**Update (2019-06-25):** changing title from *"Do non-invertible MA models make sense?"* to distinguish it from [Question 333802](https://stats.stackexchange.com/questions/333802/why-do-we-care-if-an-ma-process-is-invertible).

While reviewing MA($q$) models, I came across [these slides](http://www.etsii.upm.es/ingor/estadistica/Carol/TSAtema4petten.pdf) (Alonso and Garcia-Martos, 2012). The authors state that, while all MA processes are stationary, if they are **not invertible** you have 

> 
> "*the paradoxical situation in which the effect of past observations
>  increases with the distance.*"
> 
> 
> 

This can be seen in by the decomposition of the MA(1) process:
$$ y\_t = \epsilon\_t - \theta \epsilon\_{t-1} $$
into
$$ y\_t = \epsilon\_t -\sum\_{i=1}^{t-1} \theta ^i y\_{t-i} - \theta^t \epsilon\_0,$$
where clearly $|\theta|>1$ translates into history having more and more influence over the present. Two things about this bother me:

1. It's not hard to imagine a situation where there's a one-time period lag in the effects of something
2. This [Cross Validated Post](https://stats.stackexchange.com/questions/333802/why-do-we-care-if-an-ma-process-is-invertible) has an answer which claims:

> 
> "*Invertibility is not really a big deal because almost any Gaussian, non-invertible MA(q) model can be changed to an invertible MA(q) model representing the same process*"
> 
> 
> 

Is it true that the effect of past observations increases with the distance? If so, does that make the models unfit for describing real world phenomena?

**Update (2019-11-09)** Found this in the text *Time Series Analysis and Its Applications* (Shumway and Stoffer, page 85) which also supports the case that it doesn't really matter if an MA model in non-invertible, but we may want to chose the non-invertible version of the model for convenience.
[![Time Series Analysis and Its Applications Page 85](https://i.stack.imgur.com/ubSuA.png)](https://i.stack.imgur.com/ubSuA.png)