Partial Autocorrelation in Markov Regime-Switching Processes
Summary
The document asks whether the partial autocorrelation function can be expressed in closed form for a Markov regime-switching process whose observations have state-dependent means and variances. Regime probabilities evolve through a transition matrix, so the overall series may exhibit serial dependence even when observations within each state are independent and normally distributed.
The response points to a hidden Markov model paper as a source for a closed-form result in the simple case described, but it does not reproduce the derivation or verify the mathematics. It also notes that state-specific observations need not be independent and identically distributed; richer within-state dynamics, including ARMA structure, can be considered. The discussion is therefore a pointer to references rather than a worked formula, and it gives no market application or empirical evidence.
Key ideas
- A Markov regime-switching series can exhibit autocorrelation even when observations within states are independent.
- A cited paper is offered as a source for a closed-form partial autocorrelation result in the simple case.
- The answer does not show the formula or verify its derivation.
- Regime-switching models can include richer within-state dynamics such as ARMA processes.
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Full text
# Is there a closed-form solution for the partial autocorrelation function of a Markov regime-switching process?
# 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?
## Answer by Zarbouzou (score 5, accepted)
https://quant.stackexchange.com/a/3493
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.