GARCHProcessCopy to clipboard.
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GARCHProcess
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GARCHProcess[κ,{α1,…,αq},{β1,…,βp}]
represents a generalized autoregressive conditionally heteroscedastic process of orders p and q, driven by a standard white noise.
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GARCHProcess[κ,{α1,…,αq},{β1,…,βp},init]
represents a GARCH process with initial data init.
Details
- 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.
Examples
open allclose allBasic Examples (3)Summary of the most common use cases
Simulate a GARCHProcess:
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In[1]:=1
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https://wolfram.com/xid/0v51restwkdde-se76vc
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Out[1]=1
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In[2]:=2
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https://wolfram.com/xid/0v51restwkdde-4iewur
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Out[2]=2
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In[3]:=3
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https://wolfram.com/xid/0v51restwkdde-0lg2yf
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Out[3]=3
Unconditional mean and variance of a weakly stationary process:
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In[1]:=1
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https://wolfram.com/xid/0v51restwkdde-b7z11p
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Out[1]=1
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In[2]:=2
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https://wolfram.com/xid/0v51restwkdde-p99hv2
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Out[2]=2
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In[3]:=3
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https://wolfram.com/xid/0v51restwkdde-t8vwwp
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Out[3]=3
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In[4]:=4
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https://wolfram.com/xid/0v51restwkdde-lqhuff
Direct link to example
Out[4]=4
The observations are uncorrelated but dependent:
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In[1]:=1
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https://wolfram.com/xid/0v51restwkdde-kd62j1
Direct link to example
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In[2]:=2
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https://wolfram.com/xid/0v51restwkdde-2hau7
Direct link to example
Out[2]=2
The squared values of the data are correlated:
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In[3]:=3
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https://wolfram.com/xid/0v51restwkdde-4m7h01
Direct link to example
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In[4]:=4
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https://wolfram.com/xid/0v51restwkdde-ia2afh
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Out[4]=4
Scope (13)Survey of the scope of standard use cases
Basic Uses (8)
Simulate an ensemble of paths:
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In[1]:=1
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https://wolfram.com/xid/0v51restwkdde-bi9bqw
Direct link to example
Out[1]=1
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In[2]:=2
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https://wolfram.com/xid/0v51restwkdde-iv2byr
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Out[2]=2
Simulate with arbitrary precision:
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In[1]:=1
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https://wolfram.com/xid/0v51restwkdde-big92j
Direct link to example
Out[1]=1
Simulate a weakly stationary process with given initial values:
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In[1]:=1
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https://wolfram.com/xid/0v51restwkdde-qz5qde
Direct link to example
Copy to clipboard.
In[2]:=2
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https://wolfram.com/xid/0v51restwkdde-h1b72a
Direct link to example
Copy to clipboard.
In[3]:=3
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https://wolfram.com/xid/0v51restwkdde-l3esgs
Direct link to example
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In[4]:=4
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https://wolfram.com/xid/0v51restwkdde-2vkjih
Direct link to example
Out[4]=4
A non-weakly stationary process:
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In[5]:=5
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https://wolfram.com/xid/0v51restwkdde-849us2
Direct link to example
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In[6]:=6
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https://wolfram.com/xid/0v51restwkdde-gq6qdl
Direct link to example
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In[7]:=7
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https://wolfram.com/xid/0v51restwkdde-t1m0n8
Direct link to example
Out[7]=7
An integrated GARCHProcess:
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In[1]:=1
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https://wolfram.com/xid/0v51restwkdde-bp9na8
Direct link to example
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In[2]:=2
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https://wolfram.com/xid/0v51restwkdde-lqdz53
Direct link to example
Out[2]=2
Explosive GARCHProcess:
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In[1]:=1
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https://wolfram.com/xid/0v51restwkdde-vdahex
Direct link to example
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In[2]:=2
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https://wolfram.com/xid/0v51restwkdde-ptahjr
Direct link to example
Out[2]=2
Such a process is not second-order stationary:
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In[3]:=3
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https://wolfram.com/xid/0v51restwkdde-b12vkp
Direct link to example
Out[3]=3
Conditions for a GARCHProcess to be covariance-stationary:
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In[4]:=4
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https://wolfram.com/xid/0v51restwkdde-mutsnx
Direct link to example
Out[4]=4
Region of second-order stationarity for a GARCHProcess[1,1]:
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In[1]:=1
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https://wolfram.com/xid/0v51restwkdde-vxckai
Direct link to example
Out[1]=1
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In[2]:=2
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https://wolfram.com/xid/0v51restwkdde-i15pmg
Direct link to example
Out[2]=2
Estimate a GARCHProcess:
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In[1]:=1
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https://wolfram.com/xid/0v51restwkdde-ppxlwo
Direct link to example
Copy to clipboard.
In[2]:=2
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https://wolfram.com/xid/0v51restwkdde-fsgvlg
Direct link to example
Out[2]=2
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In[3]:=3
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https://wolfram.com/xid/0v51restwkdde-x957ms
Direct link to example
Out[3]=3
Use maximum conditional likelihood:
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In[4]:=4
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https://wolfram.com/xid/0v51restwkdde-7kln61
Direct link to example
Out[4]=4
Copy to clipboard.
