I am trying to estimate covariance in multiple time series. However, I want to do this using a regime-switching framework. So, I start with fitting a GARCH(1,1) model and then de-volatalize the series. Using this series I try to fit an HMM. Suppose I have only 3 regimes ie: high, medium and low. Now I can assume that correlation is constant across a regime in which case I can use all sample data from a particular regime and fit 3 separate correlation matrices to multiple time series. Also, entire data series was used to fit the GARCH model which implies both correlation and convariance remain constant across a regime for the entire period in consideration. I have a few questions
Is it a problem to have a constant correlation and convariance matrix for a particular regime for the entire time series?
In the above procedure if I want the correlation in a time period for a regime to not be constant, then I will can fit a dynamic correlation model. What regime data should be used to fit the dynamic model? Should it be all data for the regime or just data for this regime since the last switch? What if the dynamic model fails to fit?
How does correlation one time period affect the correlation in another regime in the next period?
How does correlation one time period affect the correlation in the same regime in the next period, ie after a switch?