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keywords=conditionally heteroscedastic process, conditionally heteroscedastic time series, GARCHProcess, generalized ARCH, generalized autoregressive conditionally heteroscedastic, generalized conditionally heteroscedastic process, generalized conditionally heteroscedastic time series;
description=GARCHProcess[\[Kappa], \[Alpha]1, ..., \[Alpha]q , \ \[Beta]1, ..., \[Beta]p ] represents a generalized autoregressive conditionally heteroscedastic process of orders p and q, driven by a standard white noise. GARCHProcess[\[Kappa], \[Alpha]1, ..., \[Alpha]q , \ \[Beta]1, ..., \[Beta]p , init] represents a GARCH process with initial data init.;

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forms on wolfram websites learn how wolfram language system documentation center close garchprocess see also archprocess armaprocess arprocess maprocess timeseriesmodelfit related guides time series processes see also archprocess armaprocess arprocess maprocess timeseriesmodelfit related guides time series processes garchprocess κ α 1 α q β 1 β p represents a generalized autoregressive conditionally heteroscedastic process of orders p and q driven by a standard white noise garchprocess κ α 1 α q β 1 β p init represents a garch process with initial data init details examples basic examples scope basic uses process slice properties properties relations see also related guides history cite this page built in symbol see also archprocess armaprocess arprocess maprocess timeseriesmodelfit related guides time series processes see also archprocess armaprocess arprocess maprocess timeseriesmodelfit related guides time series processes garchprocess garchprocess κ α 1 α q β 1 β p represents a generalized autoregressive conditionally heteroscedastic process of orders p and q driven by a standard white noise garchprocess κ α 1 α q β 1 β p 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 α i and β j 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 all close all basic examples 3 simulate a garchprocess wolfram language code randomfunction garchprocess 2 1 2 0 10 wolfram language code path wolfram language code listplot randomfunction garchprocess 2 1 2 0 100 filling axis unconditional mean and variance of a weakly stationary process wolfram language code mean garchprocess κ subscript α 1 subscript α 2 subscript β 1 subscript β 2 subscript β 3 t wolfram language code variance garchprocess κ subscript α 1 subscript α 2 subscript β 1 subscript β 2 subscript β 3 t with fixed initial values wolfram language code mean garchprocess 1 1 3 1 4 subscript x 1 t wolfram language code variance garchprocess 1 1 3 1 4 subscript x 1 t the observations are uncorrelated but dependent wolfram language code data randomfunction garchprocess 1 1 3 2 10 3 corr correlationfunction data 20 wolfram language code listplot corr filling axis plotrange all the squared values of the data are correlated wolfram language code corr2 correlationfunction data 2 20 wolfram language code listplot corr2 filling axis plotrange all scope 13 basic uses 8 simulate an ensemble of paths wolfram language code data randomfunction garchprocess 3 5 2 0 30 4 wolfram language code listlineplot data filling axis simulate with arbitrary precision wolfram language code randomfunction garchprocess 1 3 1 10 1 4 5 workingprecision 20 path simulate a weakly stationary process with given initial values wolfram language code sproc x_ garchprocess 03 4 3 x wolfram language code pts 1 0 2 wolfram language code samples table seedrandom 4 randomfunction sproc x 20 x pts wolfram language code listlineplot samples datarange 0 12 plotlegends stringjoin x tostring pts a non weakly stationary process wolfram language code tproc x_ garchprocess 03 7 8 x wolfram language code tsamples table seedrandom 4 randomfunction tproc x 20 x pts wolfram language code listlineplot tsamples datarange 0 12 plotrange all plotlegends stringjoin x tostring pts an integrated garchprocess wolfram language code α 4 proc garchprocess 1 α 1 α wolfram language code listplot randomfunction proc 0 100 filling axis explosive garchprocess wolfram language code proc garchprocess 2 3 7 4 2 wolfram language code listplot randomfunction proc 0 100 filling axis such a process is not second order stationary wolfram language code