Joint MGF of ${\bf z}$.

The joint MGF of ${\bf z}$ is, for $\lambda$$=[\lambda_1 \lambda_2 \ldots \lambda_M]^\prime$,

$$\begin{array}{rcl} g_z(\mbox{\boldmath$\lambda$}) & \stackrel... ...\lambda_{M-1} y_{t_{M-1}} + \lambda_M r) \right\} . \end{array}$$ (8.10)

This can be written

$$g_z($$$$\mbox{\boldmath$\lambda$}$$$$) = g_0({\bf A}$$$$\mbox{\boldmath$\lambda$}$$$$)$$

where ${\bf A}$ is the $N$-by-$M$ matrix that has 1's everywhere in the $M$-th column except for 0's in rows $t_1, t_2 \ldots t_{M-1}$, and ${\bf A}$ has 1's in row $t_1$, column 1; row $t_2$, column 2; etc. Therefore, from (8.9),

$$g_z($$$$\mbox{\boldmath$\lambda$}$$$$) = {1 \over {\rm prod}({\underline{\bf 1}}- {\bf Q} {\bf A}\mbox{\boldmath$\lambda$})}.$$

Note that ${\bf Q}$ can be a large $N$-by-$N$ matrix if $N$ is large. However, if we define

$${\bf P}\stackrel{\mbox{\tiny$\Delta$}}{=}{\bf Q}{\bf A},$$

${\bf P}$ is a reasonable $N$-by-$M$ size matrix and can be easily formed directly. The final simplified form for the joint MGF is

$$g_z($$$$\mbox{\boldmath$\lambda$}$$$$) = {1 \over {\rm prod}({\underline{\bf 1}}- {\bf P}\mbox{\boldmath$\lambda$})}.$$