1[2 {"id":"C1","text":"At n=30, rho=0.8 the Monte Carlo estimate of the OLS bias of rho_hat (with intercept) is about -0.124 (mean rho_hat about 0.676), with MC standard error about 0.001.","results_keys":["bias_mc","rho_hat_mean","bias_mc_se"],"status":"supported","caveat":"Single design point; R=20000, seed 20261029."},3 {"id":"C2","text":"The Marriott-Pope/Kendall approximation -(1+3rho)/n gives -0.1133, which under-states the magnitude of the true bias by about 9%; the gap is about -9.9 MC standard errors and so is not Monte Carlo noise.","results_keys":["bias_analytical","bias_mc","discrepancy_se_units","supp_ratio_bias_mc_to_analytical"],"status":"supported","caveat":"The discrepancy is consistent with the O(n^-2) remainder of a first-order asymptotic expansion; it does not refute the formula's asymptotic validity."},4 {"id":"C3","text":"Using n-1=29 in the formula does not close the gap (about -6.1 MC standard errors remain).","results_keys":["supp_bias_analytical_n_minus_1","bias_mc","bias_mc_se"],"status":"supported","caveat":"The standardized gap for the n-1 variant is computed from these keys (recorded in checks); it is not a separate results key."},5 {"id":"C4","text":"The finite-sample sd of rho_hat (0.149) exceeds the asymptotic s.e. sqrt((1-rho^2)/n)=0.110 by a factor of about 1.36, and the bias is about the same size as the asymptotic s.e.","results_keys":["rho_hat_sd_mc","se_asymptotic","supp_ratio_sd_mc_to_se_asymptotic","bias_mc"],"status":"supported","caveat":"Descriptive, at this single design."}6]7