Variability of ARMA Processes

In this chapter, the numerical and pictorial interpretation of the dependence of the standard deviation of the forecast error for the different types and orders of univariate autoregressive-moving average (ARMA) processes on the lead time and on the autocorrelations (in the domains of the permissible autocorrelations) are given. While the convenience of fitting a stochastic model enables us to estimate its accuracy for the only time series under consideration, the graphs in this chapter demonstrate such accuracy for all possible models of the first and second order. Such a study can help in evaluating the appropriateness of the presupposed model, in earring out the model identification procedure, in designing an experiment, and in optimally organizing computations (or electing not to do so). A priori knowledge of the theoretical values of a forecast’s accuracy indicates the reasonable limits of complicating the model and facilitates evaluation of the consequences of certain preliminary decisions concerning its application. The approach applied is similar to the methodology developed in Chapters 1 and 2. Because the linear process theory has been thoroughly discussed in the statistical literature (see, for example, Box and Jenkins, 1976; Kashyap and Rao, 1976; and so on), its principal concepts are presented in recipe form with the minimum of details necessary for understanding the computational aspects of the subject. Consider a discrete stationary random process zt with null expected value [E(zt) = 0] and autocovariance function . . . M(T) = σ2 ρ(T), (4.1) . . . where σ2 is the variance and ρ(T) is the autocorrelation function of zt. Let at be a discrete white noise process with a zero mean and a variance σ2a. Let us assume that processes zt and at are normally distributed and that their cross-covariance function Mza(T) = 0 if T > 0.