# Wolfram Language & System 10.3 (2015)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.
BUILT-IN WOLFRAM LANGUAGE SYMBOL

# ARCHProcess

ARCHProcess[κ,{α1,,αq}]
represents an autoregressive conditionally heteroscedastic process of order q, driven by a standard white noise.

ARCHProcess[κ,{α1,,αq},init]
represents an ARCH process with initial data init.

## DetailsDetails

• ARCHProcess is a discrete-time and continuous-state random process.
• A process is an ARCH process if the conditional mean Expectation[x[t] {x[t-1], }]=0 and the conditional variance given by Expectation [x[t]2{x[t-1, }] satisfies the equation .
• The initial data init can be given as a list or a single path TemporalData object with time stamps understood as .
• A scalar ARCH process can have non-negative coefficients and a positive coefficient κ.
• ARCHProcess[q] represents an ARCH process of order q for use in EstimatedProcess and related functions.
• ARCHProcess can be used with such functions as RandomFunction, CovarianceFunction, and TimeSeriesForecast.

## ExamplesExamplesopen allclose all

### Basic Examples  (3)Basic Examples  (3)

Simulate an ARCHProcess:

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Unconditional mean and variance of a weakly stationary process:

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With fixed initial values:

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The observations are uncorrelated but dependent:

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The squared values of the data are correlated:

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