Autoregressive Moving-Average Process (ARMA)

Explained:

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. ARMA(1,1) processes are popular.

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]

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ARCH A category of conditionally heteroskedastic stochastic processes.

autoregressive process A type of stochastic process.

heteroskedasticity A condition where a stochastic process has non-constant second moments.

moving-average process A type of stochastic process.

random walk A discrete stochastic process whose increments form a white noise.

stochastic volatility model A category of conditionally heteroskedastic stochastic processes.

time series and stochastic processes An introductory article.

volatility A metric of  variability in a stochastic process.

white noise A simple form of stochastic process.

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