# Markov Switching GARCH conditional distribution

Given a Markov switching GARCH model

$\epsilon_n=Z_n \sqrt{\sigma_{\Delta_n,n}}$

where $\epsilon_n$ is a time-series of residuals, $Z_n$ is a sequence independent and identically distributed random variables with mean zero and unit variance, and $\left\{ \Delta_n,n\in \mathbb{Z}\right\}$ is a Markov chain.

If we assume that $Z_n$ follow a t-distribution does this mean that each regime will have the degrees of freedom $v$ and only changing the conditional variance $\sigma_{\Delta_n,n}$ or is the degrees of freedom changing with the regime as well.

• From the equation you wrote down it does not look that the distribution of $Z_n$ would vary with time. Doesn't that answer the question? Oh, I guess you are asking about the distribution of $\epsilon_n$. – Richard Hardy Mar 7 '18 at 14:05
• Yes the distribution of Z_n does not change i.e. it is always t for each regime with mean 0 variance 1. On the other hand, the distribution of $\epsilon_n$ is t with mean zero and variance equivalent to the conditional distribution depending on the regime. What I am asking is regarding to the shape parameter. – Anna Mar 7 '18 at 14:56
• The degrees of freedom parameter? Or what shape parameter? – Richard Hardy Mar 7 '18 at 15:14
• Yes degrees of freedom sorry for the confusion – Anna Mar 7 '18 at 15:16