Innovation algorithm in ARMA process

Abstract

Most of the works in Time Series Analysis are based on the Auto Regressive Integrated Moving Average (ARIMA) models presented by Box and Jenkins(1976). If the data exhibits no apparent deviation from stationarity and if it has rapidly decreasing autocorrelation function then a suitable ARMA(p,q) model is fit to the given data. Selection of the orders of p and q is one of the crucial steps in Time Series Analysis. Most of the methods to determine p and q are based on the autocorrelation function and partial autocorrelation function as suggested by Box and Jenkins (1976). Many new techniques have emerged in the literature and it is found that most of them are of very little use in determining the orders of p and q when both of them are non-zero. The Durbin-Levinson algorithm and Innovation algorithm (Brockwell and Davis, 1987) are used as recursive methods for computing best linear predictors in an ARMA(p,q) model. These algorithms are modified to yield an effective method for ARMA model identification so that the values of order p and q can be determined from them. The new method is developed and its validity and usefulness is illustrated by many theoretical examples. This method can also be applied to any real world data.