Explicit weighting coefficients for predicting ARMA time series from the finite past

摘要

Explicit formulas are given for the weighting coefficients in the linear minimum variance predictor of a wide sense stationary autoregressive-moving average time series k steps ahead, given n + 1 successive observations of a realization of the process. The formulas involve determinants whose entries are the values of certain polynomials related to the autoregressive part of the process at the zeros of the polynomial that defines the moving average part. The number of observations n + 1 enters into the formulas as parameter, and not so as to increase their complexity as n grows large. Formulas are also given for the variance of the prediction.