1[2 {"id":"C1","text":"The Marriott–Pope/Kendall approximation gives a bias of −0.1133 at n=30, rho=0.8.","results_keys":["bias_analytical"],"status":"supported","caveat":"Evaluated with n = series length 30; with n-1=29 it is −0.1172 (bias_analytical_neff)."},3 {"id":"C2","text":"The Monte Carlo finite-sample bias of OLS rho_hat is −0.1237 (MC SE 0.0011), i.e. mean rho_hat ≈ 0.676 versus true 0.8.","results_keys":["bias_mc","bias_mc_se","rho_hat_mean","n_analysed"],"status":"supported","caveat":"Valid only for this single design point and seed; MC error only."},4 {"id":"C3","text":"The analytical approximation understates the bias magnitude by about 9%; the gap is about −9.9 MC standard errors and the MC 95% CI excludes the analytical value.","results_keys":["discrepancy_se_units","ratio_bias_mc_to_analytical","bias_mc_ci95_low","bias_mc_ci95_high","bias_analytical"],"status":"supported","caveat":"Statistically detectable O(n^-2) remainder of a first-order approximation, not evidence the formula is wrong; using n-1 the gap is −6.1 SE."},5 {"id":"C4","text":"The textbook asymptotic SE (0.110) understates the Monte Carlo sd of rho_hat (0.149) at n=30.","results_keys":["se_asymptotic","rho_hat_sd_mc","se_asymptotic_over_sd_mc"],"status":"supported","caveat":"Single design point."},6 {"id":"C5","text":"The sampling distribution of rho_hat is left-skewed, with ~79% of estimates below the true rho.","results_keys":["rho_hat_median","share_rho_hat_below_rho"],"status":"supported","caveat":"Descriptive, single design point."}7]8