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Thread: Backshift operators

  1. #1
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    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) $?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by southprkfan1 View Post
    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
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