GARCH-M model with interaction term

[PeterZhu]

Hi, I am going to run a regression by using the GARCH-M model. I have done it before, but this time the mean funtion is Rt=α+θσt^2+(φ0+φ1*σt^2)*R(t-1)+εt, where σ^2 is the conditional variance and Rt is the stock returns. The conditional variance should use GARCH-M model and I know that. But once there is a joint term in the mean function (specifically, the production of conditional variance and the one-lagged return) I do not know how to make this regression work. I have seen a same question posted a couple of years ago and their solution is to use the AR(1) model as a mean funtion and run the GARCH-M once and store the conditional variance as a new series and then regress again and this time they put the conditional variance series into the mean function to get the final result.
A part of the code they used has been shown below.
...
equation eq1.arch(1,1, archm=VAR,backcast=1) y c
eq1.makegarch garchm
equation eq1.arch(1,1, archm=VAR,backcast=1) y c garchm*y(-1)
...

I understand what this code means and I can do the same job by using the Eviews tools (not using code). But there still a few things that I want to know.
First, we set AR(1) (with a constant form) as the mean function to run the GARCH-M first and store the variance. But this conditional variances are not coming from the target model , and will it influences the result? I have tried a few times if I change the mean function form the conditional variance WILL change. So are these differences small enough to be ignored?
Second, the example uses the AR(1) combining GARCH-M to generate the conditional variance. What if the AR(1) is not the best model? What if the best model is, say, AR(2)? Should I change the target model as well? I mean why exactly they use AR(1) to get the conditional variance? is it because the target model is a AR(1) model or it does not matter what model is to generate the conditional variance?
Third, after I get the GARCH-M result, I may need to use EGARCH-M to make a comparison (with same mean function). Should I still go through this procudure again? (I mean back to step one run the regression based on AR(1) and EGARCH-M and get a new serise of conditional variance)

[diggetybo]

Hey. Sometimes it helps to post the workfile with what you got, just for reference.