# ARMA and Normality

In this book Zucchinni et al on page 3 the authors state "Consider, as an example, the series of annual counts of major earthquakes for the years 1900-2006, both inclusive... For this series, the application of standard models such as ARMA would be inappropriate, because such models are based on the normal distribution. Instead, the usual model for unbounded counts is the Poisson distribution...".

Is this statement correct? If it is, in what sense are ARMA models based on the Normal distribution? ARMA models are defined without invoking any Normality assumption, so then where does the Normality assumption come in?

• If you're asking whether " the usual model for unbounded counts is the Poisson distribution..." is correct, I'd say that's not established that it's the case. I'd think models other that the Poisson (Negative binomial, ZIP, COM Poisson etc) would between them probably be in the majority but the Poisson probably has the larger share (is it reasonable to call a plurality but probably not a majority "the usual case"? I wouldn't.) Nov 12 '21 at 1:27
• @Glen_b: Oh, interesting. I didn't even think that this part of the statement may be contested as well. Thank you for the insight. Nov 12 '21 at 10:30

The ARMA equation is just a recursive equation for the time series, but it is usually paired with an assumption that the error terms in the model are white noise (i.e., IID normal with zero mean). Moreover, if you want the model to be strongly stationary the resulting joint distribution of the time-series values should be normal. (You can find more on the latter aspect of the process in related questions here and here). So as a practical matter, while it is possible to form ARMA models that are not normal, these are also non-stationary and so they are not in common use.

In any case, there is no stationary ARMA process with a non-empty auto-regressive part that can model a count variable (i.e., a non-negative integer). The reason for this is that any stationary non-empty AR-part will always have a coefficient that is not an integer, so it will multiply previous integer count values by a non-integer, which will give a non-integer output in at least some cases. (As an exercise, you might want to have a look at the ARMA model form and see if you can demonstrate this.)

As jbowman points out in the comments, what you can do is to have a hierarchical model where you specify a distribution for the observable count values and then model some parameter of this distribution (e.g., the logarithm of the mean of the count) as a latent variable using an underlying ARMA model. This type of model would allow you to have count values as your observable variable while still exhibiting latent auto-regressive behaviour.

• I realize this is out of the scope which the OP's statement and your answer are referring to, but you can model, say, a Poisson process with an ARMA process on, for example, the log of the mean, akin to a GLM. (You are of course correct about modeling the counts directly with an ARMA process.) Nov 11 '21 at 1:54
• @jbowman: Yes, indeed --- the problem comes in with trying to have an auto-regressive equation operting directly on the integer value itself (as opposed to the log-mean of its distribution). I have edited the answer to add your suggestion.
– Ben
Nov 11 '21 at 3:18
• @Ben: +1: Thank you for this great answer, Ben. A clarification: Wold's decomposition tells us that every weakly stationary series is either a linear process (an infinite sum) or can be transformed to one by subtracting a deterministic component. Then doesn't it mean that we can choose the error term to have any distribution out there and by CLT the resultant series will be Normally distributed. Nov 11 '21 at 13:36
• @Ben: But nonstationary series do not necessarily have a Wold decomposition so there could be a nonstationary series with a Poisson distribution. But we can first-difference some series [that are I(1)] to make them stationary and then we can model the first-differenced series with ARMA (effectively modelling with ARIMA). This would suggest to me you may still be able to use ARIMA models to model series that are not Normally distributed. Where did I get that wrong? Nov 11 '21 at 13:39
• When you are dealing with a time-series with observations having a countable range (e.g., count values), I suspect that you can decompose any weakly stationary series as a linear process simply by virtue of the fact that linear sums allow a countable number of terms. So I suspect you would get some kind of trivial correspondence there that would satisfy the Wold theorem, but it wouldn't be ARMA.
– Ben
Nov 11 '21 at 14:17