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I am having trouble understanding why we care if an MA process is invertible or not.

Please correct me if I'm wrong, but I can understand why we care whether or not an AR process is causal, ie if we can "re-write it," so to speak, as the sum of some parameter and white noise - ie a moving average process. If so, we can easily see that the AR process is causal.

However, I'm having trouble understanding why we care whether or not we can represent an MA process as an AR process by showing that it is invertible. I don't really understand why we care.

Any insight would be great.

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    $\begingroup$ casual $\rightarrow$ causal $\endgroup$ – Richard Hardy Feb 28 '19 at 10:33
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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 by changing the parameter values. This is mentioned in most textbooks for the MA(1) model but it is true more generally.

As an example, consider the MA(2) model $$ z_t = (1-0.2B)(1-2B)w_t, \tag{1} $$ where $w_t$ is white noise with variance $\sigma_w^2$. This is not an invertible model because $\theta(B)$ has one root equal to 0.5 inside the unit circle. However, consider the alternative MA(2) model obtained by changing this root to its reciprocal value of 2 such that model takes the form $$ z_t = (1-0.2B)(1-0.5 B)w_t' \tag{2} $$ where $w_t'$ has variance $\sigma_w'^2=4\sigma_w^2$. You can easily verify that models (1) and (2) both have the same autocovariance functions and hence specify the same distribution for the data if the process is Gaussian.

To make the model identifiable such that there is a one-to-one mapping from $\theta_1,\theta_2,\dots,\theta_q,\sigma_w^2$ to the distribution of the data, the parameter space is therefore by convention restricted to that of invertible models. This particular convention is preferred since the model then can be put directly in AR$(\infty)$ form with coefficients $\pi_1,\pi_2,\dots$ satisfying the simple difference equation $\theta(B)\pi_i=0$.

If we didn't impose this restriction on the parameter space, the likelihood function of an MA$(q)$ would in general have up to $2^q$ local optima (if the MA polynomial has $q$ distinct real roots) which is something we want to avoid.

You can always move roots from inside to outside the unit circle with a corresponding change in the white noise variance using the above technique, except in cases where the MA-polynomial has one or more roots exactly on the unit circle.

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  • $\begingroup$ Very interesting! $\endgroup$ – Richard Hardy Feb 28 '19 at 10:36
  • $\begingroup$ Yes, I don't know why this isn't stated more clearly in textbooks. You can see this "trick" being used by function maInvert inside R's arima function to ensure that the parameter estimates correspond to a invertible model. $\endgroup$ – Jarle Tufto Feb 28 '19 at 10:40
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There are many reasons. In addition to an answer for a similar question I had a while back (What are the properties of invertible Moving Average), I will also say that knowing whether or not a process is invertible can help you figure out how it relates to another process. When you calculate a sample cross correlation, that random estimator will be extremely noisy if both of the series have nonzero autocorrelation functions. If $X_t$ is invertible, then you can write $$ X_t = \theta(B)Z_t $$ as $$ \xi(B)X_t = Z_t $$ nonambiguously, which means, you could filter/pre-whiten your data using $\xi(B)$. For more details, see this: interpreting the cross correlation function between this process and another one.

Also, judging by the title in the reference in the first link, those authors have a lot more to say about this matter. Unfortunately I cannot find a copy of that book/paper on the internet. If anyone can find that, please let me know.

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