# Is it redundant to use moving average AND auto-regressive terms in an ARIMA model?

Seems to me they are by the following equation:

$$(1-\phi_1B)(1-B)X_t = (1-\theta_1B)\epsilon_t$$

We could just divide both sides by $1-\theta_1B$ and not ever have to deal with lagged error? I'm sure I'm missing something so I'm obviously throwing up a straw-man here...

• Note that if you do that division, the l.h.s. will be a polynomial in $B$ with an infinite number of terms unless $\theta_1 = \phi_1$. Commented Nov 23, 2015 at 19:49
• Makes sense -- I suppose "B" shift operator is equivalent to the z-transform case $z^{-1}$ so if $\phi_1 \approx \theta_1$ its a cancellation, but otherwise no dice. Thanks!
– JPJ
Commented Nov 23, 2015 at 19:52
• @jbowman, I was writing the answer before you posted the comment and did not notice it. Otherwise I would have given credit to you. Commented Nov 23, 2015 at 19:53

Dividing both sides by $1-\theta_1 B$ would result in a nasty polynomial of infinite order on the left hand side, unless $\phi_1=\theta_1$. Thus you would have an equivalent representation of the process that has an infinite number of parameters instead of two parameters $\phi_1$ and $\theta_1$. That may not be a tradeoff worth making.