1. Backshift operators

Hi All, I have a question about using a back shift operator.

The Back shift operator B, is defined such that:

$\displaystyle B^{-1} E_t(X_t) = E_t(X_{t+1})$

where $\displaystyle E_t$ is defined as the expectation given information at time t and X is a matrix.

My question is, what if we had something like the following:
$\displaystyle B^{-1}*A* E_t(X_t)$ for some constant matrix A. Would this equal
$\displaystyle A* E_t(X_{t+1})$ or $\displaystyle A* E_t(X_t)$?

2. Originally Posted by southprkfan1
Hi All, I have a question about using a back shift operator.

The Back shift operator B, is defined such that:

$\displaystyle B^{-1} E_t(X_t) = E_t(X_{t+1})$

where $\displaystyle E_t$ is defined as the expectation given information at time t and X is a matrix.

My question is, what if we had something like the following:
$\displaystyle B^{-1}*A* E_t(X_t)$ for some constant matrix A. Would this equal
$\displaystyle A* E_t(X_{t+1})$ or $\displaystyle A* E_t(X_t)$?
Assuming you are using $\displaystyle$$"*"$ to denote matrix multiplication we have:

$\displaystyle A*E_t(X_t)=E_t(A*X_t)=E_t(Y_t)$

where $\displaystyle Y_t=A*X_t$

Then:

$\displaystyle B^{-1}\left(A*E_t(X_t) \right)=B^{-1}E_t(Y_t)=E_t(Y_{t+1})=A*E_t(X_{t+1})$

CB