Why are MA(q) time series models called "moving averages"? When I read "moving average" in relation to a time series, I think something like $\frac{(x_{t-1} + x_{t-2} + x_{t-3})}3$, or perhaps a weighted average like $0.5x_{t-1} + 0.3x_{t-2} + 0.2x_{t-3}$.
(I realize these are actually AR(3) models, but these are what my brain jumps to.)
Why are MA(q) models formulas of error terms, or "innovations"?  What does $\{\epsilon\}$ have to do with a moving average?
I feel like I'm missing some obvious intuition.
 A: A footnote in Pankratz (1983), on page 48, says:

The label "moving average" is technically incorrect since the MA
  coefficients may be negative and may not sum to unity. This label is
  used by convention.

Box and Jenkins (1976) also says something similar. On page 10:

The name "moving average" is somewhat misleading because the weights
  $1, -\theta_{1}, -\theta_{2}, \ldots, -\theta_{q}$, which multiply the
  $a$'s, need not total unity nor need that be positive. However, this
  nomenclature is in common use, and therefore we employ it.

I hope this helps.
A: If you look at a zero-mean MA process:
$X_t =  \varepsilon_t + \theta_1 \varepsilon_{t-1} + \cdots + \theta_q \varepsilon_{t-q} \,$
then you could regard the right hand side as akin to a weighted moving average of the $\varepsilon$ terms, but where the weights don't sum to 1.
For example, Hyndman and Athanasopoulos (2013) [1] say:

Notice that each value of $y_t$ can be thought of as a weighted mov­ing aver­age of the past few fore­cast errors. 

Similar explanations of the term may be found in numerous other places. (In spite of the popularity of this explanation, I don't know for certain that this is the origin of the term, however; for example perhaps there was originally some connection between the model and moving-average smoothing.)
Note that Graeme Walsh points out in comments above that this may have originated with Slutsky (1927) "The Summation of Random Causes as a Source of Cyclical Processes"
[1] Hyn­d­man, R.J. and Athana­sopou­los, G. (2013)  Fore­cast­ing: prin­ci­ples and prac­tice. Sec­tion 8/4.  http://otexts.com/fpp/8/4. Accessed on 22 Sept 2013.
