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.
 A: Your model
$$
(1-\phi_1B-\phi_2B^2)y_t=u_t+(1-B)\epsilon_t \tag{1}
$$
is ARMA(2,1) since the right hand side is clearly an MA(1) process since its autocovariance function cuts off for lags larger than 1. Hence, the right hand side can be represented by
$$
(1-\theta_1 B)v_t \tag{2}
$$
where $v_t$ is another white noise process.
Equating the autocovariance of the right hand side of (1) to that of (2) at lag 0 and 1 yields a set of two non-linear equations
\begin{align}
\sigma_u^2+2 \sigma_\epsilon^2 &= \sigma_v^2(1+\theta_1^2)
\\ -\sigma_\epsilon^2 &= -\theta_1\sigma_v^2
\end{align}
which can be straightforwardly solved for $\sigma_v^2$ and $\theta_1$.  There are two solutions. Only the one for which $|\theta_1|<1$ is invertible and relevant.
A: 
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?

It is possible to show that in general
$ARMA(p_1,q_1) + ARMA(p_2,q_2) = ARMA(p_3,q_3)$
now, "noise" is a special case for ARMA. Then your ARMA plus noise admit an ARMA representation.
Moreover, the fact that in your case the coefficient of MA component is imposed to $1$ is not a problem. You can obtain something like that:
$u_t + \varepsilon_t = r_t$
$$ y_t = \phi_1 y_{t-1} + \phi_2 y_{t-2}
 + r_t + \alpha r_{t-1}, $$
Regarding estimation problems I think that ML procedure is a better idea.
A: 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.
