# F5-conf-02 — Finite-sample bias of OLS for an AR(1) coefficient

## Question
In the Gaussian AR(1) y_t = c + ρ y_{t−1} + ε_t (c = 0 in the DGP, σ = 1), how large is the finite-sample bias of the OLS estimator ρ̂ from a regression with intercept, at n = 30 and ρ = 0.8, and how well does the Marriott–Pope/Kendall approximation E[ρ̂] − ρ ≈ −(1+3ρ)/n describe it?

## Data and exclusions
Only `data/design.csv` (7 rows; `n_input_rows`) was used; SHA-256 and audit are in `out/state/data_audit.md`. Design: n = 30, ρ = 0.8, σ = 1, intercept yes, stationary start, R = 20000 replications (`n_replications`), seed 20261029 (`random_seed`). No replication was excluded (`n_analysed` = 20000; `out/exclusions.csv`); `data_issues` is empty. One ambiguity: "n" in the formula could mean the series length (30) or the number of regression pairs (29). The main value uses n = 30, the design value; the n−1 version is given as a sensitivity check.

## Method
- **Baseline:** the textbook asymptotic standard error √((1−ρ²)/n) (`se_asymptotic`) and the naive "unbiased" benchmark (`baseline_naive_bias` = 0).
- **Analytical approximation:** `bias_analytical` = −(1+3ρ)/n.
- **Monte Carlo:** each replication draws y_1 ~ N(0, σ²/(1−ρ²)), then y_2..y_30 recursively with fresh N(0, σ²) innovations, using `numpy.random.default_rng(20261029)`. ρ̂ is the OLS slope of y_t on (1, y_{t−1}) for t = 2..n. `bias_mc` = mean(ρ̂) − ρ, with Monte Carlo SE sd(ρ̂)/√R (`bias_mc_se`).

**What kind of approximation the formula is.** −(1+3ρ)/n is the leading term of an asymptotic expansion of the bias in powers of 1/n: E[ρ̂] − ρ = −(1+3ρ)/n + O(n^−2). It is accurate in the sense that the remainder shrinks faster than 1/n as n → ∞ for fixed |ρ| < 1. It is not an exact finite-sample result and gives no error bound at any particular n. The neglected higher-order terms grow as ρ approaches 1, because the expansion is not uniform in ρ near the unit root. So the formula should be expected to be least accurate for small n and large ρ, which is the case in this design.

## Results
- Mean of ρ̂: 0.6763 (`rho_hat_mean`); median 0.6987 (`rho_hat_median`).
- Monte Carlo bias: −0.1237 (`bias_mc`), with MC SE 0.00105 (`bias_mc_se`).
- Analytical approximation: −0.1133 (`bias_analytical`).
- The difference is −9.85 MC standard errors (`discrepancy_se_units`). The formula understates the size of the bias by about 8.4% of the MC bias (`relative_error_analytical_vs_mc` = −0.084).
- **Sensitivity:** using the 29 effective pairs gives −0.1172 (`bias_analytical_effective_n_minus_1`), which is still −6.14 MC SEs away (`discrepancy_se_units_n_minus_1`).
- The bias is about 1.13 times the asymptotic SE of 0.1095 (`se_asymptotic`). The MC standard deviation of ρ̂ is 0.1489 (`rho_hat_sd_mc`), 1.36 times the asymptotic SE (`rho_hat_sd_ratio_to_asymptotic`).

**Interpretation.** The approximation gets the sign and order of magnitude right: it captures about 92% of the bias. The gap that remains is statistically unambiguous and is what one would expect from the O(n^−2) terms the formula leaves out at n = 30, ρ = 0.8. The bias is also large relative to sampling noise, so ignoring it would distort inference on ρ.

## Uncertainty
The only uncertainty is Monte Carlo error. `bias_mc_se` = 0.00105 gives an approximate 95% MC interval for the true bias of about −0.1237 ± 0.0021, i.e. [−0.1258, −0.1216]. The analytical value lies outside this interval. The analytical value and `se_asymptotic` are deterministic. The MC SE treats replications as iid, which holds by construction.

## Limitations and open issues
- The Monte Carlo result applies only to this one design point: n = 30, ρ = 0.8, Gaussian innovations, stationary start, intercept included, c = 0.
  - It cannot show how the formula's error behaves at other (n, ρ) values, with non-Gaussian errors, or with other initial conditions. Note that σ and c do not affect the distribution of ρ̂ when an intercept is included.
  - It does not prove that the formula is "wrong". The formula is a leading-order asymptotic approximation, and the simulation simply measures its finite-sample error at this point.
- The size of the discrepancy depends on whether "n" is read as 30 or 29. Both readings are reported, and both are rejected relative to MC precision.
- No bias-corrected estimator or higher-order expansion was evaluated; the task did not require it.