Moving-Average Process

Explained:

moving-average process
 
   

An n-dimensional moving-average process of order q, MA(q), is a stochastic process of the form

 [1]

where a is an n-dimensional vector, the are nn matrices, and W is n-dimensional white noise (see the notation conventions documentation). The coefficients of [1] induce autocorrelations in an MA process. In applications, MA(1) and MA(2) processes are common.

Exhibit 1 indicates a realization of the univariate MA(2) process

[2]
 
   

where W is variance 1 Gaussian white noise.

Univariate Moving-Average Process
Exhibit 1

A realization of the MA(2) 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 MA(2) process

[e1]

where tW is a variance 1 Gaussian white noise. [spreadsheet solution]

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Related Internal Links

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autoregressive moving-average process A type of stochastic process.

autoregressive process A type of stochastic process.

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

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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copyright © Glyn A. Holton, 2006

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