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Thread: 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 $\displaystyle {Y_t}$ and let $\displaystyle D$ be first difference operator $\displaystyle Dy_t = y_t - y_{t-1}$
    1.If $\displaystyle y_t$ is a polynomial in $\displaystyle t$ of order $\displaystyle k$ then prove that $\displaystyle Dy$ is polynomial of order $\displaystyle k-1$.

    2. Suppose $\displaystyle Y_t = p(t) + U_t$ where $\displaystyle p(t)$ is a polynomial of degree $\displaystyle k$ and $\displaystyle U_t$ is a stationary noise process. Show that $\displaystyle D^kY$ 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 $\displaystyle {Y_t}$ and let $\displaystyle D$ be first difference operator $\displaystyle Dy_t = y_t - y_{t-1}$
    1.If $\displaystyle y_t$ is a polynomial in $\displaystyle t$ of order $\displaystyle k$ then prove that $\displaystyle Dy$ is polynomial of order $\displaystyle k-1$.
    If $\displaystyle y_t$ is a polynomial of order $\displaystyle k$ in $\displaystyle t$, then it may be written:

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

    Then:

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

    so:

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

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

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

    $\displaystyle D^kY=D^kU$.

    RonL
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