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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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up vote 5 down vote accepted

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:

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.

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I'm aware of the point about ARMA models, but wanted to get to a simpler case just to understand it better. Let me take a look at that paper. –  John May 14 '12 at 21:01
    
+1 for simplicity. Are you planning on applying this model to financial time series prediction? –  Zarbouzou May 14 '12 at 21:13
    
I've played around with MRS models a little in econometric and financial applications, but I wasn't positive on this property. –  John May 14 '12 at 21:26
    
@John: Any out-sample predictive ability? I haven't found them very usefull in the past. –  Zarbouzou May 15 '12 at 9:57
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Tricky question. An MRS model estimated on something like the S&P500 with switching means and variances will be primarily dominated by the regime-switching variance. Regime-switching variance tends to look like Garch, so you can often capture a lot of the effect with that. Also, the regime-switching forecasted means tend to bounce around a lot, so you need to be sure to account for the transaction costs of the switching strategies. Nevertheless, Ang and Bekaert report positive results for international asset allocation. www2.gsb.columbia.edu/faculty/aang/papers/inquire.pdf –  John May 15 '12 at 15:45
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