# 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}

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}