Garch and Multiplicative error models

Hello,

I'm trying to fit a multiplicative error model: carr model.

It is a model for estimating volatility that uses range instead of lagged squared returns.

Basically, CARR model can be summarized using these 3 equations:

, where λ is the conditional mean of the range of price of a financial asset.

, where is the innovation term. I want to include three cases:

As you can see from the basic details I provided, it is quite related to garch models.

Thus I tried to modify the garch.m class and estimate.m related to garch models to generate my carr models. However I don't understand if the estimation of and also in garch depends on the Maximum Likelihood function or not. Can you please explain how the coefficients are generated?

Moreover I'm a bit concerned about how to include Weibull and Gamma: their expected value is not directly in the likelihood function but depends on the scale and the shape parameters, which are in the likelihood. How should I deal with this issue on the code?

Thank you very much.

Michele