Conditional and unconditional variance in ARMA processes

Question

How can I compute the conditional and unconditional variance of $x_t$ given its past if $\{x_t\}$ is an ARMA$(p,q)$ process. I'm literally struggling over that.

Cheers

Answer 1

If it is a standard stationary ARMA process then it will have mean zero and the form

$$x_t = \sum_{j=1}^p \alpha_j x_{t-j} + \sum_{j=0}^q \beta_j u_{t-j},$$

where $\beta_0=0$ and the $u_t$'s are iid.

The easiest way to get the objects that you want is to express the ARMA process as an infinite order MA process.