Autoregressive Moving-Average Process

An n-dimensional autoregressive moving-average process of orders p and q, ARMA(p,q), is a stochastic process of the form

[1]

where a is an n-dimensional vector, the and are nn matrices, and W is n-dimensional white noise (see the notation conventions documentation). As the name suggests, this combines an AR(p) model with an MA(q) model of the same dimension n. In applications, ARMA(1,1) processes are common.

Exhibit 1 indicates a realization of the univariate ARMA(1,1) process

[2]

where W is variance 1 Gaussian white noise.

ARMA Process Exhibit 1

A realization of the ARMA(1,1) process [2].

Exercises

Below are indicated a realization of 50 consecutive terms of a variance 1 Gaussian white noise. 0.293 0.317 0.047 -0.286 -1.237 -0.554 0.535 -1.640 -0.899 -0.704 -1.886 0.271 0.418 1.651 0.078 0.528 1.013 2.296 0.086 1.471 -0.580 -1.776 -2.217 0.502 -1.104 -1.211 0.205 0.110 0.011 0.778 -1.036 1.195 -1.169 -0.162 -0.504 -0.679 -1.366 0.885 -0.476 1.644 -1.665 0.129 2.882 0.978 0.054 -0.396 0.685 1.403 -0.009 0.918 Realization of 50 consecutive terms of a variance 1 Gaussian white noise. Use this to generate a corresponding realization of the ARMA(1,1) process [e1] where tW is a variance 1 Gaussian white noise. Initialize the realization with term 0x = 0. [spreadsheet solution]