# 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 estimator rho_hat (regression with intercept, t=2..n), and how does it compare with the Marriott–Pope/Kendall approximation -(1+3rho)/n?

Estimands. (i) bias_analytical = -(1+3rho)/n at design (n=30, rho=0.8) — a deterministic number; (ii) bias_mc = E[rho_hat]-rho estimated by MC mean over R replications, with MC SE = sd(rho_hat)/sqrt(R). Meaningful under: Gaussian iid innovations, stationary start (|rho|<1), design followed exactly. The analytical formula is a first-order (O(1/n)) asymptotic expansion of the bias; its error is O(n^-2), so at small n and rho near 1 a discrepancy is expected; it is not an exact finite-sample result.

Outputs. out/results.json (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 (Question; Data and exclusions; Method; Results; Uncertainty; Limitations and open issues); out/analysis.py; out/claims.json; plus protocol files out/exclusions.csv, out/state/*.

Information rules. Only data/design.csv; follow design exactly (n, rho, sigma, intercept, stationary init, R, seed with numpy default_rng); c=0; fresh series of n obs per replication; OLS of y_t on (1, y_{t-1}), t=2..n.

Invalidity / impossibility. Missing/inconsistent design parameters, |rho|>=1 with stationary start, non-reproducible seed usage, degenerate regressions (zero variance of y_{t-1}) — none expected.