An Improved Estimation Method for Univariate Autoregressive Models

Abstract

Autoregressive models are important in describing the behaviour of the observed time series. One of the reasons is that a covariance stationary process can be approximated by an autoregressive model. Thus, e.g., the spectrum of a covariance stationary time series can be approximated by the spectrum of an autoregressive process. The estimation of the autoregressive parameters is therefore of special importance in time series analysis. Several methods have been introduced to estimate autoregressive models. The most popular method has been the Yule-Walker method. The Yule-Walker estimates for the autoregressive parameters are known to have poor statistical properties in certain cases. On the other hand, the Burg estimates have better statistical properties. For example the Burg estimates are less biased than the Yule-Walker estimates. In this paper an alternative to the Burg estimates will be introduced. In the proposed method the true correlation matrix of the lagged variables is calculated for the lags 1, 2,…. From each correlation matrix the corresponding partial autocorrelation can be calculated. These, on the other hand, will lead to autocorrelation estimates with improved statistical properties. From the autocorrelation estimates the autoregressive parameters can be estimated by solving the Yule-Walker equations. The statistical properties of the new estimates are studied by simulations.