Hi

The différence is major !

If you have two time series X, Y.

Y is a GARCH so you specify the mean equation of Y (as example a constant or an ARMA process) and after the variance equation with GARCH equation.

If you add a external regressor in the mean for eg X the mean variance is Y=a+b.X +E

Where E is residuals folliwing a GARCH process. you incorporate the effect of X on Y mean and the resid of this is a GARCH

Now if you add X on the variance equation you consider that the variance of Y is influenced by X but not the mean.

Moreover Parameters estimates and interpretation are not similar.

I hope my respond is useful to you

Best regards

Thanks for your response Roman.

If you add the exogenous regressors in the conditional mean model only, you get different fitted and predicted values of μt. Hence, you get different point forecasts but the same variance forecasts around the different points.

If you add the exogenous regressors in the conditional variance model only, you get different fitted and predicted values of σ2t. Hence, you get different variance forecasts around the same point forecasts.

If you add the exogenous regressors in both the conditional mean and the conditional variance models, for different values of regressors you get different fitted and predicted values of μt and σ2t. Hence, you get different point forecasts and different variance forecasts around the different point forecasts.

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Hope this help.

Goodluck my brother

Ok, can one further say that the mean equation models the spill over effect from the last period volatility of the exogenous(X) variable on the current mean of the dependent variable (y)(i.e. volatility to mean)? Or does it model the spill over effect from the last period mean of the exogenous(x) variable to the mean of the dependent variable(Y)?

Thank you

Well , there has to be some theoretical base as to whether you feel that the exogenous variables impact the returns or the volatility , if you are of the opinion that they impact the returns , you have to include them in the mean equation and accordingly if you think they have consequences for volatility, then include them in the volatility equation.

If you have two time series X, Y. Y is a GARCH so you specify the mean equation of Y (as example a constant or an ARMA process) and after the variance equation with GARCH equation. ... If you add the exogenous regressors in the conditional mean model only, you get different fitted and predicted values of μt.