1[2 {"id":"C1","text":"At n=30, rho=0.8 the OLS estimator of rho (with intercept) is biased downward by about 0.124 (mean rho_hat about 0.676), with MC SE about 0.001.","results_keys":["bias_mc","bias_mc_se","rho_hat_mean"],"status":"supported","caveat":"Specific to this design and the stated seed."},3 {"id":"C2","text":"The Marriott-Pope/Kendall approximation -(1+3rho)/n = -0.1133 captures the sign and roughly 92% of the size of the bias (MC bias / analytical about 1.09).","results_keys":["bias_analytical","bias_mc","supp_ratio_bias_mc_to_analytical"],"status":"supported","caveat":"Single design point."},4 {"id":"C3","text":"The gap between the MC bias and the analytical approximation (about -9.9 MC SEs) is statistically distinguishable from Monte Carlo noise; the first-order formula understates the bias size at this design.","results_keys":["discrepancy_se_units","bias_mc_se","supp_bias_mc_ci95"],"status":"supported","caveat":"Interpreting the gap as O(n^-2) higher-order terms is consistent with theory but not tested here (would require varying n)."},5 {"id":"C4","text":"Using n-1 instead of n in the formula reduces but does not eliminate the discrepancy (about -6.1 MC SEs).","results_keys":["supp_bias_analytical_n_minus_1","supp_discrepancy_se_units_n_minus_1"],"status":"supported","caveat":"Supplementary; the task defines the formula at n."},6 {"id":"C5","text":"The MC sd of rho_hat exceeds the asymptotic SE sqrt((1-rho^2)/n) by about 36%, and the distribution is left-skewed (median above mean).","results_keys":["rho_hat_sd_mc","se_asymptotic","supp_ratio_sd_mc_to_se_asymptotic","supp_rho_hat_median","rho_hat_mean"],"status":"supported","caveat":"Single design point."}7]8