# F5-conf-02 — Finite-sample bias of the OLS AR(1) estimator

# Question

In the Gaussian AR(1) y_t = c + ρ y_{t−1} + ε_t (c = 0 in the DGP), how large is the finite-sample bias of the OLS slope ρ̂ from regressing y_t on (1, y_{t−1}), t = 2..n, and how does it compare with the Marriott–Pope/Kendall approximation E[ρ̂] − ρ ≈ −(1 + 3ρ)/n at the design n = 30, ρ = 0.8?

# Data and exclusions

Single input data/design.csv (7 rows; n_input_rows), audited in out/state/data_audit.md: n = 30, ρ = 0.8, σ = 1, intercept included, stationary initial condition, R = 20000 replications, seed 20261029 (random_seed). No data issues were found (data_issues is empty). The exclusion ledger (out/exclusions.csv) has one rule, which drops replications with a non-finite ρ̂. It removed 0, so all n_replications = 20000 replications are analysed (n_analysed).

# Method

Each replication draws a fresh series. First y_1 ~ N(0, σ²/(1−ρ²)), the stationary distribution. Then y_t = ρ y_{t−1} + ε_t with ε_t ~ N(0, 1) for t = 2..30, using numpy.random.default_rng(20261029). All R initial values are drawn first, then an R×29 innovation matrix. ρ̂ is the OLS slope with an intercept, computed in closed form as the demeaned cross-product ratio over the 29 pairs. In the script, it is checked against numpy.linalg.lstsq for one replication. bias_mc = mean(ρ̂) − ρ. bias_mc_se = sd(ρ̂)/√R. discrepancy_se_units = (bias_mc − bias_analytical)/bias_mc_se. Baseline quantities are bias_analytical = −(1+3ρ)/n with n = 30 (the series length given in the design) and the textbook se_asymptotic = √((1−ρ²)/n).

What the analytical formula is. −(1+3ρ)/n is the leading term of an asymptotic expansion of the bias: E[ρ̂] − ρ = −(1+3ρ)/n + O(n⁻²). It is accurate in the sense that n × (error) → 0 as n → ∞ for fixed |ρ| < 1. It is not an exact finite-sample result and comes with no error bound at any given n. The neglected higher-order terms grow as ρ approaches 1, because the expansion is not uniform in ρ near the unit root. Whether n is the series length or the number of regression pairs (n − 1) makes no difference at first order. At n = 30, however, that choice shifts the formula by about 0.004 (supp_bias_analytical_n_minus_1).

# Results

• Analytical approximation: bias_analytical = −0.1133.
• Monte Carlo: rho_hat_mean = 0.6763, so bias_mc = −0.1237 with bias_mc_se = 0.00105. The 95% MC interval is [−0.1258, −0.1216] (supp_bias_mc_ci95_lower, supp_bias_mc_ci95_upper).
• Discrepancy: discrepancy_se_units = −9.85. The true bias is larger in magnitude than the first-order approximation, and the gap is far beyond Monte Carlo noise. The approximation understates the magnitude of the bias by about 8% (supp_relative_error_analytical = −0.084). With n − 1 = 29 in the formula, the gap shrinks but is still clearly detectable (−6.14 SE; supp_discrepancy_se_units_n_minus_1).
• The bias is large relative to the sampling spread. |bias_mc| ≈ 0.83 × rho_hat_sd_mc (0.149). The MC sd of ρ̂ is about 1.36 times the textbook se_asymptotic = 0.110 (supp_sd_ratio_mc_to_asymptotic). The median of ρ̂ is 0.699 (supp_rho_hat_median), so the distribution is left-skewed.

# Uncertainty

The only uncertainty is Monte Carlo error. The MC SE of the bias is bias_mc_se = 0.00105, roughly 1% of the bias. The comparison uses a normal approximation for the replication mean, which is well justified with R = 20000 i.i.d. replications. Results are exactly reproducible with the fixed seed. A different seed would move bias_mc by about ±0.002 (2 SE) and would not change the conclusion.

# Limitations and open issues

• The Monte Carlo evidence applies to this single design point only: n = 30, ρ = 0.8, Gaussian innovations, stationary start, c = 0 and an intercept in the regression. It shows that, at this point, the first-order formula misses an O(n⁻²) remainder that is statistically detectable and of practical size (about 0.01, or 8%). It cannot show how the error behaves at other n or ρ, under non-Gaussian errors, or with other initial conditions. It also cannot prove the asymptotic validity of the expansion, which is a theoretical result. A finding at one point does not mean the formula is "wrong": as a first-order approximation it is expected to be off by O(n⁻²).
• The formula's n was taken to be the series length (30), as stated in the design. The n − 1 variant is reported as a supplementary result only.
• The draw order (initial values first, then innovations) is one documented choice. Other valid orderings give statistically equivalent but not identical numbers.