Are conditional mean in an AR(1)-GARCH(1,1) equal for different GARCH(1,1) processes of the same data?

Question

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.

Answer 1

You can assume a state-dependent equation for $y_t$, $\sigma_t^2$ or both depending on what you think will best approximate the data generating process.

If you assume the equation for $y_t$ not to be state dependent (the same in all states) and the equation for $\sigma_t^2$ to be state dependent (different in different states), you would get the same point forecast $\hat y_t$ for $y_t$ regardless of the state as the point forecast is determined by the equation for $y_t$ alone. (This holds if you take the conditional mean as the point forecast; if you take, say, a quantile instead, the forecast will generally depend also on the conditional variance.) What would differ between the states would be the forecast interval, since it depends on a future value of $\sigma_t$ obtained from a state-dependent equation for $\sigma_t^2$.