1[2 {"id":"C1","text":"At n=30, rho=0.8 the Monte Carlo bias of OLS rho_hat is about -0.124 (MC SE 0.001), i.e. rho_hat underestimates rho on average (mean 0.676).","results_keys":["bias_mc","bias_mc_se","rho_hat_mean"],"status":"supported","caveat":"Single design point; Monte Carlo error only."},3 {"id":"C2","text":"The Marriott-Pope/Kendall approximation -(1+3rho)/n gives -0.1133 at the design, matching the sign and roughly the size of the MC bias (ratio 1.09).","results_keys":["bias_analytical","ratio_bias_mc_to_analytical"],"status":"supported","caveat":"Evaluated with n=30 (series length) as specified."},4 {"id":"C3","text":"The approximation understates the bias by about 0.010, a gap of about 9.9 MC standard errors, so it is not simulation noise; with n-1=29 the gap is about 6.1 SEs.","results_keys":["discrepancy_se_units","bias_mc_minus_analytical","discrepancy_se_units_neff"],"status":"supported","caveat":"Consistent with omitted O(n^-2) terms of a first-order expansion; does not show the formula is wrong."},5 {"id":"C4","text":"The bias is comparable in size to the asymptotic SE sqrt((1-rho^2)/n)=0.110, and the MC sampling SD (0.149) is about 1.36 times that SE.","results_keys":["se_asymptotic","rho_hat_sd_mc","ratio_sd_mc_to_se_asymptotic","bias_mc"],"status":"supported","caveat":"At this design only."},6 {"id":"C5","text":"The sampling distribution of rho_hat is left-skewed: the median (0.699) exceeds the mean and about 79% of estimates fall below rho.","results_keys":["rho_hat_median","rho_hat_mean","share_rho_hat_below_rho"],"status":"supported","caveat":"Descriptive, single design."}7]8