The exercise is the following:
(b) (4 points) Consider the following model $$ \begin{array}{r} X_t=c+X_{t-1}+\phi_1\left(X_{t-1}-X_{t-2}\right)+\phi_2\left(X_{t-2}-X_{t-3}\right) \\ +\phi_3\left(X_{t-3}-X_{t-4}\right)+Z_t, \end{array} $$ where $X_t$ is the population in year $t$ and $\left\{Z_t\right\} \sim \mathrm{WN}\left(0, \sigma^2\right)$. What sort of ARIMA model is this (i.e., what are $p, d$, and $q)$ ? Include an explanation.
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I'm not sure if $p = 4$ since there is a $X_{t-4}$ , or if $p = 3$ since the phi's go up to $\phi_3$. This is the only value that I'm unsure of, since I'm pretty sure $d=1$ since the differencing is done at lag 1 and $q=0$ since there the only white noise is $Z_{t}$
Hi: You can write your model in the following manner:
Let $Y_t = X_{t} - X_{t-1}$.
Then, the original model can be re-written as $Y_t = \phi Y_{t-1} + \phi_{2} Y_{t-2} + \phi_{3} Y_{t-3} + Z_t$.
So, this is an ARIMA(3,0,0) for $Y_t$ because there are 3 lagged $Y_t$ variables on the RHS along with the error term.
But then that can be re-written as an ARIMA(3,1,0) on $X_t$ since we differenced once.
So, the model for $X_t$ is an ARIMA(3,1,0). I hope that makes sense.
ADDENDUM: I would write it out, in terms of ARIMA parameters, as
$(1 - B) X_{t} = \mu + \phi_1 B(1-B)(X_{t}) + \phi_2 B^2 (1-B) (X_{t}) + \phi_{3} B^3 (1-B)(X_{t}) + Z_{t} = \mu + (1-B)(\phi_{1} B + \phi_{2} B^2 + \phi_{3} B^3) X_{t} + Z_{t}$