of β̂ | 0.2680 |\n| MC SD of β̂ (R−1 divisor) | 0.2696 |\n| Mean se_conv | 0.1903 |\n| Mean se_HC1 (diagnostic) | 0.2408 |\n| Mean β̂ (diagnostic) | 0.4962 |\n\nThe conventional SE is badly biased downward. On average it is about 0.190, against a true SD of 0.268, so it understates the SD by roughly 29%. As a result, the conventional interval covers only about 84% of the time. Its shortfall from 95% is 10.6 percentage points, about 13 MCSEs, so `conventional_ci_undercovers = true`: **the conventional interval does not attain its nominal 95% coverage.**\n\nHC1 does much better, at about 90%. It still undercovers, though. At n = 30 the robust SE is itself biased downward (mean 0.241 vs 0.268) and noisy, especially here, where the observations with the largest variance also have high leverage. The MC SD of β̂ agrees with the exact value to within simulation error, which supports the correctness of the simulation.\n\n## Uncertainty\nBinomial Monte Carlo standard errors at R = 2000:\n- **Conventional coverage:** 0.844, MCSE 0.0081, approximate 95% MC interval [0.828, 0.860].\n- **HC1 coverage:** 0.902, MCSE 0.0067, approximate 95% MC interval [0.889, 0.915].\n- **MCSE at the nominal 0.95:** 0.0049.\n\nThe MC SD of β̂ has a relative standard error of about 1/√(2R) ≈ 1.6%. All of these values are reproducible with seed 20261030; a different seed would change them within these MC errors.\n\n## Limitations and open issues\n- The results hold for this single fixed design and for normal errors only. They should not be generalised to other x/σ configurations or to other sample sizes.\n- HC1 was not compared with HC2/HC3 or with wild-bootstrap intervals. Those might bring coverage closer to 95%, but they were outside the task.\n- R = 2000 limits coverage precision to about ±1.6 percentage points (2·MCSE).\n- Additional diagnostics are written to `out/diagnostics.json`. They are not part of the required results.\nEOF\ncat > out/claims.json <<'EOF'\n[\n {\"id\":\"C1\",\"text\":\"The conventional OLS 95% interval does not attain nominal coverage under this design: estimated coverage 0.844, more than three MC standard errors below 0.95.\",\"results_keys\":[\"coverage_conventional\",\"conventional_ci_undercovers\"],\"status\":\"supported\",\"caveat\":\"Specific to this fixed design, normal errors, n=30; MCSE ~0.008 at R=2000.\"},\n {\"id\":\"C2\",\"text\":\"The HC1 robust interval covers better (0.902) but still falls short of 95% at n=30.\",\"results_keys\":[\"coverage_hc1\"],\"status\":\"supported\",\"caveat\":\"MCSE ~0.0067; HC2/HC3 not evaluated.\"},\n {\"id\":\"C3\",\"text\":\"The conventional SE is biased downward: its mean (0.190) is well below the exact SD of the slope estimator (0.268).\",\"results_keys\":[\"mean_se_conventional\",\"sd_beta_exact\"],\"status\":\"supported\",\"caveat\":\"Mean se_conv is a Monte Carlo estimate.\"},\n {\"id\":\"C4\",\"text\":\"The simulated SD of the slope estimator (0.270) agrees with the analytic exact SD (0.268) within Monte Carlo error, validating the simulation.\",\"results_keys\":[\"sd_beta_mc\",\"sd_beta_exact\"],\"status\":\"supported\",\"caveat\":\"Relative MC error of the SD is about 1.6%.\"}\n]\nEOF\n~/wp29/venv/bin/python out/analysis.py >/dev/null && md5 out/results.json && ~/wp29/venv/bin/python out/analysis.py >/dev/null && md5 out/results.json","description":"Write report and claims, verify 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conventional 95% confidence interval does not reach its nominal coverage in this design: it contains the true slope in only about 84% of the 2000 simulations. The heteroskedasticity-robust (HC1) interval does better but still falls short, at about 90%.