1[2 {"id":"C1","text":"OLS with an intercept underestimates rho at n=30, rho=0.8: the MC bias is about -0.122 (mean rho_hat 0.678).","results_keys":["bias_mc","bias_mc_se","rho_hat_mean"],"status":"supported","caveat":"Valid for this single design point only."},3 {"id":"C2","text":"The analytical approximation -(1+3rho)/n = -0.1133 captures most of the bias, but the MC bias is about 8% larger in magnitude.","results_keys":["bias_analytical","bias_mc","supp_ratio_bias_mc_to_analytical"],"status":"supported","caveat":"The approximation is first-order; it is not intended to be exact at finite n."},4 {"id":"C3","text":"The discrepancy between MC and analytical bias is about -8.6 Monte Carlo SEs, so it is not simulation noise; it reflects neglected higher-order O(n^-2) terms.","results_keys":["discrepancy_se_units","bias_mc_se","supp_bias_analytical_neff"],"status":"supported","caveat":"Attributing the gap to O(n^-2) terms is an interpretation consistent with the theory; a grid over n would be needed to verify the rate."},5 {"id":"C4","text":"The bias is large relative to sampling variability (about 0.83 sd of rho_hat), and the MC sd of rho_hat exceeds the textbook asymptotic SE by about 35%.","results_keys":["supp_bias_mc_over_sd","rho_hat_sd_mc","se_asymptotic","supp_ratio_sd_mc_to_se_asymptotic"],"status":"supported","caveat":"Single design point."}6]7