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

# Question

In y_t = c + ρ y_{t−1} + ε_t, ε_t ~ N(0, σ²), how large is the finite-sample bias of the OLS estimator ρ̂ (regression on a constant and y_{t−1}) at the design n = 30, ρ = 0.8, and how does it compare with the Marriott–Pope/Kendall approximation E[ρ̂] − ρ ≈ −(1 + 3ρ)/n?

# Data and exclusions

Only data/design.csv (7 rows) was used: n = 30, ρ = 0.8, σ = 1, intercept included, stationary initial condition, R = 20,000 replications, seed 20261029 (numpy.random.default_rng). The data-generating process has c = 0. No data problems were found. No replications were excluded: all 20,000 estimates were finite.

# Method

For each replication, y_1 is drawn from N(0, σ²/(1−ρ²)) and y_2..y_30 are generated recursively with N(0, σ²) innovations. ρ̂ is the OLS slope of y_t on (1, y_{t−1}) for t = 2..30, so each regression uses 29 observations. Draw order: first all R initial values, then an R×(n−1) matrix of innovations. Bias_mc = mean(ρ̂) − ρ, with MC standard error sd(ρ̂)/√R.

What kind of formula the analytical one is. −(1+3ρ)/n is a first-order asymptotic expansion of the bias. It keeps the O(1/n) term, so E[ρ̂] − ρ = −(1+3ρ)/n + O(n⁻²). It is accurate in the sense that n·(bias) → −(1+3ρ) as n → ∞; the leftover error shrinks faster than 1/n. It is not an exact finite-sample formula. At a fixed small n, and especially for ρ near 1, the neglected higher-order terms can matter. The "n" in the formula is also defined only up to O(1) shifts: using n or n−1 gives the same first-order result.

# Results

Supplementary: median ρ̂ = 0.699; 79.3% of the estimates fall below ρ. The MC bias is 1.09 times the analytical value. With n−1 = 29 in the denominator, the approximation gives −0.1172, which is still about 6 MC SEs from the MC estimate.

Interpretation: the bias is large and downward, about −0.12. That is roughly 15% of ρ and about 0.8 of a sampling standard deviation. The first-order formula gets the sign and size right, capturing about 92% of the bias. It understates the bias by about 0.010, and this gap is statistically clear (≈ 9.9 MC SEs). A gap of this size is what the O(n⁻²) remainder would produce: 1/n² ≈ 0.0011, so a coefficient of order 10 is enough, which is plausible for ρ = 0.8. The test rejects "the approximation is exact at n = 30". It does not show that the approximation is wrong as an asymptotic statement.

The sampling spread of ρ̂ (0.149) is also about 36% larger than the textbook asymptotic SE (0.110). So at this design the first-order theory understates both the bias and the dispersion.

# Uncertainty

Bias_mc has MC standard error 0.00105 (approximate 95% interval −0.1258 to −0.1216). This is the only uncertainty in the MC number: it is simulation noise, and it can be made arbitrarily small with more replications. The analytical value is deterministic. The discrepancy is ~10 SEs and is therefore not simulation noise. The result comes from one seed. A different seed would move bias_mc by about ±0.002.

# Limitations and open issues

• What the Monte Carlo can and cannot show. It estimates the exact bias at one design point: n = 30, ρ = 0.8, Gaussian innovations, stationary start, intercept included. It cannot show how the approximation performs at other (n, ρ), for non-Gaussian errors, under other initial conditions, or near the unit root. It also cannot confirm or refute the asymptotic validity of the formula. That would need a sequence of n values, e.g. checking that n²·(bias_mc − bias_analytical) stabilises; this was not done.
• Because the formula is first-order, a significant discrepancy at a fixed n is expected and does not contradict it. The discrepancy's "SE units" grow without bound as R increases, so their size reflects simulation precision, not how bad the approximation is. The size of the absolute gap (≈0.010, ≈8% of the bias) is the more meaningful measure.
• Bias is a mean. ρ̂ is left-skewed (median 0.699 > mean 0.676), so the bias does not describe the whole sampling distribution.
• Whether "n" means T = 30 or the 29 regression observations is a convention. We followed the task (n = 30) and reported the n−1 variant as supplementary.