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

# Question

Gaussian AR(1) process y_t = c + ρ y_{t−1} + ε_t, with c = 0, ρ = 0.8, σ = 1, n = 30. The slope ρ̂ comes from OLS of y_t on (1, y_{t−1}). We compare two things: (i) the Marriott–Pope/Kendall approximation to the bias, −(1+3ρ)/n, and (ii) the actual bias E[ρ̂] − ρ, estimated by Monte Carlo (R = 20,000).

# Data and exclusions

The only input is data/design.csv (7 rows). It contains no real data, only simulated values. I followed the design exactly: y_1 is drawn from N(0, σ²/(1−ρ²)), the stationary distribution; innovations are N(0, 1); the seed is 20261029 with numpy.random.default_rng. Random draws happen in this order: first the vector of y_1 for all R replications, then an R×(n−1) matrix of innovations. Each replication gives a new series of 30 observations, and the regression uses t = 2,…,30 (29 observations). No replication was excluded: all 20,000 ρ̂ are finite. I found no data issues.

# Method

• ρ̂ is the OLS slope with an intercept, computed in vectorised form from demeaned (y_{t−1}, y_t). On replication 1 I checked it against an explicit least-squares fit with lstsq.
• bias_mc = mean(ρ̂) − ρ. Its Monte Carlo standard error is sd(ρ̂)/√R.
• bias_analytical = −(1+3ρ)/n, evaluated at n = 30 (the design's series length), as the task specifies.
• What kind of formula this is. −(1+3ρ)/n is the leading O(1/n) term of an asymptotic expansion of E[ρ̂] − ρ. The full bias is −(1+3ρ)/n + O(n⁻²). The formula is accurate only in the sense that the leftover error shrinks faster than 1/n as n → ∞ with ρ held fixed, |ρ| < 1. It is not an exact finite-sample result, and it gives no guaranteed 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. At n = 30 the formula is also ambiguous to order O(n⁻²) depending on whether n means the series length (30) or the number of regression observations (29). I report the n = 30 version as the main figure and the n − 1 version as a supplement.

# Results

The bias is large in practical terms. It equals about 1.13 times the textbook asymptotic standard error and about 0.83 times the Monte Carlo sd of ρ̂. The Monte Carlo sd (0.149) is also about 36% larger than the asymptotic s.e. (0.110), so at this design the asymptotic s.e. understates the actual sampling variability.

The analytical formula has the right sign and gets the order of magnitude right: it accounts for roughly nine tenths of the bias. However, it understates the size of the bias, and the gap of about 10 MC standard errors is far too large to be Monte Carlo noise. This fits the formula's nature as a first-order approximation. At n = 30 with ρ = 0.8, the O(n⁻²) terms it leaves out are about 0.01, and they push in the same direction (more downward bias). The approximation is therefore useful for rough work at this design but not exact.

# Uncertainty

The Monte Carlo s.e. of bias_mc is 0.00105, so a 95% interval for the true bias is about −0.1237 ± 0.0021, or [−0.1258, −0.1216]. The analytical value −0.1133 lies well outside this interval. The result depends on a single seed, as the design requires. Another seed would move bias_mc by an amount on the order of 0.001 and would not change the conclusion.

# Limitations and open issues

• The Monte Carlo evidence applies to one design point only (n = 30, ρ = 0.8, Gaussian errors, stationary start, c = 0, intercept included). It cannot show how accurate the formula is at other values of n or ρ, under non-Gaussian errors, or with other initial conditions. It also cannot by itself confirm the asymptotic O(n⁻²) rate of the leftover error, which would require a grid over n. What it does show is that, at this design, the first-order approximation is detectably biased toward zero by about 0.010, with an MC error of about 0.001.
• ρ̂ does not depend on σ, so the choice σ = 1 does not matter. ρ̂ also does not depend on c, because the regression includes an intercept and the start is stationary.
• The formula is ambiguous between n and n − 1. Even the more favourable n − 1 version is rejected at about 6 MC s.e.
• Bias is only one feature of the distribution. ρ̂ is left-skewed (median 0.699 > mean 0.676), so bias-corrected point estimates or normal-approximation intervals built from the asymptotic s.e. would still be inaccurate at this sample size.