Note on the bias in the estimation of the serial correlation coefficient of AR(1) processes

Abstract

We derive approximating formulas for the mean and the variance of an autocorrelation estimator which are of practical use over the entire range of the autocorrelation coefficient ρ. The least-squares estimator

$$\Sigma _{i = 1}^{n - 1} \in _i \in _{i = 1} /\Sigma _{i = 1}^{n - 1} \in _i^2 $$

(1)

is studied for a stationary AR(1) process with known mean. We use the second order Taylor expansion of a ratio, and employ the arithmetic—geometric series instead of replacing partial Cesaro sums. In case of the mean we derive Marriott and Pope’s (1954) formula, with (n - 1)^-1 instead of (n)^-1, and an additional term ∝ (n - 1)^-2. This new formula produces the expected decline to zero negative bias as ρ approaches unity. In case of the variance Bartlett’s (1946) formula results, with (n — 1)^-1 instead of (n)^-1. The theoretical expressions are corroborated with a simulation experiment. A comparison shows that our formula for the mean is more accurate than the higher-order approximation of White (1961), for ¦ρ¦> 0.88 and n ≥ 20. In principal, the presented method can be used to derive approximating formulas for other estimators and processes.