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I've got the following problem. I have two AR(1) processes (which are the returns on assets)-

$$r_{1t} = \phi_1r_{1,t-1} + u_{1t},$$ $$r_{2t} = \phi_2r_{2,t-1} + u_{2t},$$

We have the following weighted portfolio of these two returns as:

$$rp_t = \frac{r_{1t} + r_{2t}}{2}$$

And we must represent $rp_t$ as an ARMA. Obviously I have subbed in the original definitions and got

$$rp_t = \frac{\phi_1r_{1,t-1} + \phi_2r_{2,t-2}}{2} + \frac{u_{1t} + u_{2t}}{2}$$

but I guess I really need to express $rp_t$ in terms of $rp_{t-1}$ or previous terms plus some error terms - i.e. how do I get rid of the $\phi_1$ and $\phi_2$ in the expansion immediately above?

Any help greatly appreciated as I've been struggling with this for ages, and I will be sure to vote up any helpful answers.

Apologies if this question is a duplicate, but I can't seem to find it already.


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Apologies for the poor tagging. I really want to tag with the following: Autoregressive process, ARMA, moving average, econometrics. – Phil Whittington Mar 4 '12 at 11:37
(...but I don't have enough reputation to add new tag categories, unfortunately. Thanks.) – Phil Whittington Mar 4 '12 at 11:43
What makes you think this is possible (except when $\phi_1=\phi_2$)? – Did Mar 4 '12 at 12:16
@DidierPiau, it's a question in a past exam paper for my course, but this topic doesn't seem to be covered in the course material. My instinct is that there must be some way of representing $rp_t$ as a linear combination of ($rp_{t-1}, rp_{t-2}...$) with everything else being simply a linear combination of error terms. The exact question is how to represent $rp_t$ as an ARMA. – Phil Whittington Mar 4 '12 at 12:37

See also Helmut Lütkepohl "Linear transformations of vector ARMA processes" (1984). Journal of Econometrics 26 p 283-293. (I know this is only helpful if you have access somehow to the article :( ). Goes something like this. By some rewriting we get

$$ rp_t - (\phi_1+\phi_2)rp_{t-1}+\phi_1\phi_2rp_{t-2}=(1/2) (u_{1t}-\phi_2u_{1(t-1)})+(1/2)(u_{2t}-\phi_1u_{2(t-1)}) $$

If the right hand side is an MA process then this equation gives an ARMA process. The article mentioned above proves that the RHS is indeed again an MA process.

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