A Fixed Point Characterization for Bias of Autoregressive Estimators

Abstract

Least squares estimators of the coefficients of an autoregression of known, finite order are biased to order $1/T$, where $T$ is the sample length, unless the observed time series is generated by a unique model for that order. The coefficients of this special model are the fixed point of a linear mapping defined by the bias of the least squares estimator. Separate results are given for models with known mean and unknown mean. The "fixed point models" for different orders of autoregression are least squares approximations to an infinite-order autoregression which is unique but for arbitrary scaling. Explicit expressions are given for the coefficients of the fixed point models at each order. The autocorrelation function and spectral density of the underlying infinite-order process are also presented. Numerical calculations suggest similar properties hold for Yule-Walker estimators. Implications for bootstrapping autoregressive models are discussed.