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I have read this article which says

Invertible models are not simply to enable us to convert from MA models to AR models. They also have some mathematical properties that make them easier to use in practice.

What are those properties and why would we want our MA model to be invertible.

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Here's one I just came across:

If $X_t$ is an invertible MA(q) process $$ X_t = Z_t + \theta_1 Z_{t-1} + \cdots + \theta_q Z_{t-q} $$ where $\{Z_t\} \sim \text{IID}(0,\sigma^2)$, and $EZ_t^4 < \infty$, then the parameter estimates from an innovations algorithm are asymptotically normal. For more details, check out "Applications of Innovation Representations in Time Series Analysis" by Brockwell and Davis (1988).

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