# If $X_t$ is an AR(2) process, what is $Y_t := X_t - X_{t-1}$?

Q: If $$X_t$$ is an AR(2) process, what is $$Y_t := X_t - X_{t-1}$$?

Attempted solution:

$$X_t = \phi_1 X_{t-1} + \phi_2 X_{t-2} + W_t$$, where $$W_t$$ is white noise.

$$$$\begin{split} Y_t &:= X_t - X_{t-1} = \phi_1 X_{t-1} + \phi_2 X_{t-2} + W_t - \phi_1 X_{t-2} - \phi_2 X_{t-3} - W_{t-1} \\ \end{split}$$$$

But what can we say about $$Y_t$$?

• this sounds like a self study question. if you agree, consider adding self-study tag :-)
– Ute
Commented May 29, 2023 at 13:23
• Reorder your terms such that you have a linear combination of $X_{t-i}$ terms and a linear combination of $W_{t-j}$ terms. Then have a look at ARMA models
– Ute
Commented May 29, 2023 at 13:28

If we write it with characteristic polynomials the relations will look like below $$Y_t=(1-B)X_t, \ \ \ \ \ \ \ X_t(1-\phi_1B-\phi_2B^2)=W_t$$
Then, the relation between $$Y_t$$ and $$W_t$$ can be written with $$Y_t=\frac{1-B}{1-\phi_1B-\phi_2B^2}W_t \rightarrow Y_t(1-\phi_1B-\phi_2B^2)=(1-B)W_t$$
Converting it back yields the following ARMA(2,1) model. $$Y_t-\phi_1 Y_{t-1}-\phi_2 Y_{t-2}=W_t-W_{t-1}$$