An Algorithm of Identification Method of Autoregressive – Moving-Average Model, Generalizing the Yule–Walker Method, and its Implementation on Python

Abstract

The paper presents a detailed 11-step algorithm for practical implementation of a new identification method of autoregressive – moving-average model ARMA(n_ap, n_kc)  of prediction of stationary time series with arbitrary values of the orders n_ap, n_kc . The new method, published in the previous authors' works, is a generalization of the well-known Yule–Walker method. The algorithm is implemented under the condition, proven by the authors in the previous publications, that the optimal structure of the ARMA(n_ap, n_kc) model is the ARMA(3,3). A feature of this algorithm is that the parameters of the autoregressive component of the ARMA(3,3) model are determined using the fourth, fifth, and sixth autocovariances, which significantly distinguishes it from the traditional algorithm for identifying this class of models using the Yule–Walker method, which uses only autocovariances of the first, second, and third orders. Another feature of the algorithm is a straightforward procedure of determining the parameters of the moving average that does not require renewing the minimizing the residual sum of squares procedure when moving to other orders of the autoregressive and the moving average components, unlike the traditional approaches. The article presents a Python program implementation of the proposed identification algorithm and a demonstration of its effectiveness in solving the problem of identifying the ARMA(3,3) model for a specific time series given by an experimental implementation. The paper also determines the conditions for the experimental implementation of the time series to provide more accurate forecasting compared to the traditional approach.