What ARMA model is this? \begin{aligned}
Y_t &= a  Y_{t-1} + e_t, \\
Z_t &= Y_t + H_t, \\
\end{aligned}
where $H_t$ is independent of $Y_t$.
I'm trying to understand what ARMA model $Z_t$ corresponds to but I'm not really sure.
Can someone provide a quick explanation?
 A: We can see that $Y_t$ is an AR(1) process with $a$ parameter.
We can find the autocorrelation function of $Z_t$ by first calculating its autocovariance. 
$$\text{cov}(Z_t, Z_{t-k}) = \text{cov}(Y_t + h_t, Y_{t-k} + H_{t-k})$$
this gives
$$\text{cov}(Z_t, Z_{t-k}) = \text{cov}(Y_t, Y_{t-k}) + \text{cov}(Y_t, H_{t-k}) + \text{cov}(H_t, Y_{t-k}) + \text{cov}(H_t, H_{t-k})$$
therefore when k = 0, we get $$ \gamma_Z(0) = \sigma^2_Y + \sigma^2_H $$ 
and when k > 0, we get $$\gamma_Z(k) = \gamma_Y(k) $$
From the equation for autocovariance for AR(1) models (since $Y_t$ is and AR(1)), 
$$ \gamma_Y(k) = a * \gamma_Y(0) \text{ and since } \gamma_Y(0) = \sigma^2_Y$$
this gives us an autocorrelation function of 
$$\rho_Z(k) = \frac{a * \sigma^2_Y}{\sigma^2_Y + \sigma^2_H}$$
this has the form $$\rho_Z(k) = A a^{k-1}$$
which is typical of ARMA$(1,1)$ models and therefore implies that $Z_t$ is an ARMA$(1,1)$ model.
A: $Z_t$ is not described by an autoregressive model because
$$ Z_t = a  Y_{t-1} + e_t + H_t $$
No lagged values of $Z_t$ are present in the right-hand-side, and so there is no auto-regression. The fact that the variable $Y_t$ appears lagged, makes the model dynamic, but not autoregressive also.
(I presume that $e_t$ is not assumed to be a function of lagged values of $Z$)
