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

## 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 slope ρ̂ (regression on a constant and y_{t−1}) at the design (n = 30, ρ = 0.8), and how does the Marriott–Pope/Kendall approximation −(1+3ρ)/n compare with a Monte Carlo estimate of the actual bias?

## Data and exclusions
Only `data/design.csv` (7 rows) was used: n = 30, ρ = 0.8, σ = 1, intercept included, stationary initial condition y_1 ~ N(0, σ²/(1−ρ²)), R = 20,000 replications, seed 20261029 (`numpy.random.default_rng`), c = 0. No data problems were found. All 20,000 replications gave a finite ρ̂, so none were excluded.

## Method
- **Analytical approximation:** −(1+3ρ)/n with n = 30, the series length, as specified in the task. This gives −0.11333. Kendall (1954) and Marriott & Pope (1954) derived it as a **first-order asymptotic expansion**: it is the O(1/n) term of the bias, and the terms it drops are O(n⁻²). It is not an exact finite-sample result. Its accuracy holds only in the limit: the error shrinks faster than 1/n as n → ∞. At any fixed n there is no guarantee about how accurate it is, and the guarantee gets weaker as ρ approaches 1, because the expansion is not uniform in ρ near the unit root. It is also ambiguous whether "n" means the T = 30 observations or the T−1 = 29 regression observations, and the two choices differ at O(n⁻²). With n−1 = 29 the approximation would be −0.11724; this is reported only as a supplementary value.
- **Monte Carlo:** The draws come from one (R × n) standard-normal matrix. Column 1 is scaled to give the stationary y_1, and the other columns are the innovations, applied with the AR recursion. In each replication, ρ̂ is the OLS slope of y_t on (1, y_{t−1}) for t = 2..n, computed as the within-sample demeaned cross-product ratio. bias_mc = mean(ρ̂) − ρ, and its MC standard error is sd(ρ̂)/√R.

## Results
| quantity | value |
|---|---|
| analytical bias −(1+3ρ)/n | −0.11333 |
| Monte Carlo bias | −0.12234 |
| MC standard error | 0.00105 |
| 95% MC interval | [−0.12439, −0.12029] |
| mean ρ̂ | 0.67766 (median 0.69898) |
| sd of ρ̂ across replications | 0.14792 |
| asymptotic SE √((1−ρ²)/n) | 0.10954 |
| (bias_mc − bias_analytical)/MC SE | −8.61 |

ρ̂ is strongly biased downward: on average it falls about 0.12 below the true value of 0.8, and 78.9% of the estimates are below ρ. The bias is about 0.83 times the sampling sd of ρ̂ and about 1.1 times the textbook asymptotic SE. The analytical approximation explains most of the bias, about 93% of it. The approximation understates the bias by about 0.009 (around 7–8% in relative terms). This gap is statistically distinguishable from zero in MC terms (8.6 MC SEs). It is not a Monte Carlo artefact. It is consistent with the size of the O(n⁻²) terms the first-order formula leaves out (1/n² ≈ 0.001 times a coefficient that grows with ρ). The sampling sd of ρ̂ is also clearly larger than the asymptotic SE (0.148 vs 0.110), so at this design the first-order asymptotics also understate the dispersion.

## Uncertainty
The MC standard error of the bias estimate is 0.00105, which makes the bias precise to about ±0.002 (95%). The approximation is a deterministic number with no sampling error. The discrepancy test therefore checks only the MC noise against the approximation's deterministic error. Results depend on the seed only through MC error of that size.

## Limitations and open issues
- The Monte Carlo result covers **only this design point** (n = 30, ρ = 0.8, Gaussian errors, stationary start, intercept included). It cannot show how accurate the formula is at other (n, ρ), with non-Gaussian errors, with other initial conditions, or with a trend in the regression. It cannot prove the formula's asymptotic properties either. Those come from the analytical derivation.
- A rejection at 8.6 MC SEs does not mean the approximation is "wrong". A first-order expansion is expected to have a non-zero error at finite n. With enough replications, the MC test will always detect that error eventually. What matters is the size of the error: here it is about 0.009, small compared with the bias itself (0.12) and with the sampling sd (0.15).
- The choice between n = 30 (used, as the task specifies) and n−1 = 29 changes the approximation by about 0.004. Even with 29, the gap from the MC estimate is still about −0.005 (around 4.9 MC SEs).
- Higher-order or exact bias computations, such as second-order expansions or numerical integration of the exact moments, were not performed.