text: # 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
explanation: The document is a forum thread where a user named "kleenex" asks for help with a math problem related to the difference operator in time series analysis. The user poses two questions:
  
  1. Prove that if $\displaystyle y_t$ is a polynomial in $\displaystyle t$ of order $\displaystyle k$, then $\displaystyle Dy$ is a polynomial of order $\displaystyle k-1$.
  2. Show that if $\displaystyle Y_t = p(t) + U_t$ where $\displaystyle p(t)$ is a polynomial of degree $\displaystyle k$ and $\displaystyle U_t$ is a stationary noise process, then $\displaystyle D^kY$ is a stationary process.
  
  Another user, "RonL," provides answers to both questions.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # 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
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a forum thread where a user named "kleenex" asks for help with a math problem related to the difference operator in time series analysis. The user poses two questions:
  
  1. Prove that if $\displaystyle y_t$ is a polynomial in $\displaystyle t$ of order $\displaystyle k$, then $\displaystyle Dy$ is a polynomial of order $\displaystyle k-1$.
  2. Show that if $\displaystyle Y_t = p(t) + U_t$ where $\displaystyle p(t)$ is a polynomial of degree $\displaystyle k$ and $\displaystyle U_t$ is a stationary noise process, then $\displaystyle D^kY$ is a stationary process.
  
  Another user, "RonL," provides answers to both questions.
  
  Label: multiple-question-and-answer