# 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 $\displaystyle {Y_t}$ 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.

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 $\displaystyle {Y_t}$ 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

3. Originally Posted by kleenex
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