weakstationarity proc conditions for a garchprocess to be covariance stationary wolfram language code weakstationarity garchprocess κ subscript α 1 subscript α 2 subscript β 1 subscript β 2 region of second order stationarity for a garchprocess 1 1 wolfram language code cond weakstationarity garchprocess κ α β wolfram language code regionplot cond α 0 1 β 0 1 framelabel automatic estimate a garchprocess wolfram language code seedrandom 34 data randomfunction garchprocess 3 2 4 6 10 2 wolfram language code tsm timeseriesmodelfit data garch 1 1 wolfram language code tsm process use maximum conditional likelihood wolfram language code estimatedprocess data garchprocess 1 1 processestimator maximumconditionallikelihood forecast wolfram language code proc garchprocess 3 5 4 data randomfunction proc 10 2 find the forecast 20 steps ahead wolfram language code forecast timeseriesforecast proc data 20 wolfram language code forecast path find the mean squared errors of the forecast wolfram language code errors forecast meansquarederrors the forecasted states are equal to zero hence the forecasted standard deviation bounds are wolfram language code ubound timeseriesmap sqrt errors lbound timeseriesmap sqrt errors plot the values with mean squared errors wolfram language code listlineplot data forecast lbound ubound plotstyle automatic automatic red red filling 3 4 process slice properties 5 moments of a weakly stationary garch of orders wolfram language code proc garchprocess k α β wolfram language code moment proc t 4 simplify wolfram language code cumulant proc 2 4 moment of a garch process with given initial conditions wolfram language code discreteplot moment garchprocess 1 4 1 t 4 t 0 4 wolfram language code moment garchprocess 06 0 5 3 2 4 wolfram language code cumulant garchprocess 3 1 4 2 2 4 skewness wolfram language code skewness garchprocess κ α β t wolfram language code skewness garchprocess κ α β x t kurtosis wolfram language code kurtosis garchprocess κ α β t region where kurtosis is defined wolfram language code regionplot not β 0 105 α 4 60 α 3 β 18 α 2 β 2 4 α β 3 β 4 1 α 0 1 β 0 1 framelabel automatic simulate slice distribution wolfram language code proc α_ garchprocess 3 α 2 sample α_ randomvariate proc α 3 10 4 probability density function of the sample wolfram language code r 1 3 5 7 histogram sample automatic pdf plotlabel stringjoin α tostring r use the monte carlo method to calculate nprobability for slice distribution wolfram language code proc garchprocess 1 3 2 wolfram language code nprobability x 1 3 x proc method montecarlo samplingincrement 10 4 calculate nexpectation wolfram language code nexpectation x 2 2 x proc method montecarlo samplingincrement 10 4 compare to the second moment wolfram language code moment proc 2 2 properties relations 3 the values of a garchprocess are uncorrelated wolfram language code correlation garchprocess 2 3 2 2 1 1 2 3 matrixform corresponding armaprocess wolfram language code armaprocess garchprocess 2 3 1 for a process with given initial values wolfram language code armaprocess garchprocess 2 3 1 3 2 squared values of a garchprocess follow an armaprocess wolfram language code proc garchprocess 1 2 3 4 wolfram language code data randomfunction proc 10 6 correlationfunction and partialcorrelationfunction of squared values wolfram language code datasq data 2 listplot datasq 1 30 filling axis plotrange 2 1 plotlabel correlationfunction partialcorrelationfunction the corresponding arma process wolfram language code arma armaprocess proc correlationfunction and partialcorrelationfunction of the arma process wolfram language code listplot arma 1 30 filling axis plotrange 2 1 plotlabel correlationfunction partialcorrelationfunction see also archprocess armaprocess arprocess maprocess timeseriesmodelfit related guides time series processes history introduced in 2014 10 0 cite this as 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 cms 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 bibtex misc reference wolfram_2026_garchprocess author wolfram research title garchprocess year 2014 howpublished url https reference wolfram com language ref garchprocess html note accessed 03 october 2026 biblatex online reference wolfram_2026_garchprocess organization wolfram 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