# Thread: Need help on difference operator

1. ## 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.

2. Originally Posted by kleenex
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

3. Originally Posted by kleenex
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