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2.4 Univariate ARCH and GARCH Models

Autoregressive 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; ht is the conditional variance of Zt conditional on all the information up to time t-1, It-1:
When p=0, we have an ARCH(q) model. A GARCH-in-mean (or ARCH-in-mean) model is defined by
That is, the conditional mean is also a function of conditional variance ht, where f(h) is usually or h. An ARCH (or GARCH) regression model is
or
 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] ARCH-in-mean model GARCHModel[alphalist, betalist, , f] GARCH-in-mean model

ARCH and GARCH models.

Note that the function f should be a symbol representing a Mathematica built-in 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.

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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 AR-ARCH or AR-GARCH series data LogLikelihood[data, archmodel, ARModel[philist], ] give the logarithm of the Gaussian likelihood of an AR-ARCH or AR-GARCH 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, h0} 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 GARCH-in-mean models.
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 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 AR-ARCH or AR-GARCH model to data ConditionalMLEstimate[data, archmodel, ARModel[philist], ] fit nonzero mean AR-ARCH or AR-GARCH 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

Options for ConditionalMLEstimate.

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The Lagrange multiplier (LM) test is commonly used to test for the existence of ARCH. The null hypothesis is no ARCH, that is, 1=2==q=0 and the LM statistic has a distribution asymptotically under the null hypothesis.
 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 AR-ARCH or AR-GARCH series LMStatistic[data, archmodel, ARModel[philist], ] calculate LM statistic of an AR-ARCH or AR-GARCH series with nonzero mean

LM statistic.

 option name default value PresampleValue Automatic presample value

Option for LMStatistic.

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