I have an original identity, that has been transformed into a different form as follows:
$z_t=a+\Gamma F_t +v_t$
$F_t=\mu +AF_{t-1}+u_t$ $\;\; u_t \sim N(0,Q)$
$v_t=B v_{t-1}+ \xi_t$ $\;\;\;\;\;\;\;\;\;\; \xi_t \sim N(0,R)$
where
$z_t=(n*1)$ vector
$a=(n*1)$vector
$\Gamma=(n*p)$ vector
$F=(p*1)$ vector
$v_t=(n*1)$ vector
$u_t=(p*1)$ vector
$\xi_t=(n*1)$ vector
This is then transformed into this form
$z_t=\Gamma^* F_t^* +v_t^* $ $\;\;\;\;\;\;\;\;\;\; u_t \sim N(0,R^*)$
$F_t^*=A^*F_{t-1}^* +u_t^*$ $\;\;\;\;\;\; \xi_t \sim N(0,Q^*)$
where
$ \Gamma^*_t = \left( \begin{array} {c} \Gamma \\a \\ I_n \end{array} \right) $
$ F_t^* = \left( \begin{array} {c} F_t \\c_t \\v_t\end{array} \right) $
$\def\iddots{ {\kern3mu\raise1mu{.}\kern3mu\raise6mu{.}\kern3mu\raise12mu{.}}} A^* = \left( \begin{array} {c,|c|,c} A\;\;\;\mu &\cdots& 0 \\ \hline \vdots\;\;\; \iddots &1&\vdots \\ \hline 0 \;\;\;\cdots&\cdots &B \end{array} \right) $
$ u_t^* = \left( \begin{array} {c} u_t \\v_t \\xi_t\end{array} \right) $
$ Q^* = \left( \begin{array} {c,|c|,c} Q &\cdots& 0 \\ \hline \vdots & \epsilon&\vdots \\ \hline 0 & \cdots & R \end{array} \right) $
where $R^*_t=\epsilon I_n$
The part I don't understand is $A^*F^*_{t-1}$ how do I handle the fact that the first column contains two elements?
It seems that you should take the first element of $A^*$ i.e. the (1*2) submatrix $(A\;\;\; \mu)$ and mutiply it by the first two elements of $F^*_{t-1}$ but what happens to the element [1,2] of $A^*$ should it also be multiplied by c in $F_{t-1}^*$ looking at the next two rows of $A^*$ this is what should happen but there appears to be an element of double counting? Can someone tell me if I am doing this correctly and what is the correct term for a matrix like this (I realize it's partitioned but it also has one column with two sub-columns what is this part called?
Lastly what do the $\cdots, \vdots $ etc represent? Do they all signify that those elements with dots in should be considered zero? Or something else?
Kind Regards
Baz
