Statistica Sinica 29 (2019), 1831-1849

THE BIAS MAPPING OF THE YULE-WALKER

ESTIMATOR IS A CONTRACTION

Philip A. Ernst and Paul Shaman

Rice University and the University of Pennsylvania

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.