Is this an ARMA(2, 1) process? I am puzzled by an equation,
$$
y_t = \phi_1 y_{t-1} + \phi_2 y_{t-2} + u_t + \varepsilon_t - \varepsilon_{t-1},
$$
where $u_t$ and $\varepsilon_t$ are independent white-noise processes.
Is this an ARMA(2, 1) process? Or does the fact that there are two shocks at each timestep (but only one carried over) change this?
Secondly, can $\phi_1$ and $\phi_2$ be estimated consistently using 2SLS? I thought that $y_{t-2}$ would be a valid instrument for $y_{t-1}$, but then $y_{t-2}$ is also a regressor variable.

To answer the second part of my question: using $y_{t-3}$ as an instrument, it is indeed possible to estimate $\phi_1$ and $\phi_2$ consistently by 2SLS. I verified this by computer simulation.
