Doubt in state space representation
Your FIR state space representation seems to be doing too much.

The way I would write it is to have the process noise is $\epsilon(t)$ as your input, and your state as two time-delayed copies of it:

$$x(t+1) = \left[ \begin{array}{c}
\epsilon(t+1)\\
\epsilon(t) \\
\epsilon(t-1) 
\end{array} \right] = \left[ \begin{array}{ccc}
0 &amp; 0 &amp; 0\\
1 &amp; 0 &amp; 0\\
0 &amp; 1 &amp; 0 
\end{array} \right] x(t) + \left[\begin{array}{c}
1\\
0\\
0
\end{array} \right] \epsilon(t + 1)
$$

then your output equation is just:
$$
y(t) = \left[ \begin{array}{ccc}
1 &amp; h_1 &amp; h_2 \end{array} \right] x(t) + v(t)
$$

Your IIR representation is too confused for me to make head or tail out of. Can you clarify that you have the right information there?

Other questions:


What is $w(t)$? It does not appear to take part in any of your equations?
What are the parameters for $u(t)$?