# 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 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.

• Thanks. That takes me from "the name is a mystery" to "the name is inaccurate", but doesn't take me as far as "the name is arbitrary". I'd be most comfortable with the latter. I still don't understand why it's called moving average rather than e.g. lagged error regressive. – Stats newb May 6 '13 at 4:06
• I checked out Box and Jenkins (1976) and found they say the same thing as Pankratz (1983). I have to say, I have had moments of confusion when switching from reading "moving average" in the time-series analysis literature to "moving average" in the technical analysis literature! It'd be nice to know who made the first reference to the term. Track that information down and you might get the "why" answer that you're looking for. – Graeme Walsh May 6 '13 at 6:46
• @Statsnewb Update: According to Spanos' "Statistical Foundations of Econometric Modelling" (1986), Slutsky's 1927 paper "The Summation of Random Causes as a Source of Cyclical Processes" gave rise to the moving average (MA) model. That said, I doesn't seem to be the case that this is the source of the term "moving average" since Slutsky uses the term "moving summation". One step closer to finding this one out! :) – Graeme Walsh May 31 '13 at 11:47

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.