State-space model with contemporaneous effects I have the following system of equations:
$$
\begin{align}
y_t^{(1)}&=y_t^{(2)}-x_t+\epsilon_t\\
y_t^{(2)}&=x_t+\nu_t\\
x_t&=\alpha x_{t-1}+u_t
\end{align}
$$
where $y_t^{(1)}, y_t^{(2)}$ are observed and $x_t$ is not.
I'm having some issues putting this into the state-space formulation. The issue I have is that in order to get $y_t^{(2)}$ in the (measurement) equation for $y_t^{(1)}$ I need to put it in the state vector. But I need $y_t^{(2)}$ in the measurement equation in order to get $x_t$ in there. So what do I do with $y_t^{(2)}$ in the state vector? Can I simply skip that row in the state equation? That would give me:
$$
\begin{align}
\begin{pmatrix}y_t^{(1)}\\ y_t^{(2)}\end{pmatrix}&=\begin{pmatrix}-1 & 1 \\
0 & 1\end{pmatrix}\begin{pmatrix}x_t \\ y_t^{(2)}\end{pmatrix}+\begin{pmatrix}\epsilon_t\\\nu_t\end{pmatrix}\\
x_t&=\begin{pmatrix}\alpha & 0 \end{pmatrix}\begin{pmatrix}x_{t-1} \\ y_{t-1}^{(2)}\end{pmatrix}+u_t.
\end{align}
$$
 A: Substitute the second equation into the first and you have
\begin{align}
y_t^{(1)}&=y_t^{(2)}-x_t+\epsilon_t\\
&=x_t+\nu_t-x_t+\epsilon_t\\
&=\nu_t+\epsilon_t.
\end{align}
So it's
\begin{align}
\begin{pmatrix}y_t^{(1)}\\ y_t^{(2)}\end{pmatrix}&=\begin{pmatrix}0 \\
1\end{pmatrix}x_t+\begin{pmatrix}1 & 1 \\ 0 & 1\end{pmatrix}\begin{pmatrix}\epsilon_t\\\nu_t\end{pmatrix}\\
x_t&=\alpha x_{t-1}+u_t.
\end{align}
Another way to see it is to rewrite the measurement equations as
\begin{align}
y_t^{(1)}-y_t^{(2)}&=-x_t+\epsilon_t\\
y_t^{(2)}&=x_t+\nu_t,
\end{align}
which is equivalent to
$$\begin{pmatrix}1 & -1 \\ 0 & 1\end{pmatrix}\begin{pmatrix}y_t^{(1)}\\ y_t^{(2)}\end{pmatrix}=\begin{pmatrix}-1 \\
1\end{pmatrix}x_t+\begin{pmatrix}\epsilon_t\\\nu_t\end{pmatrix}.$$
To isolate the observables on the left hand side, note that 
$$\begin{pmatrix}1&-1\\0&1\end{pmatrix}^{-1}=\begin{pmatrix}1&1\\0&1\end{pmatrix}.$$
Multiply both sides by that and you get
\begin{align}
\begin{pmatrix}1&1\\0&1\end{pmatrix}\begin{pmatrix}1 & -1 \\ 0 & 1\end{pmatrix}\begin{pmatrix}y_t^{(1)}\\ y_t^{(2)}\end{pmatrix}&=\begin{pmatrix}1&1\\0&1\end{pmatrix}\left[\begin{pmatrix}-1 \\
1\end{pmatrix}x_t+\begin{pmatrix}\epsilon_t\\\nu_t\end{pmatrix}\right]\\
\begin{pmatrix}y_t^{(1)}\\ y_t^{(2)}\end{pmatrix}&=\begin{pmatrix}1&1\\0&1\end{pmatrix}\begin{pmatrix}-1 \\
1\end{pmatrix}x_t+\begin{pmatrix}1&1\\0&1\end{pmatrix}\begin{pmatrix}\epsilon_t\\\nu_t\end{pmatrix}\\
\begin{pmatrix}y_t^{(1)}\\ y_t^{(2)}\end{pmatrix}&=\begin{pmatrix}0 \\
1\end{pmatrix}x_t+\begin{pmatrix}1&1\\0&1\end{pmatrix}\begin{pmatrix}\epsilon_t\\\nu_t\end{pmatrix}.
\end{align}
