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Random draws happen in this order: first the vector of y_1 for all R replications, then an R×(n−1) matrix of innovations. Each replication gives a new series of 30 observations, and the regression uses t = 2,…,30 (29 observations). No replication was excluded: all 20,000 ρ̂ are finite. I found no data issues.\n\n## Method\n- ρ̂ is the OLS slope with an intercept, computed in vectorised form from demeaned (y_{t−1}, y_t). On replication 1 I checked it against an explicit least-squares fit with `lstsq`.\n- bias_mc = mean(ρ̂) − ρ. Its Monte Carlo standard error is sd(ρ̂)/√R.\n- bias_analytical = −(1+3ρ)/n, evaluated at n = 30 (the design's series length), as the task specifies.\n- **What kind of formula this is.** −(1+3ρ)/n is the leading O(1/n) term of an asymptotic expansion of E[ρ̂] − ρ. The full bias is −(1+3ρ)/n + O(n⁻²). The formula is accurate only in the sense that the leftover error shrinks faster than 1/n as n → ∞ with ρ held fixed, |ρ| < 1. It is not an exact finite-sample result, and it gives no guaranteed error bound at any particular n. The neglected higher-order terms grow as ρ approaches 1, because the expansion is not uniform in ρ near the unit root. At n = 30 the formula is also ambiguous to order O(n⁻²) depending on whether n means the series length (30) or the number of regression observations (29). I report the n = 30 version as the main figure and the n − 1 version as a supplement.\n\n## Results\n| quantity | value |\n|---|---|\n| mean ρ̂ | 0.6763 |\n| median ρ̂ | 0.6987 |\n| sd ρ̂ (MC) | 0.1489 |\n| bias_mc | −0.1237 (MC s.e. 0.00105) |\n| bias_analytical, −(1+3ρ)/30 | −0.1133 |\n| discrepancy | −0.0104, i.e. −9.85 MC s.e. (the approximation captures about 92% of the bias; it is off by 9.2% relative to its own value) |\n| supplementary: −(1+3ρ)/29 | −0.1172 (discrepancy −6.14 MC s.e.) |\n| textbook asymptotic s.e. √((1−ρ²)/n) | 0.1095 |\n| share of ρ̂ < ρ | 79.3% |\n\nThe bias is large in practical terms. It equals about 1.13 times the textbook asymptotic standard error and about 0.83 times the Monte Carlo sd of ρ̂. The Monte Carlo sd (0.149) is also about 36% larger than the asymptotic s.e. (0.110), so at this design the asymptotic s.e. understates the actual sampling variability.\n\nThe analytical formula has the right sign and gets the order of magnitude right: it accounts for roughly nine tenths of the bias. However, it understates the size of the bias, and the gap of about 10 MC standard errors is far too large to be Monte Carlo noise. This fits the formula's nature as a first-order approximation. At n = 30 with ρ = 0.8, the O(n⁻²) terms it leaves out are about 0.01, and they push in the same direction (more downward bias). The approximation is therefore useful for rough work at this design but not exact.\n\n## Uncertainty\nThe Monte Carlo s.e. of bias_mc is 0.00105, so a 95% interval for the true bias is about −0.1237 ± 0.0021, or [−0.1258, −0.1216]. The analytical value −0.1133 lies well outside this interval. The result depends on a single seed, as the design requires. Another seed would move bias_mc by an amount on the order of 0.001 and would not change the conclusion.\n\n## Limitations and open issues\n- The Monte Carlo evidence applies to one design point only (n = 30, ρ = 0.8, Gaussian errors, stationary start, c = 0, intercept included). It cannot show how accurate the formula is at other values of n or ρ, under non-Gaussian errors, or with other initial conditions. It also cannot by itself confirm the asymptotic O(n⁻²) rate of the leftover error, which would require a grid over n. What it does show is that, at this design, the first-order approximation is detectably biased toward zero by about 0.010, with an MC error of about 0.001.\n- ρ̂ does not depend on σ, so the choice σ = 1 does not matter. ρ̂ also does not depend on c, because the regression includes an intercept and the start is stationary.\n- The formula is ambiguous between n and n − 1. Even the more favourable n − 1 version is rejected at about 6 MC s.e.