Properties of Predictors for Autoregressive Time Series

Abstract

The prediction of the (n + s)th observation of the pth order autoregressive process is investigated. The mean square of the predictor error through terms of order n ^—1, conditional on Y_n, Y _{n — 1}, …, Y _{n — p + 1}, is obtained for the stationary normal process. The mean squared error expression is similar to the usual regression formula for the variance of the predictor error. The usual regression formula for the estimated variance of a predictor error and its generalization to s-period prediction is shown to provide a consistent estimator of the mean squared error of the least squares predictor for both stationary and non-stationary processes.

Key Words:

• Prediction
• Autoregression
• Time series
• Nonstationary