I have created a Markov-Switching GARCH model, where the volatility is defined to be switching between two different GARCH(1,1) processes. The data is assumed to have zero mean, where the data is demeand with an AR(1) process.
The AR(1)-GARCH(1,1) model is defined as,
$y_t = \phi_1 y_{t-1} + a_t$
$a_t=\sigma_t\epsilon_t$
$\sigma_t^2= \omega+\alpha_1 a_{t-1}^2+\beta_1\sigma_{t-1}^2$
Therefor my question, is it correct to assume that the AR(1) therm $\phi_1$ is the same for both processes since it is based on the same time series?
And if yes, why are not the change in $\omega, \alpha_1$ and $\beta_1$ affecting the calculations of $y_t$? Where the resulting forecasted mean is the same for both GARCH processes.