0
$\begingroup$

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.

PW

  • 0
    Apologies for the poor tagging. I really want to tag with the following: Autoregressive process, ARMA, moving average, econometrics.2012-03-04
  • 0
    (...but I don't have enough reputation to add new tag categories, unfortunately. Thanks.)2012-03-04
  • 0
    What makes you think this is possible (except when $\phi_1=\phi_2$)?2012-03-04
  • 0
    @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.2012-03-04

1 Answers 1