Show that this process is an ARIMA(p,d,q) process for some values of p, d, and q

Consider stochastic process given by model equation \begin{aligned} Y_t &= \beta_0 + \beta_1 t + z_t, \\ z_t &= \phi z_{t-1} + \varepsilon_t, \\ \end{aligned}

where $|\phi|<1$.

How do I show that $Y_t$ is an ARIMA($p,d,q$) process for some values of $p, d,$and $q$?

By substituting in our values of $z_{t-1}$ into $z_t$ and continuing in this way isn't that just representing $z_t$ as an MA($\infty$) model? Where does the ARI model appear in that case? Sorry if this seems silly. I think I'm overcomplicating this problem.

$$Y_t - Y_{t-1} = \beta_1 + \phi z_{t-1} - z_{t-1}$$

So this should exist in our $Y_t$ term somewhere?

\begin{align} Y_t &= \beta_0 + \beta_1 t + \phi z_{t-1} \hspace{20mm} + \varepsilon_t \\ Y_t &= \beta_0 + \beta_1 t + \phi (\phi z_{t-2} + \epsilon_{t-1}) + \varepsilon_t \end{align}

I have definitely messed this up somewhere.

• Consider adding the self-study tag and reading its Wiki. Then show us what you have done already and where you got stuck. Also, there is no need to format ARMA as formula ($ARMA$) as it is an acronym, not a function name. – Richard Hardy May 10 '17 at 13:15
• @RichardHardy Thanks for the edit. I'm not entirely sure as I don't understand how it can have an MA part? As far as I can tell it's dependent on $z_t$ and $t$. – tryingtolearn May 10 '17 at 13:37
• If you substitute $z_{t}$ in the first equation with its expression from the second equation and continue that way for $z_{t-1}, z_{t-2},\dots$, you will see something interesting. What about the self-study tag, though? – Richard Hardy May 10 '17 at 13:46
• You need to do more than add the tag .. .show what you did and ask about where you got stuck – Glen_b May 10 '17 at 13:50
• But isn't that just representing $z_t$ as an MA(\$\infinity$) model? Where does the ARI model appear in that case? Sorry if this seems silly. I think I'm overcomplicating this problem. – tryingtolearn May 10 '17 at 13:51