In[1]:=1
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https://wolfram.com/xid/0v51restwkdde-eirvpx
Direct link to example
Find the forecast 20 steps ahead:
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In[2]:=2
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https://wolfram.com/xid/0v51restwkdde-dwbfbx
Direct link to example
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In[3]:=3
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https://wolfram.com/xid/0v51restwkdde-qqw7j5
Direct link to example
Out[3]=3
Find the mean squared errors of the forecast:
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In[4]:=4
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https://wolfram.com/xid/0v51restwkdde-zqnnuw
Direct link to example
Out[4]=4
The forecasted states are equal to zero, hence the forecasted standard deviation bounds are:
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In[5]:=5
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https://wolfram.com/xid/0v51restwkdde-wux3zt
Direct link to example
Plot the values with mean squared errors:
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In[6]:=6
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https://wolfram.com/xid/0v51restwkdde-pksy44
Direct link to example
Out[6]=6
Process Slice Properties (5)
Moments of a weakly stationary GARCH of orders 
:
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In[1]:=1
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https://wolfram.com/xid/0v51restwkdde-u1d2k5
Direct link to example
Copy to clipboard.
In[2]:=2
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https://wolfram.com/xid/0v51restwkdde-6vu1p7
Direct link to example
Out[2]=2
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In[3]:=3
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https://wolfram.com/xid/0v51restwkdde-1z4n08
Direct link to example
Out[3]=3
Moment of a GARCH process with given initial conditions:
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In[1]:=1
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https://wolfram.com/xid/0v51restwkdde-4vo40k
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
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https://wolfram.com/xid/0v51restwkdde-c76l3j
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
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https://wolfram.com/xid/0v51restwkdde-djic8g
Direct link to example
Out[3]=3
Copy to clipboard.
In[1]:=1
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https://wolfram.com/xid/0v51restwkdde-nz2z6e
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
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https://wolfram.com/xid/0v51restwkdde-vw5ylw
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
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https://wolfram.com/xid/0v51restwkdde-diexmz
Direct link to example
Out[3]=3
Region where kurtosis is defined:
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In[4]:=4
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https://wolfram.com/xid/0v51restwkdde-y5r0ud
Direct link to example
Out[4]=4
Copy to clipboard.
In[1]:=1
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https://wolfram.com/xid/0v51restwkdde-zmyzm4
Direct link to example
Probability density function of the sample:
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In[2]:=2
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https://wolfram.com/xid/0v51restwkdde-s3nc1d
Direct link to example
Out[2]=2
Use the Monte Carlo method to calculate NProbability for slice distribution:
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In[1]:=1
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https://wolfram.com/xid/0v51restwkdde-60em1m
Direct link to example
Copy to clipboard.
In[2]:=2
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https://wolfram.com/xid/0v51restwkdde-00997i
Direct link to example
Out[2]=2
Calculate NExpectation:
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In[3]:=3
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https://wolfram.com/xid/0v51restwkdde-n91pf6
Direct link to example
Out[3]=3
Compare to the second Moment:
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In[4]:=4
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https://wolfram.com/xid/0v51restwkdde-yzt5wc
Direct link to example
Out[4]=4
Properties & Relations (3)Properties of the function, and connections to other functions
The values of a GARCHProcess are uncorrelated:
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In[1]:=1
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https://wolfram.com/xid/0v51restwkdde-8tryi9
Direct link to example
Corresponding ARMAProcess:
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In[1]:=1
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https://wolfram.com/xid/0v51restwkdde-dfyenx
Direct link to example
Out[1]=1
For a process with given initial values:
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In[2]:=2
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https://wolfram.com/xid/0v51restwkdde-z1tydk
Direct link to example
Out[2]=2
Squared values of a GARCHProcess follow an ARMAProcess:
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In[1]:=1
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https://wolfram.com/xid/0v51restwkdde-rhmavw
Direct link to example
Copy to clipboard.
In[2]:=2
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https://wolfram.com/xid/0v51restwkdde-inxryx
Direct link to example
CorrelationFunction and PartialCorrelationFunction of squared values:
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In[3]:=3
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https://wolfram.com/xid/0v51restwkdde-ntjl10
Direct link to example
Out[3]=3
The corresponding ARMA process:
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In[4]:=4
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https://wolfram.com/xid/0v51restwkdde-x7lwn4
Direct link to example
Out[4]=4
CorrelationFunction and PartialCorrelationFunction of the ARMA process:
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In[5]:=5
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https://wolfram.com/xid/0v51restwkdde-sc2499
Direct link to example
Out[5]=5
Wolfram Research (2014), GARCHProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/GARCHProcess.html.
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Wolfram Research (2014), GARCHProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/GARCHProcess.html.Text
Wolfram Research (2014), GARCHProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/GARCHProcess.html.
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Wolfram Research (2014), GARCHProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/GARCHProcess.html.CMS
Wolfram Language. 2014. "GARCHProcess." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/GARCHProcess.html.
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Wolfram Language. 2014. "GARCHProcess." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/GARCHProcess.html.APA
Wolfram Language. (2014). GARCHProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GARCHProcess.html
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Wolfram Language. (2014). GARCHProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GARCHProcess.htmlBibTeX
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@misc{reference.wolfram_2025_garchprocess, author="Wolfram Research", title="{GARCHProcess}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/GARCHProcess.html}", note=[Accessed: 02-April-2025
]}BibLaTeX
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@online{reference.wolfram_2025_garchprocess, organization={Wolfram Research}, title={GARCHProcess}, year={2014}, url={https://reference.wolfram.com/language/ref/GARCHProcess.html}, note=[Accessed: 02-April-2025
]}