Abstract: This paper addresses a long-standing conjecture that order 1/T bias mappings arising from Yule–Walker estimation of autoregressive coefficients are contractions, and that iteration of the order 1/T bias mapping gives convergence to a unique set of fixed-point process coefficients. The conjecture is easily proved for processes of order 1. We provide a proof and resolve this conjecture for order 2 processes. Although it is well-known that the Yule–Walker estimator can have substantial bias, the nature of the bias has often been only partially understood, and sometimes even misunderstood, in the literature. We argue that Yule–Walker fixed-point processes are key to understanding the nature of the bias. These processes provide essentially maximal separation of spectral peaks, and bias pulls Yule–Walker estimated coefficients toward those of the fixed-point process for the given order of autoregression and degree of polynomial trend for the process mean. In addition, we illustrate with a simulation that, in addition to unacceptable bias, the distribution of the Yule–Walker estimator can exhibit strong skewness and excessive kurtosis. This departure from normality can occur for very large sample sizes.

Key words and phrases: Autoregressive process, bias mapping, contraction, fixed-point process, Yule–Walker estimation.