# F5-conf-02 — Finite-sample bias of OLS ρ̂ in a Gaussian AR(1)

# Question

In the Gaussian AR(1) y_t = c + ρ y_{t−1} + ε_t, ε_t ~ N(0, σ²), OLS of y_t on (1, y_{t−1}) is biased downward in small samples. We compare the Marriott–Pope/Kendall approximation E[ρ̂] − ρ ≈ −(1+3ρ)/n with a Monte Carlo estimate of the actual bias at the design n = 30, ρ = 0.8 (design_n, design_rho).

# Data and exclusions

The only input is data/design.csv (7 rows, n_input_rows; SHA-256 in design_sha256): n = 30, ρ = 0.8, σ = 1 (design_sigma), intercept included, stationary initial condition, R = 20000 replications (n_replications), seed 20261029 (random_seed). No data problems were found (data_issues is empty). One exclusion rule (drop replications with a non-finite ρ̂) removed 0 replications, so n_analysed = 20000 (out/exclusions.csv).

# Method

• Analytical approximation (bias_analytical): −(1+3ρ)/n with n = 30, the series length given in the design. This formula is the leading term of an asymptotic expansion of E[ρ̂] in powers of 1/n, derived for a stationary Gaussian AR(1) with an estimated mean/intercept. Its error is O(n⁻²), so it is accurate as n → ∞ (relative error shrinks like 1/n). It is not an exact finite-sample result and gives no guarantee at any fixed n, especially when ρ is near 1, where higher-order terms are larger. Because the regression actually uses n − 1 = 29 pairs, and both n and n − 1 are consistent with the same first-order expansion, I also report the formula evaluated at n − 1 as a sensitivity check (bias_analytical_neff).
• Monte Carlo (bias_mc): for each of R = 20000 replications, y_1 ~ N(0, σ²/(1−ρ²)), then y_t = 0.8 y_{t−1} + ε_t for t = 2…30 (c = 0). ρ̂ is the OLS slope of y_t on (1, y_{t−1}) for t = 2…30. One (R × n) standard-normal matrix is drawn from numpy.random.default_rng(20261029). bias_mc = mean(ρ̂) − ρ; its MC standard error is sd(ρ̂)/√R (bias_mc_se). Baseline reference: the textbook asymptotic SE √((1−ρ²)/n) (se_asymptotic).
• Checks: a statsmodels OLS fit reproduces the first replications exactly, and a clean re-run gives byte-identical output (out/state/checks.md).

# Results

Interpretation:

• Size and direction of the bias. The OLS estimator is clearly biased downward. On average ρ̂ is about 0.68 rather than 0.8, a bias of −0.122 (bias_mc). That is larger than the asymptotic SE itself (se_asymptotic = 0.110). About 79% of the ρ̂ values fall below the true ρ (share_rho_hat_below_rho).
• How well the approximation does. It gets the sign and most of the size right: the actual bias is about 8% larger than predicted (bias_ratio_mc_to_analytical = 1.079).
• The gap is real, not simulation noise. The difference is −8.6 MC standard errors (discrepancy_se_units). Even with n − 1 in the formula, the gap is still −4.9 SE (discrepancy_se_units_neff).
• Why the gap is expected. A first-order O(1/n) approximation should leave a remainder of order 1/n². With n = 30 and ρ = 0.8, a remainder of about −0.009 is consistent with that. The gap is statistically significant but small in substantive terms. It does not show that the formula is "wrong"; it is the expected size of the higher-order terms.
• Spread of ρ̂. The sampling sd of ρ̂ is 1.35 times the asymptotic SE (sd_ratio_mc_to_asymptotic). The distribution is left-skewed: its 2.5%–97.5% range is 0.331 to 0.902 (rho_hat_q025, rho_hat_q975). So first-order asymptotics also understate the dispersion at this design.

# Uncertainty

The MC standard error of the bias is 0.00105 (bias_mc_se), giving a 95% interval for the true finite-sample bias of [−0.1244, −0.1203] (bias_mc_ci95). This interval excludes the analytical value −0.1133 (bias_analytical) and also its n−1 version −0.1172 (bias_analytical_neff). The analytical quantities are deterministic and have no sampling uncertainty. The results are exact for the fixed seed; a different seed would move bias_mc by roughly ±0.002 (2 × bias_mc_se).

# Limitations and open issues

• What the analytical formula is. −(1+3ρ)/n is a first-order asymptotic (large-n) approximation to the mean bias, with an O(n⁻²) remainder. It is accurate in the sense that its relative error vanishes as n grows. It carries no exact finite-sample guarantee, and its accuracy gets worse as ρ → 1 and as n gets smaller.

• What the Monte Carlo can establish. It estimates the bias at exactly one design point: n = 30, ρ = 0.8, Gaussian errors, stationary start, intercept included, c = 0. It cannot establish:

  • the approximation's accuracy at other (n, ρ) values;
  • behaviour with non-Gaussian errors or non-stationary starting values;
  • the general size of the higher-order terms.

  Making such claims would need a grid of designs or analytical higher-order expansions, which this task did not require.

• Ambiguity in "n". Whether n in the formula means the series length (30) or the number of regression pairs (29) is a matter of convention. I used 30 as the task specifies, and both versions are reported. Neither version falls within the MC confidence interval.

• Mean bias only. The bias is a statement about the mean. Because the ρ̂ distribution is skewed, the median bias (rho_hat_median − 0.8 ≈ −0.10) differs from the mean bias.