\n- Bias is only one feature of the distribution. ρ̂ is left-skewed (median 0.699 > mean 0.676), so bias-corrected point estimates or normal-approximation intervals built from the asymptotic s.e. would still be inaccurate at this sample size.\nEOF\ncat > out/claims.json <<'EOF'\n[\n {\"id\":\"C1\",\"text\":\"At n=30, rho=0.8 the OLS estimator of rho is substantially downward biased: Monte Carlo bias -0.124 (MC s.e. 0.001), mean rho_hat 0.676.\",\"results_keys\":[\"bias_mc\",\"bias_mc_se\",\"rho_hat_mean\"],\"status\":\"supported\",\"caveat\":\"Single design point and single seed.\"},\n {\"id\":\"C2\",\"text\":\"The Marriott-Pope/Kendall approximation -(1+3rho)/n = -0.113 has the correct sign and captures roughly 90% of the bias.\",\"results_keys\":[\"bias_analytical\",\"bias_mc\"],\"status\":\"supported\",\"caveat\":\"Uses n=30 (series length); with n-1=29 the value is -0.117.\"},\n {\"id\":\"C3\",\"text\":\"The approximation understates the magnitude of the bias by about 0.010, a discrepancy of about -9.9 Monte Carlo standard errors, which is not attributable to simulation noise and is consistent with omitted O(n^-2) terms.\",\"results_keys\":[\"discrepancy_se_units\",\"bias_mc\",\"bias_analytical\",\"bias_mc_se\"],\"status\":\"supported\",\"caveat\":\"Attribution to O(n^-2) terms is an interpretation; MC at one n cannot verify the rate.\"},\n {\"id\":\"C4\",\"text\":\"The bias is of the same order as the sampling uncertainty: it exceeds the textbook asymptotic standard error (0.110), and the MC sd of rho_hat (0.149) is larger than that asymptotic s.e.\",\"results_keys\":[\"bias_mc\",\"se_asymptotic\",\"rho_hat_sd_mc\"],\"status\":\"supported\",\"caveat\":\"Design-specific.\"},\n {\"id\":\"C5\",\"text\":\"The first-order formula is accurate in general for other (n, rho) designs.\",\"results_keys\":[],\"status\":\"not_evaluated\",\"caveat\":\"Only one design point was simulated; no evidence on other designs.\"}\n]\nEOF\n~/wp29/venv/bin/python -c \"import json;json.load(open('out/claims.json'))\"; ls out"},"caller":{"type":"direct"}}],"container":null,"stop_reason":"tool_use","stop_sequence":null,"stop_details":null,"usage":{"input_tokens":2,"cache_creation_input_tokens":251,"cache_read_input_tokens":6141,"output_tokens":2834,"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":251},"inference_geo":"global","iterations":[],"speed":"standard","fallback_credit":null},"input_transformations":[],"diagnostics":null,"context_management":null},"apiBlockIndex":0,"requestId":"req_011CgXrBMcBen62a2rzSqn1Y","requestRef":"20528c9e-96ee-40d8-8d6f-adf768bb786c","requestedModel":"claude-opus-5-5","type":"assistant","uuid":"4c6e65db-9c11-420c-982d-a93dc84e45e6","timestamp":"2026-10-30T05:13:03.122Z","effort":"medium","perTurnEffort":"medium","userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0248","sessionId":"af504bbe-3d43-4f25-89cc-e2d040b54bbc","version":"2.1.314","gitBranch":"HEAD"}
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Random draws happen in this order: first the vector of y_1 for all R replications, then an R×(n−1) matrix of innovations. Each replication gives a new series of 30 observations, and the regression uses t = 2,…,30 (29 observations). No replication was excluded: all 20,000 ρ̂ are finite. I found no data issues.\n\n## Method\n- ρ̂ is the OLS slope with an intercept, computed in vectorised form from demeaned (y_{t−1}, y_t). On replication 1 I checked it against an explicit least-squares fit with `lstsq`.\n- bias_mc = mean(ρ̂) − ρ. Its Monte Carlo standard error is sd(ρ̂)/√R.\n- bias_analytical = −(1+3ρ)/n, evaluated at n = 30 (the design's series length), as the task specifies.\n- **What kind of formula this is.** −(1+3ρ)/n is the leading O(1/n) term of an asymptotic expansion of E[ρ̂] − ρ. The full bias is −(1+3ρ)/n + O(n⁻²). The formula is accurate only in the sense that the leftover error shrinks faster than 1/n as n → ∞ with ρ held fixed, |ρ| < 1. It is not an exact finite-sample result, and it gives no guaranteed error bound at any particular n. The neglected higher-order terms grow as ρ approaches 1, because the expansion is not uniform in ρ near the unit root. At n = 30 the formula is also ambiguous to order O(n⁻²) depending on whether n means the series length (30) or the number of regression observations (29). I report the n = 30 version as the main figure and the n − 1 version as a supplement.