Open Access
Translator Disclaimer
September, 1989 A Fixed Point Characterization for Bias of Autoregressive Estimators
Robert A. Stine, Paul Shaman
Ann. Statist. 17(3): 1275-1284 (September, 1989). DOI: 10.1214/aos/1176347268


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.


Download Citation

Robert A. Stine. Paul Shaman. "A Fixed Point Characterization for Bias of Autoregressive Estimators." Ann. Statist. 17 (3) 1275 - 1284, September, 1989.


Published: September, 1989
First available in Project Euclid: 12 April 2007

zbMATH: 0681.62075
MathSciNet: MR1015150
Digital Object Identifier: 10.1214/aos/1176347268

Primary: 62M10

Keywords: autoregressive process , bias , contraction , Durbin-Levinson recursion , fixed point , least squares estimator , Yule-Walker estimator

Rights: Copyright © 1989 Institute of Mathematical Statistics


Vol.17 • No. 3 • September, 1989
Back to Top