Bias of Autoregressive Spectral Estimators

Abstract

Bias of the least squares estimator of the log of the spectral density of an autoregression attenuates the peaks of the estimator. Under the assumption of an autoregressive generating process of known finite order, we obtain an expression for the order 1/T bias, where T is the sample length of the observed series. This approximation is a sum of several simple functions of the unknown coefficients. When the spectral density has sharp peaks, one of these functions dominates the bias. The attenuation from this dominant component can be substantial when the spectral peak is well defined, and several examples illustrate this effect. Since the integral of the order 1/T bias components that are frequency dependent is 0, unbiased estimation of entropy to this order is possible for autoregressive processes. These bias expressions extend to autoregressive models in which the mean is a polynomial function of time. Similar results obtain for the log of the Yule-Walker spectral estimator, for which the order 1/T bias is typically larger than that of the least squares estimator. Simulation results using 10,000 Gaussian series confirm the accuracy of the analytic bias approximations.

Key Words:

• Autoregressive process
• Bias
• Entropy
• Least squares autoregressive spectral estimator
• Yule-Walker estimator