I am reading the section about moving average models in Hyndman & Athanasopoulos Forecasting: principles and practice. I am trying to understand the MA(q) model in words.

What is white noise? Is this a differenced series which is normally distributed with mean zero? Is it the difference between an observation and the mean of all observations? I do not know what the book is talking about when it says "white noise".

I can understand what a differenced series is. I can understand what sum of square error means. But what is this "white noise" and where did it come from? What is an error term? What does it mean? Who made this up? Can I see an actual example that I can work out in Excel?

When forecasting with an MA(q) model, do you add the moving average series to the mean to get a forecast? How does it actually work? An Excel document or an example involving actual numbers would really help.

I am having a lot of difficulty understanding what is actually going on in the formula (reproduced below). Some examples with actual numbers would be great. $$ y_t=c+e_t+θ_1e_{t−1}+θ_2e_{t−2}+⋯+θ_qe_{t−q} $$

  • $\begingroup$ Colors of noise: en.wikipedia.org/wiki/Colors_of_noise $\endgroup$
    – Alexis
    Jun 5, 2014 at 22:50
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    $\begingroup$ There is an awful lot here, can you focus your question somewhat? What is "the formula" you are referring to? Can you list it? What is "the book" you are referring to? We'd be happy to help you, but I'm not sure this is answerable at the moment. $\endgroup$ Jun 5, 2014 at 22:57
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    $\begingroup$ The text you cite defines white noise earlier: otexts.org/fpp/2/2 You just need to search in the same document. $\endgroup$
    – Nick Cox
    Jun 5, 2014 at 23:12
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    $\begingroup$ There is an excellent reference book called Applied Time Series Analysis for the Social Sciences by McCleary and Hay or Forecasting with Univariate Box - Jenkins Models: Concepts and Cases by Pankratz that you can consult to learn more about what MA means. $\endgroup$
    – forecaster
    Jun 6, 2014 at 0:47
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    $\begingroup$ I'd suggest if you're looking for intuition, first begin by trying to understand an MA(1). $\endgroup$
    – Glen_b
    Jun 6, 2014 at 2:38

1 Answer 1


It sounds like you are reading about statistical models. Such models include:

  1. A deterministic part (i.e. something that looks like an algebraic relationship; e.g. a line like $y = a + bx$ is a deterministic relationship where $y$ is determined by a linear function of $x$); and

  2. A random part (i.e. something, like noise, that is more or less unknowable... or only knowable in an aggregate sense, like a normal distribution, or some other distribution.). The random part may be called 'noise' or 'error' or something else, depending on the conventions of talking about statistics in a particular discipline. The difference between an observation and the mean of all observations (e.g. $X_{i} - \bar{X}$) is often termed error.

In a moving average($q$) model—e.g. $y_{t} = \mu + \varepsilon_{t} + \theta_{1}\varepsilon_{t-1} + \theta_{2}\varepsilon_{t-2} + \dots + \theta_{q}\varepsilon_{t-q}$—you are explaining $y$ as determined by some mean $\mu$ plus some amount of noise (i.e. a random quantity), plus some amount ($\theta_{1}$) of noise ($\varepsilon_{t-1}$) from last time ($t-1$), plus some (possibly different) amounts of noises to $t-q$ times ago.

I do not know the history of who made the MA(q) model up. Some jerk? Some awesome person? No idea.

I am not gonna post an excel spreadsheet, but it's not too hard to apply. Suppose the contribution of noise at time $t$ is inversely proportional to how long ago the noise happened. Then $\theta_{1} = 1$, $\theta_{2} = 1/2$, $\dots$, and $\theta_{q} = 1/q$, and the MA(q=3) is:

$y_{t} = \mu + \varepsilon_{t} + \varepsilon_{t-1} + \frac{1}{2}\varepsilon_{t-2} + \frac{1}{3}\varepsilon_{t-3}$

Estimating this model is trickier than with a straight up least squares regression... but that's the basic idea of it.

  • $\begingroup$ Does this mean that et = yt - u ? Does et need to come from a differenced series ? Or a stationary series ? is this talking about the change in yt or the actual value of yt ? If I make up a pretend series that follows a perfectly predictable pattern of 2,3,4,2,3,4,2,3,4 and use an order one model, should this method predict the series to continue ? Apologies because this is probably confusing and not using the right terminology I'm just trying to understand the basics a bit better. Thanks. $\endgroup$ Jun 6, 2014 at 2:39
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    $\begingroup$ But what is the εt ? If I need calculate y, I need know εt , what is the value of εt?? $\endgroup$
    – Mithril
    Oct 25, 2018 at 6:56
  • $\begingroup$ @Mithril Ask this question, but not as a comment. $\endgroup$
    – Alexis
    Oct 25, 2018 at 16:09

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