\n\n| Quantity | Value |\n|---|---|\n| Coverage, conventional interval | 0.844 (Monte Carlo standard error 0.0081) |\n| Coverage, HC1 interval | 0.902 (Monte Carlo standard error 0.0067) |\n| Exact standard deviation of the slope estimate (analytic) | 0.2680 |\n| Simulated standard deviation of the slope estimate | 0.2696 |\n| Average conventional standard error | 0.1903 |\n\n- **Why the conventional interval fails:** its standard error averages 0.190 against a true value of 0.268, so it understates the uncertainty by about 29%. The coverage gap from 95% is about 13 Monte Carlo standard errors, so `conventional_ci_undercovers` is true.\n- **Why HC1 still undercovers:** in a sample of 30 its standard error is also too small on average (0.241 vs 0.268). The observations with the largest error variance are also the most extreme x values.\n- **Checks:** the simulated standard deviation of the slope matches the exact value within simulation error. For one sample, my standard error formulas matched statsmodels exactly. Running the script twice gave identical results with the fixed seed 20261030.\n- **Data:** all 30 rows of the design were used. There were no missing values or other problems, and nothing was excluded.\n\n**Limitations:**\n- The results hold only for this one design with normal errors and n = 30.\n- I did not compare other robust standard errors (HC2, HC3) or bootstrap intervals, which were outside the task.\n- With 2000 replications, each coverage estimate is precise to roughly ±1.6 percentage points.\n\nEverything is in `out/`:\n- `analysis.py` (rerun with `~/wp29/venv/bin/python out/analysis.py`)\n- `results.json`\n- `report.md`\n- `claims.json` (four claims, including whether the conventional interval reaches 95%)\n- `diagnostics.json` (extra figures, such as the average HC1 standard error, outside the required outputs)"}],"container":null,"stop_reason":"end_turn","stop_sequence":null,"stop_details":null,"usage":{"input_tokens":2,"cache_creation_input_tokens":2396,"cache_read_input_tokens":6941,"output_tokens":820,"output_tokens_details":{"thinking_tokens":59},"server_tool_use":{"web_search_requests":0,"web_fetch_requests":0},"service_tier":"standard","cache_creation":{"ephemeral_1h_input_tokens":0,"ephemeral_5m_input_tokens":2396},"inference_geo":"global","iterations":[],"speed":"standard","fallback_credit":null},"input_transformations":[],"diagnostics":null,"context_management":null},"apiBlockIndex":1,"requestId":"req_011CgXmx193hx32JyF3ivGDM","requestRef":"d47184e7-5df6-4d25-8516-f561d1b1d7f0","requestedModel":"claude-opus-5-5","type":"assistant","uuid":"fea3a431-bae8-4633-99f3-144ba3d6181b","timestamp":"2026-10-30T04:17:15.915Z","effort":"medium","perTurnEffort":"medium","userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0056","sessionId":"fe3d3c52-0b32-42a3-a868-c59731699afe","version":"2.1.314","gitBranch":"HEAD"}39{"type":"last-prompt","lastPrompt":"Begin","leafUuid":"fea3a431-bae8-4633-99f3-144ba3d6181b","sessionId":"fe3d3c52-0b32-42a3-a868-c59731699afe"}40{"type":"atis-latch","atis":"","sessionId":"fe3d3c52-0b32-42a3-a868-c59731699afe"}41{"type":"cost-state","sessionId":"fe3d3c52-0b32-42a3-a868-c59731699afe","totalCostUSD":0.15271500000000002,"totalAPIDuration":48263,"totalAPIDurationWithoutRetries":48260,"totalToolDuration":2172,"totalLinesAdded":0,"totalLinesRemoved":0,"totalDuration":50554,"startTime":1793333785387,"modelUsage":{"claude-opus-5-5":{"inputTokens":14,"outputTokens":5420,"thinkingTokens":93,"cacheReadInputTokens":25470,"cacheCreationInputTokens":7833,"cacheCreationEphemeral5mInputTokens":7833,"cacheCreationEphemeral1hInputTokens":0,"webSearchRequests":0,"costUSD":0.15271500000000002}},"hasUnknownModelCost":false}42