# Is there a closed-form solution for the partial autocorrelation function of a Markov regime-switching process?

Consider a Markov Regime-switching process $X_{t}$ with $k$ regimes represented by $s_{t}$ such that

$$X_{t}=\mu\left(s_{t}\right)+\epsilon_{t}$$

and

$$\epsilon_{t}\sim N\left(0,\sigma^{2}\left(s_{t}\right)\right)$$

with the probability of being in state $s_{t}$ represented by $p_{t}=Qp_{t-1}$ where $p_{t}$ is a $k \cdot 1$ vector containing the probabilities and Q is a transition matrix conforming based on the number of regimes.

Each state separately would be considered i.i.d. normal, but the regime-switching process exhibits autocorrelation. Is it possible to derive a closed-form solution for the partial autocorrelation function of $X_{t}$? If so, what is it?

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