Moments of a Serial Correlation Coefficient

Abstract

Summary For a discrete first-order auto-regressive scheme (with initial value zero) xt = αxt-1 + ϵt, expressions have been found for the first four moments of α̂=∑1nxtxt−1/∑1n−1xt2. The expressions are of two kinds : (i) series in ascending powers of α2, (ii) asymptotic series in descending powers of n. A table of values of μ1‱,σ(α̂),β1(α̂),β2(α̂) is included. For the special case α2 = 1 the dominant terms in the first four moments are stated.