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Math Help - Need help on difference operator

  1. #1
    Junior Member
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    Need help on difference operator

    I just started on this topic, but when i come the polynomial. I don't have any idea where to start.

    Consider the time series {Y_t} and let D be first difference operator Dy_t = y_t - y_{t-1}
    1.If y_t is a polynomial in t of order k then prove that Dy is polynomial of order k-1.

    2. Suppose Y_t = p(t) + U_t where p(t) is a polynomial of degree k and U_t is a stationary noise process. Show that D^kY is stationary process.

    Please help me, thank you.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by kleenex View Post
    I just started on this topic, but when i come the polynomial. I don't have any idea where to start.

    Consider the time series {Y_t} and let D be first difference operator Dy_t = y_t - y_{t-1}
    1.If y_t is a polynomial in t of order k then prove that Dy is polynomial of order k-1.
    If y_t is a polynomial of order k in t, then it may be written:

    y_t=a_k t^k+ a_{k-1}t^{k-1}+ ... + a_0, \ \ \ ...\ a_k \ne 0

    Then:

    (Dy)_t=y_{t+1}-y_{t}

    so:

    (Dy)_t=[a_k (t+1)^k+ a_{k-1}(t+1)^{k-1}+ ... + a_0] - [a_k t^k+ a_{k-1}t^{k-1}+ ... + a_0]

    Now the coeficient of t^k when this is expanded is a_k-a_k=0, so (Dy)_t is a polynomial in t of order no greater than k-1

    RonL
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by kleenex View Post
    2. Suppose Y_t = p(t) + U_t where p(t) is a polynomial of degree k and U_t is a stationary noise process. Show that D^kY is stationary process.
    Because p(t) is a polynomial of degree k; D^kp is a constant so:

    D^kY=D^kU.

    RonL
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