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Time Series (2011)

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1.10.2 ARCH-in-Mean Models

The ARCH-regression model can be generalized to include the cases where the conditional mean mt is a function of ht. This is the so-called ARCH-in-mean model (see Engle, Lilien, and Robins (1987) and Domowitz and Hakkio (1985)). For example, for an ARCH-in-mean (or GARCH-in-mean) regression model, (10.4) is generalized to
where ht is given by (10.2) (or (10.3)) and f(h) is usually or h. (Other functions like lnh are also possible.) The ARCH-in-mean or GARCH-in-mean models are represented by
ARCHModel[alphalist, , f]
or
GARCHModel[alphalist, betalist, , f],
respectively. Note that the function f should be input as a symbol representing a built-in function (e.g., Sqrt, Identity, Log, represent , f(h)=h, and f(h)=lnh, respectively) or as a pure function (e.g., Sqrt[#]&, etc.). The functions introduced in the previous sections can all be used for ARCH-in-mean models. However, the default value for PresampleValue is 0/(1-i-i) if the model is ARCH-in-mean or GARCH-in-mean with q≠0 and 0/(1-i)+0.001 if q=0. (Note the small number 0.001 in the case of q=0. It is added to prevent a constant ht that would have resulted from using 0/(1-i) as the presample value.)
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Note that if there is a constant in the regression or AR model, the constant term and may not be distinguished as the following example shows.
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As we have said earlier, the result can depend on the presample values that are specified in the beginning; in this case, the presample values are determined by the initial parameter values. We can iterate the estimation to make sure that the result has converged.
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