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}

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.