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
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.)
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.
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.