The Yule-Walker Equations as a Least Squares Problem and the Need for Tapering

The most commonly used method for estimating the time domain parameters of an autoregressive process is to use the Yule-Walker equations. The Yule-Walker estimates of the parameters of an autoregressive process are known to often be highly biased. There is a Fourier transform relationship between the autocovariance sequence for an autoregressive process (the estimates of which are used in the Yule-Walker equations to estimate the time domain parameters) and the spectrum, a frequency domain representation of the autoregressive process. Tapering has been proven to reduce the bias of both the periodogram, a naive estimator of the spectrum, as well as the time domain parameter estimates. The reason for the reduction in bias in the periodogram is intuitive, but no intuitive reason for the similar reduction in the bias of the time domain estimators has previously been given. We provide insightful reasoning for tapering to reduce bias in the Yule-Walker estimates of autoregressive time series, and exploit this insight to form new data tapers. Finally, we perform a Monte Carlo simulation of various autoregressive processes and compare the traditional Yule-Walker parameter estimates formed with no taper to those estimated with traditional tapers and the tapers derived in this paper. keywords: autoregression; taper; Yule-Walker.