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Math Help - 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:

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

    where  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:
     B^{-1}*A* E_t(X_t) for some constant matrix A. Would this equal
     A* E_t(X_{t+1}) or 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:

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

    where  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:
     B^{-1}*A* E_t(X_t) for some constant matrix A. Would this equal
     A* E_t(X_{t+1}) or A* E_t(X_t) ?
    Assuming you are using $$"*" to denote matrix multiplication we have:

    A*E_t(X_t)=E_t(A*X_t)=E_t(Y_t)

    where Y_t=A*X_t

    Then:

    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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