\n\n## Results\n| quantity | value |\n|---|---|\n| mean ρ̂ | 0.6763 |\n| median ρ̂ | 0.6987 |\n| sd ρ̂ (MC) | 0.1489 |\n| bias_mc | −0.1237 (MC s.e. 0.00105) |\n| bias_analytical, −(1+3ρ)/30 | −0.1133 |\n| discrepancy | −0.0104, i.e. −9.85 MC s.e. (the approximation captures about 92% of the bias; it is off by 9.2% relative to its own value) |\n| supplementary: −(1+3ρ)/29 | −0.1172 (discrepancy −6.14 MC s.e.) |\n| textbook asymptotic s.e. √((1−ρ²)/n) | 0.1095 |\n| share of ρ̂ < ρ | 79.3% |\n\nThe bias is large in practical terms. It equals about 1.13 times the textbook asymptotic standard error and about 0.83 times the Monte Carlo sd of ρ̂. The Monte Carlo sd (0.149) is also about 36% larger than the asymptotic s.e. (0.110), so at this design the asymptotic s.e. understates the actual sampling variability.\n\nThe analytical formula has the right sign and gets the order of magnitude right: it accounts for roughly nine tenths of the bias. However, it understates the size of the bias, and the gap of about 10 MC standard errors is far too large to be Monte Carlo noise. This fits the formula's nature as a first-order approximation. At n = 30 with ρ = 0.8, the O(n⁻²) terms it leaves out are about 0.01, and they push in the same direction (more downward bias). The approximation is therefore useful for rough work at this design but not exact.\n\n## Uncertainty\nThe Monte Carlo s.e. of bias_mc is 0.00105, so a 95% interval for the true bias is about −0.1237 ± 0.0021, or [−0.1258, −0.1216]. The analytical value −0.1133 lies well outside this interval. The result depends on a single seed, as the design requires. Another seed would move bias_mc by an amount on the order of 0.001 and would not change the conclusion.\n\n## Limitations and open issues\n- The Monte Carlo evidence applies to one design point only (n = 30, ρ = 0.8, Gaussian errors, stationary start, c = 0, intercept included). It cannot show how accurate the formula is at other values of n or ρ, under non-Gaussian errors, or with other initial conditions. It also cannot by itself confirm the asymptotic O(n⁻²) rate of the leftover error, which would require a grid over n. What it does show is that, at this design, the first-order approximation is detectably biased toward zero by about 0.010, with an MC error of about 0.001.\n- ρ̂ does not depend on σ, so the choice σ = 1 does not matter. ρ̂ also does not depend on c, because the regression includes an intercept and the start is stationary.\n- The formula is ambiguous between n and n − 1. Even the more favourable n − 1 version is rejected at about 6 MC s.e.\n- Bias is only one feature of the distribution. ρ̂ is left-skewed (median 0.699 > mean 0.676), so bias-corrected point estimates or normal-approximation intervals built from the asymptotic s.e. would still be inaccurate at this sample size.\nEOF\ncat > out/claims.json <<'EOF'\n[\n {\"id\":\"C1\",\"text\":\"At n=30, rho=0.8 the OLS estimator of rho is substantially downward biased: Monte Carlo bias -0.124 (MC s.e. 0.001), mean rho_hat 0.676.\",\"results_keys\":[\"bias_mc\",\"bias_mc_se\",\"rho_hat_mean\"],\"status\":\"supported\",\"caveat\":\"Single design point and single seed.\"},\n {\"id\":\"C2\",\"text\":\"The Marriott-Pope/Kendall approximation -(1+3rho)/n = -0.113 has the correct sign and captures roughly 90% of the bias.\",\"results_keys\":[\"bias_analytical\",\"bias_mc\"],\"status\":\"supported\",\"caveat\":\"Uses n=30 (series length); with n-1=29 the value is -0.117.\"},\n {\"id\":\"C3\",\"text\":\"The approximation understates the magnitude of the bias by about 0.010, a discrepancy of about -9.9 Monte Carlo standard errors, which is not attributable to simulation noise and is consistent with omitted O(n^-2) terms.