Why no MAX models?

Question

I'm diving into the field of system identification, black box modeling and forecasting. A lot still has to become clear to me, but one question that came to my mind (and to which the answer might clarify much for me):

I encounter models like AR, MA, ARMA, ARIMA, ARX, VAR and I see some core building blocks in these acronyms:

• V: Vector
• AR: Auto-regressive
• I: Integrated
• MA: Moving average
• X: Exogenous inputs

With these you seem to be able to construct all kinds of models, all the way up to VARIMAX. My question is: Why don't I ever encounter MAX models (Moving Average with Exogenous Inputs)? They should be pretty basic, as they use few building blocks. Is there some reason why this type of model is not very useful or does not make a lot of sense?

Answer 1

Estimation
An AR(X) model is easy to estimate because one can apply OLS for that. On the other hand, MA(X) and ARMA(X) models require maximum likelihood estimation which is more computationally demanding. This becomes especially pronounced in multivariate models, which is why VAR(X) are much more popular than VARMA(X) or VMA(X) models.

Parsimony
ARMA is flexible yet parsimonious -- more so than either AR or MA. No wonder ARMA is quite popular (for univariate time series).

In total,
AR has the computational advantage; ARMA has the benefit of parsimony; while MA... I am not aware of particularly appealing benefits of MA. Of course, MA is useful when we want to look at a process as a sum of weighted past shocks, but that is not the most frequent use of ARMA type of models.