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I'm just a bit confused that when you convert an ARMA process to either a pure AR or MA representation then is it always an AR or MA infinite process that we obtain? For instance an ARMA(1,1), does it have an MA and AR infinite representation?

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Lets restrict attention to $ARMA(1,1)$ processes for the moment. I.e., we consider a process \begin{align} y_t - ay_{t-1} = y_t(1-aL) = e_t(1-bL) = e_t - be_{t-1} \end{align} Where $e_t\overset{iid}{\sim}WN(0,\sigma^2)$ and $a,b \in \mathbb{R}$. Here, $WN$ means 'White noise', i.e. we leave the distribution unspecified, but the first two moments are assumed to exist.

Consider first the invertibility of the lag polynomial $(1-aL)$. What does invertibility mean? Simply that there exists a lag polynomial - say $A(L,a)$ - such that $(1-aL)A(L,a)=1$. Define now $A(L,a) := \sum_{i=0}^{\infty}a^iL^i$. Observe that \begin{align} (1-aL)A(L,a) &= (1-aL)(\sum_{i=0}^{\infty}a^iL^i) \\ (1-aL)A(L,a) &= (1-aL)(1+aL+a^2L^2 + \dots) \\ (1-aL)A(L,a) &= 1 + (aL-aL) + (a^2L^2 - a^2L^2) + \dots \\ (1-aL)A(L,a) &= 1 \end{align} from which it follows that $A(L,a) = (1-aL)^{-1}$ (i.e., the inverse of $(1-aL)$, implying invertibility). Similarly, one can show that $A(L,b) = (1-bL)^{-1}$. More generally, for a first order lag polynomial $(1-\delta L)$, $A(L, \delta)$ will always be the inverse.

From this, you can infer that you may indeed write the above $ARMA(1,1)$ model either as $AR(\infty)$ representation ($y_t(1-aL)A(L,b) = e_t$) or as $MA(\infty)$ representation ($y_t = e_t(1-bL)A(L,a)$).

Crucially, this was linked to the invertibility of any first order lag polynomial. for $ARMA(p,q)$, one cannot in general say that one could always write them in $AR(\infty)$ or $MA(\infty)$ representation. If you are interested in higher order $ARMA(p,q)$, Wold's Representation Theorem might be interesting for you: Roughly speaking, it says that any stationary process (which includes stationary $ARMA(p,q)$ processes) can be written in a $MA(\infty)$ form.

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Any ARMA(p,q) process has an AR or MA infinite process, provided the stationary condition holds. The main purpose of this conversion is in theoretical studies. For example, it is much much easier to find the variance of an ARMA(p,q) process by a conversion to an MA($\infty$) process.

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