# Contract (P0) — F5-conf-02

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

Estimands. (i) bias_analytical = -(1+3rho)/n at design values (a deterministic number; an O(1/n) first-order asymptotic-expansion approximation, accurate up to O(n^-2) terms, and derived for |rho|<1 stationary case). (ii) bias_mc = E[rho_hat]-rho estimated by the mean over R replications, with MC SE = sd(rho_hat)/sqrt(R). Meaningful under: Gaussian iid innovations, stationary start, |rho|<1, OLS with intercept on t=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 (=sqrt((1-rho^2)/n)), n_replications, feasibility, feasibility_reason, data_issues, n_input_rows, n_analysed, random_seed. Plus out/report.md (6 sections), out/analysis.py (deterministic), out/claims.json, and 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 with numpy default_rng); fresh series each replication; OLS on (1, y_{t-1}) for t=2..n.

Invalidity / impossibility. Missing or inconsistent design parameters; |rho|>=1 (stationary start undefined); non-deterministic script; using a different n convention silently. Ambiguity noted: formula "n" is taken as the design n=30 (series length) as the task states "at the design values"; the effective regression sample n-1=29 is reported as a sensitivity only.