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I’m taking a first graduate course on Time Series Econometrics and this question regarding ARMA processes has come to my mind. I see that the ability to represent AR processes as MA or the converse has theoretical advantages when trying to establish some results and this holds true even with ARMA models. However, from the applied econometrician point of view, why one would care about inverting a process and getting a representation that has an infinite number of lags?

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I don't think the answer differs for an applied econometrician versus any other practitioner using an ARMA model. Nor is there a genuine distinction between "theoretical advantages" and advantages "from the applied [insert-profession-here] point of view". The bottom line is that if a particular form of a model yields some "theoretical advantage" by allowing some use of relevant theory, then that theoretical advantage can be deployed by any applied practitioner using the model. (This is merely a broader recognition of the illusory dichotomy between theory and practice.)

Advantage of the MA($\infty$) form: The main value of inverting to obtain an MA($\infty$) representation is that it gives the observable time-series variable as a sum of unobserved terms, without polluting the equation with auto-regressive terms. This is convenient because it allows you to obtain moments of the observable time-series variables, and possibly even its distribution (e.g., in the case of normal error terms). If you have your time-series in this form then you have:

$$Y_t = \mu + \sum_{i=0}^\infty \theta_i \varepsilon_{t-i}.$$

With a standard white-noise error process $\{ \varepsilon_t \} ... \sim \text{IID N}(0, \sigma^2)$ the MA form immediately gives you the covariances between the observable values, which are:

$$\mathbb{Cov}(Y_t, Y_{t+k}) = \sum_{i=0}^\infty \theta_i \theta_{i+k}.$$

This also gives you correlation values between the observable values, which means that you have your auto-correlation function (ACF). The MA form makes the correlation structure of your time-series clear in a way that the initial ARMA form does not. By inverting the auro-regressive aspects of the ARMA model, the MA form allows you to see the combined effect of the auto-regressive part and the moving average part in determining the auto-correlation function for the series.

The practical value of this for any applied practitioner is that it allows them to match the observed sample auto-correlation with an appropriate theoretical model of the ARMA form (which usually also involves inspection of partial auto-correlation plots and some formal estimation techniques). The theoretical advantage yields a practical advantage to the practitioner.

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