represents a generalized autoregressive conditionally heteroscedastic process of orders p and q, driven by a standard white noise.


represents a GARCH process with initial data init.


  • GARCHProcess is a discrete-time and continuous-state random process.
  • A process x[t] is a GARCH 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 {,y[-2],y[-1]} or a single-path TemporalData object with time stamps understood as {,-2,-1}.
  • A scalar GARCHProcess should have non-negative coefficients αi and βj and a positive coefficient κ.
  • GARCHProcess[q,p] represents a GARCH process of orders q and p for use in EstimatedProcess and related functions.
  • GARCHProcess can be used with such functions as RandomFunction, CovarianceFunction, and TimeSeriesForecast.


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Basic Examples  (3)

Simulate a GARCHProcess:

Unconditional mean and variance of a weakly stationary process:

With fixed initial values:

The observations are uncorrelated but dependent:

The squared values of the data are correlated:

Scope  (13)

Basic Uses  (8)

Simulate an ensemble of paths:

Simulate with arbitrary precision:

Simulate a weakly stationary process with given initial values:

A non-weakly stationary process:

An integrated GARCHProcess:

Explosive GARCHProcess:

Such a process is not second-order stationary:

Conditions for a GARCHProcess to be covariance-stationary:

Region of second-order stationarity for a GARCHProcess[1,1]:

Estimate a GARCHProcess:

Use maximum conditional likelihood:


Find the forecast 20 steps ahead:

Find the mean squared errors of the forecast:

The forecasted states are equal to zero, hence the forecasted standard deviation bounds are:

Plot the values with mean squared errors:

Process Slice Properties  (5)

Moments of a weakly stationary GARCH of orders :

Moment of a GARCH process with given initial conditions:



Region where kurtosis is defined:

Simulate slice distribution:

Probability density function of the sample:

Use the Monte Carlo method to calculate NProbability for slice distribution:

Calculate NExpectation:

Compare to the second Moment:

Properties & Relations  (3)

The values of a GARCHProcess are uncorrelated:

Corresponding ARMAProcess:

For a process with given initial values:

Squared values of a GARCHProcess follow an ARMAProcess:

CorrelationFunction and PartialCorrelationFunction of squared values:

The corresponding ARMA process:

CorrelationFunction and PartialCorrelationFunction of the ARMA process:

Wolfram Research (2014), GARCHProcess, Wolfram Language function,


Wolfram Research (2014), GARCHProcess, Wolfram Language function,


Wolfram Language. 2014. "GARCHProcess." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (2014). GARCHProcess. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2023_garchprocess, author="Wolfram Research", title="{GARCHProcess}", year="2014", howpublished="\url{}", note=[Accessed: 16-April-2024 ]}


@online{reference.wolfram_2023_garchprocess, organization={Wolfram Research}, title={GARCHProcess}, year={2014}, url={}, note=[Accessed: 16-April-2024 ]}