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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...

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    $\begingroup$ Note that if you do that division, the l.h.s. will be a polynomial in $B$ with an infinite number of terms unless $\theta_1 = \phi_1$. $\endgroup$
    – jbowman
    Commented Nov 23, 2015 at 19:49
  • $\begingroup$ 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! $\endgroup$
    – JPJ
    Commented Nov 23, 2015 at 19:52
  • $\begingroup$ @jbowman, I was writing the answer before you posted the comment and did not notice it. Otherwise I would have given credit to you. $\endgroup$ Commented Nov 23, 2015 at 19:53

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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.

However, the infinite polynomial could be approximated by a finite-order polynomial. That would introduce some inaccuracy but would allow ditching the lagged error and using OLS (fast) instead of MLE (slow, possible convergence issues). This is among the points of motivation for use of VAR models instead of VARMA models.

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  • $\begingroup$ Thanks -- I realize @jbowman also replied with the same answer. But you did extend the explanation which was very helpful. $\endgroup$
    – JPJ
    Commented Nov 23, 2015 at 19:59
  • $\begingroup$ No problems, your answer was much more extensive (+1)! $\endgroup$
    – jbowman
    Commented Nov 23, 2015 at 20:03
  • $\begingroup$ @Richard You should not worry overly about answers in comments; they don't count, answers that are answers are the ones that matter and you shouldn't be expected to check the comments to see if anyone said anything relevant first. Give your answer with a clear conscience; if commenters want credit they should give an answer. (That's not to say you can't give credit to a comment, only that you needn't be concerned about it if you don't. Comments might disappear at any time; indeed many SE sites will delete anything that even looks like an answer-in-comments without delay) $\endgroup$
    – Glen_b
    Commented Nov 23, 2015 at 23:49

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