# Contract (P0)

Question. In a Gaussian AR(1) y_t = c + rho*y_{t-1} + e_t (c=0 in DGP), how large is the finite-sample bias of the OLS rho_hat (regression on constant and y_{t-1}, t=2..n), and how well does the Marriott-Pope/Kendall approximation -(1+3rho)/n describe it?

Estimands. (i) bias_analytical = -(1+3rho)/n at design (n=30, rho=0.8): a first-order (O(1/n)) asymptotic expansion of E[rho_hat]-rho, error O(n^-2), meaningful under stationarity, |rho|<1, Gaussian errors. (ii) bias_mc = mean(rho_hat)-rho over R replications, with MC SE sd/sqrt(R). Also rho_hat_mean, rho_hat_sd_mc, discrepancy_se_units=(bias_mc-bias_analytical)/bias_mc_se, se_asymptotic=sqrt((1-rho^2)/n).

Outputs. out/results.json (keys: task_id, bias_analytical, bias_mc, bias_mc_se, rho_hat_mean, rho_hat_sd_mc, discrepancy_se_units, se_asymptotic, n_replications, feasibility, feasibility_reason, data_issues, n_input_rows, n_analysed, random_seed), out/report.md (6 sections), out/analysis.py, out/claims.json, plus protocol files (state/, exclusions.csv).

Information rules. Only data/design.csv; follow design exactly: n=30, rho=0.8, sigma=1, intercept, stationary y_1, R=20000, seed 20261029 via numpy.random.default_rng; fresh series per replication.

Invalidating conditions. Missing/inconsistent design parameters; |rho|>=1 (no stationary distribution); deviation from seed/design. Note: MC discrepancy vs the approximation is not a "test" of the formula's correctness — it's an O(1/n) approximation, and a difference at fixed n is expected from higher-order terms; MC only establishes bias at this one design point.