September 22 Q3

[Nikki89]

Hi, can someone please explain to me the Q3 answer please? I’m finding it difficult to follow it in the marking scheme.
I know for stationary we can show the E(x_t) =0
That part is fine. How are we showing the Cov ariance part?

Also how are we showing the Markov property? I thought it’s not markov as X_t depends on a series of white noise terms.

 

[Andrew Martin]

Hello

In my view, the easiest way to tackle this question is to first covert the series into its AR representation. Remember two key results that we have:

1. A stationary AR(p) can be written as an MA(infinity)
2. An invertible MA(q) can be written as an AR(infinity)

So, whenever you see a process with order infinity, I'd be thinking about these results.

Here, you should find that:

Xt = et + 0.5 Xt-1

This is an AR(1) process, which is Markov. It is also stationary, which can be checked using the characteristic polynomial of the process terms in the usual way.

Hope this helps!

Andy