\",\"results_keys\":[\"discrepancy_se_units\",\"bias_mc\",\"bias_analytical\",\"bias_mc_se\"],\"status\":\"supported\",\"caveat\":\"Attribution to O(n^-2) terms is an interpretation; MC at one n cannot verify the rate.\"},\n {\"id\":\"C4\",\"text\":\"The bias is of the same order as the sampling uncertainty: it exceeds the textbook asymptotic standard error (0.110), and the MC sd of rho_hat (0.149) is larger than that asymptotic s.e.\",\"results_keys\":[\"bias_mc\",\"se_asymptotic\",\"rho_hat_sd_mc\"],\"status\":\"supported\",\"caveat\":\"Design-specific.\"},\n {\"id\":\"C5\",\"text\":\"The first-order formula is accurate in general for other (n, rho) designs.\",\"results_keys\":[],\"status\":\"not_evaluated\",\"caveat\":\"Only one design point was simulated; no evidence on other designs.\"}\n]\nEOF\n~/wp29/venv/bin/python -c \"import json;json.load(open('out/claims.json'))\"; ls out"}}]}}
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{"parentUuid":"0395ba92-e04d-4d60-96ec-503ca2aa44a0","isSidechain":false,"message":{"model":"claude-opus-5-5","id":"msg_011CgXrDGxwMybZ7m8dQPYAz","type":"message","role":"assistant","content":[{"type":"text","text":"I ran the Monte Carlo exactly as the design specifies (n = 30, ρ = 0.8, σ = 1, intercept included, stationary start, R = 20,000, seed 20261029). The main finding is that the analytical approximation gets the sign and rough size of the bias right but understates it by an amount far too large to be simulation noise. All four deliverables are written, and re-running the script reproduces `out/results.json` byte for byte.\n\n| Quantity | Value |\n|---|---|\n| Mean of ρ̂ | 0.676 |\n| Simulated bias | −0.1237 (standard error 0.00105) |\n| Analytical bias, −(1+3ρ)/n with n = 30 | −0.1133 |\n| Gap between them | −0.0104, about −9.9 standard errors |\n| Sd of ρ̂ across replications | 0.149 |\n| Textbook asymptotic standard error | 0.110 |\n\n- **How close the formula is:** it captures about 92% of the bias, but the remaining gap of about 0.010 sits far outside the simulation's uncertainty.\n- **Why it misses:** the formula is only the leading term of a large-sample expansion. It is not exact at n = 30, and the terms it drops get larger as ρ approaches 1. Here they push the same way, making the bias more negative.\n- **Which n to use:** the series has 30 observations but the regression uses 29. I used n = 30 as the task asks. With n = 29 the formula gives −0.117, which is still about 6.1 standard errors from the simulated value; I report this as a supplement.\n- **Size relative to uncertainty:** the bias is larger than the textbook standard error, and the actual spread of ρ̂ is about 36% wider than that standard error suggests. The estimates are also skewed to the left (median 0.699 against mean 0.676).\n\nNo data problems were found and no replications were dropped. The main limitation is that this is one parameter setting with one seed. It says nothing about how accurate the formula is at other sample sizes or values of ρ, and it can't confirm how fast its error shrinks as n grows. The report states this explicitly.\n\nFiles are in `out/`:\n- `analysis.py`\n- `results.json`\n- `report.md`\n- `claims.json`"}],"container":null,"stop_reason":"end_turn","stop_sequence":null,"stop_details":null,"usage":{"input_tokens":2,"cache_creation_input_tokens":2857,"cache_read_input_tokens":6392,"output_tokens":789,"output_tokens_details":{"thinking_tokens":46},"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":2857},"inference_geo":"global","iterations":[],"speed":"standard","fallback_credit":null},"input_transformations":[],"diagnostics":null,"context_management":null},"apiBlockIndex":1,"requestId":"req_011CgXrDGh4kETf4JQd4AgbV","requestRef":"8a239731-8bb5-4ea6-9988-1c49adac764b","requestedModel":"claude-opus-5-5","type":"assistant","uuid":"c5af147e-0e4d-4984-a142-6f6d77ce2616","timestamp":"2026-10-30T05:13:11.825Z","effort":"medium","perTurnEffort":"medium","userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0248","sessionId":"af504bbe-3d43-4f25-89cc-e2d040b54bbc","version":"2.1.314","gitBranch":"HEAD"}
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