The first four algorithms (i.e., Newton Raphson, BHHH
, BHHH-2, and Steepest Ascent) are referred from the earlier research of Roh and Khan  and Train .
(9) Advantages of the BHHH
algorithm include the following: (1) it does not require computations beyond those needed to solve the likelihood equation, (2) if the function examined is not the true likelihood function it still typically ends up at the correct maximum, and (3) it is always "nonnegative definite" and highly likely to produce convergence.
As in traditional approach, we use Berndt, Hall, Hall and Hausman (hereafter BHHH
) algorithm to produce the maximum likelihood parameters and the corresponding standard errors.
Because there is no closed-form solution for the maximum likelihood estimates, nonlinear search methods, such as the Simplex or Bemdt-Hall-Halt-Hausman (BHHH
) algorithms, are needed to find the parameter estimates that maximize the likelihood function.
M (aximum) L (ikelihood) estimates of the GARCH-in-Mean model can be obtained by maximising the likelihood function using the BHHH
We estimate the parameters by Full Information Maximum Likelihood, using analytic first derivatives and scoring (BHHH
) second derivatives.
algorithm of Bernt, Hall, Hall and Hausman (1974) with numerical first derivatives is used in the optimization.
I obtained estimates using the negative binomial routine in Greene's (1990) statistical software LIMDEP set to the BHHH
The maximum likelihood estimation uses the Bemdt-Hall-Hall-Hausman (BHHH
) iterative procedure.
Sometimes this type of estimation problem does not have closed form solutions and must be estimated using iterative methods, i.e., Marquard algorithm [Box and Jenkins, 1976] or Berndt-Hall-Hall-Hausman algorithm (BHHH
algorithm) [Berndt et al., 1974].
This system can be solved with the method of Berndt, Hall, Hall, and Hausman (BHHH
The model's parameter estimates for each of the eight countries are maximum likelihood from the BHHH
algorithm in the maximum likelihood routine of TSP-386, version 4.2A.