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I'm confused because I thought $ARMA(p,q)$, has elements of autoregression $AR(p)$ and moving average $MA(q)$.

I know a series is $ARIMA$ if the differenced data is an $ARMA$.

The authors of something I'm reading say that this model:

$Y_t=0.9Y_{t-1}+W_t$ is an $ARIMA (1,1,0)$ because the differenced data are an autoregression of order one:

$Y_t=X_t-X_{t-1}$

I agree with the differenced data being an autoregression of order 1 but if a series is a $ARIMA$ if the differenced data is an $ARMA$, then where's the moving average part?.

In other words, why say that a series is a $ARIMA$ if the differenced data is an $ARMA$, why not just say that a series is a $ARIMA$ if the differenced data is an autoregression.

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    $\begingroup$ Why ever use a general word instead of the most specific word possible? $\endgroup$ – The Laconic Nov 29 '18 at 21:41
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There is no MA part .. thus it could be referred to as an ARI model . In a similar vein if there is no AR structure but differencing and an MA then it could be called an IMA model. The usage of the term ARIMA is a collective one not necessarily as precise as you wish but it is in common parlance.

Your example is an ARIMA in X and an ARMA in Y .... or more precisely ARI in X and AR in Y

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