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So, I've been looking at different ARIMA models, mainly ones with d=1, and I came across the ARI(1,1) model, which is an AR(1) model that has been differenced once.

I know that the AR(1) model is $Y_t = \phi_1 Y_{t-1} +e_t$ and so $Y_{t-1} = \phi_2 Y_{t-2} + e_{t-1}$. But when I do the first difference, I get:

$Y_t - Y_{t-1} = \phi_1 Y_{t-1} +e_t - (\phi_2 Y_{t-2} + e_{t-1}) \\ Y_t = \phi_1 Y_{t-1} - \phi_2 Y_{t-2} + Y_{t-1} + e_t - e_{t-1} $

however, the book I'm reading doesn't have that last $e_{t-1}$ term and they don't show all their steps, so I'm not sure where I went wrong, or if there's another way I should be thinking about this.

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    $\begingroup$ Look at your expression for the AR(1) model embedded in the text. Then substitute $X_t - X_{t-1}$ everywhere you see $Y_t$, and $X_{t-1}-X_{t-2}$ everywhere you see $Y_{t-1}$, etc. Is there an $e_{t-1}$ in what you get? $\endgroup$
    – jbowman
    Commented Feb 25, 2020 at 0:13
  • $\begingroup$ @jbowman oh I see now, thank you! $\endgroup$
    – clovis
    Commented Feb 25, 2020 at 1:41

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