
2.4 Univariate ARCH and GARCH ModelsAutoregressive Conditional Heteroskedasticity (ARCH) models and Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models are used to model the changes in variance as a function of time. A GARCH( p, q) model is where {_{t}} is an independently distributed Gaussian random sequence with zero mean and unit variance; h_{t} is the conditional variance of Z_{t} conditional on all the information up to time t1, I_{t1}: When p=0, we have an ARCH( q) model. A GARCHinmean (or ARCHinmean) model is defined by That is, the conditional mean is also a function of conditional variance h_{t}, where f(h) is usually or h. An ARCH (or GARCH) regression model is ARCHModel[alphalist]  ARCH( q) model with the coefficients { _{0}, _{1},..., _{q}} in alphalist  GARCHModel[alphalist, betalist]  GARCH( p, q) model with the coefficients { _{0}, _{1},..., _{q}} and { _{1},..., _{q}} in alphalist and betalist, respectively  ARCHModel[alphalist, , f]  ARCHinmean model  GARCHModel[alphalist, betalist, , f]  GARCHinmean model 
ARCH and GARCH models. Note that the function f should be a symbol representing a Mathematica builtin function or a pure function. Note also these models will be referred as archmodel to differentiate them from ARMA type models. TimeSeries[archmodel, n]  generate a time series of length n from archmodel  TimeSeries[archmodel, n, init]  generate a time series with given initial values init 
Generating ARCH time series. This generates a GARCH time series of length 10. Note that the third argument is optional. It specifies the initial values {{h_{p+1}, h_{p+2}, ..., h_{0}}, {z_{q+1}, z_{q+2}, ..., z_{0}}}, or {z_{q+1}, z_{q+2}, ..., z_{0}} as in the case of an ARCH model. Out[2]=  
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LogLikelihood[data, archmodel]  give the logarithm of the Gaussian likelihood for the given data and archmodel  LogLikelihood[data, archmodel, X, blist]  give the logarithm of the Gaussian likelihood of an ARCH or GARCH regression series data  LogLikelihood[data, archmodel, ARModel[philist]]  give the logarithm of the Gaussian likelihood of an ARARCH or ARGARCH series data  LogLikelihood[data, archmodel, ARModel[philist], ]  give the logarithm of the Gaussian likelihood of an ARARCH or ARGARCH series with a constant mean 
Log likelihood for ARCH series. option name  default value   "PresampleValue"  Automatic  presample value 
Option for LogLikelihood. Note that the X is the matrix and blist is a list of parameters "b" defined in the ARCH regression model. The presample values of and {h_{p+1}, ..., h_{1}, h_{0}} are assumed to be equal to a fixed value sigma2, and it can be specified using the option PresampleValue > sigma2. The default setting for PresampleValue is Automatic, which corresponds to using the sample equivalence of ^{2} for GARCH models and to using _{0}/(1_{i}_{i}) for GARCHinmean models. Out[5]=  
ConditionalMLEstimate[data, archmodel]  fit archmodel to data using the conditional maximum likelihood method  ConditionalMLEstimate[data, archmodel, X, blist]  fit ARCH or GARCH regression model to data  ConditionalMLEstimate[data, archmodel, ARModel[philist]]  fit ARARCH or ARGARCH model to data  ConditionalMLEstimate[data, archmodel, ARModel[philist], ]  fit nonzero mean ARARCH or ARGARCH model to data 
Conditional maximum likelihood estimations. option name  default value   MaxIterations  30  maximum number of iterations in searching for minimum  PresampleValue  Automatic  presample value 
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LMStatistic[data, archmodel]  calculate LM statistic with estimated parameters under the null inside archmodel  LMStatistic[data, archmodel, X, blist]  calculate LM statistic of an ARCH or GARCH regression series  LMStatistic[data, archmodel, ARModel[philist]]  calculate LM statistic of an ARARCH or ARGARCH series  LMStatistic[data, archmodel, ARModel[philist], ]  calculate LM statistic of an ARARCH or ARGARCH series with nonzero mean 
LM statistic. option name  default value   PresampleValue  Automatic  presample value 
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