# 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, σ = 1), how large is the finite-sample bias E[ρ̂] − ρ of the OLS estimator from a regression with intercept, at n = 30 and ρ = 0.8? How does a Monte Carlo estimate compare with the Marriott–Pope/Kendall approximation −(1 + 3ρ)/n?

# Data and exclusions

The only input is data/design.csv (7 rows): n = 30, ρ = 0.8, σ = 1.0, intercept = yes, stationary initial condition, R = 20,000 replications, seed = 20261029. I found no data issues and excluded nothing. No observed data are involved; everything is simulated.

# Method

• Simulation. All draws come from one numpy.random.default_rng(20261029) stream, as a single (R × n) array of standard normals. The first column sets the starting value, y_1 = z·σ/√(1−ρ²), which is a draw from the stationary distribution. After that, y_t = 0.8·y_{t−1} + σ·z_t for t = 2..30. Each replication is a fresh series of 30 observations.
• Estimation. For each replication, ρ̂ is the OLS slope from regressing y_t on (1, y_{t−1}) for t = 2..n, which uses 29 observations. It is computed in closed form from demeaned regressor and regressand.
• Monte Carlo quantities. bias_mc = mean(ρ̂) − ρ, with Monte Carlo SE = sd(ρ̂)/√R. Replications are independent, so this SE is valid.
• Nature of the analytical formula. −(1 + 3ρ)/n is a first-order asymptotic expansion: the leading O(1/n) term of the bias. The true bias equals −(1+3ρ)/n + O(n⁻²). The approximation is accurate only in the sense that the remainder shrinks faster than 1/n as n → ∞. It is not an exact finite-sample result. For a given n, how much the O(n⁻²) remainder matters grows as ρ approaches 1. The formula also leaves open whether "n" means the series length or the number of regression observations (n − 1). Both conventions agree to first order. Following the task, the headline uses n = 30, and n − 1 = 29 is reported as a supplement.

# Results

The OLS estimator is strongly biased downward. On average ρ̂ is about 0.68 when the true value is 0.8. The bias of about −0.12 is roughly 1.1 times the textbook asymptotic standard error, so it is large relative to sampling noise.

The analytical approximation captures most of the bias, but not all of it. The actual bias is about 8% larger in magnitude (ratio 1.08). The gap of −0.009 is 8.6 Monte Carlo standard errors, so Monte Carlo noise cannot explain it. Switching to the n − 1 convention narrows the gap to about −0.005, which is still about 5 SE. That is what you would expect from an O(n⁻²) remainder that has not vanished at n = 30 with ρ = 0.8.

The asymptotic SE also understates the actual sampling spread: 0.110 against an MC sd of 0.148.

# Uncertainty

• The Monte Carlo SE of the bias is 0.00105, giving a 95% CI of [−0.1244, −0.1203]. Replications are independent, so the plain sd/√R is the correct SE.
• The analytical value has no sampling error. Its error is approximation error.
• All results come from one seed. A different seed would move bias_mc by roughly ±0.002 (2 SE).

# Limitations and open issues

• Scope of the analytical formula. It is a large-n, first-order approximation. Here it misses at the level of the second-order term, which is what an O(1/n) expansion implies at small n and persistent ρ. It should not be read as an exact finite-sample bias.
• Scope of the Monte Carlo evidence. The simulation establishes the bias only at this one design: n = 30, ρ = 0.8, Gaussian innovations, intercept included, stationary start. It cannot say how the discrepancy behaves at other values of (n, ρ), with non-Gaussian errors, with a fixed initial condition, or with trends or other regressors. It also does not prove the analytical rate of the remainder; it only matches it at one point. Extending to a grid of (n, ρ) would be needed for general claims.
• The "n vs n − 1" convention in the formula is a matter of convention, and it changes the comparison noticeably (8.6 SE vs about 5 SE). Either way, the formula understates the magnitude of the bias.