Essentials of Time Series for Financial Applications

(3.55)

where $L$ is the lag operator discussed in Chapter 2 and $\mathbf{Ã}(L)\equiv {\mathbf{I}}_N-\mathbf{A}(L)$. At this point, let $\mathbf{\Theta }(L)\equiv {\sum }_{i=0}^{\infty }{\mathbf{\Theta }}_i{L}^i$ be an operator such that $\mathbf{\Theta }(L)\mathbf{Ã}(L)={\mathbf{I}}_N$ and postmultiply Eq. (3.55) by $\mathbf{\Theta }(L)$ to obtain

${\mathbf{y}}_t=\mathbf{\Theta }(L)\mathbf{\mu }+\mathbf{\Theta }(L){\mathbf{u}}_t,$ (3.56)

(3.56)

that is,

${\mathbf{y}}_t=\sum _{i=0}^{\infty }{\mathbf{\Theta }}_i$