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

Question

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?

Answer 1

Apparently yes, (I haven't verified the math but have no reason to doubt it). For this simple case you can find a closed form in the following paper:

• Jeff A. BILMES: What HMM can do

The closed form is given on part 4.4 of the paper but the whole thing is worth reading as it clearly shows the main properties of these models.

You can also note that contrary to your definition the observations in each state don't need to be IID (you an have other structures such as ARMA). The book by Kim and Nelson (State-Space Models with Regime-Switching) provides a lot of information on this class of models.