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What does it mean that Moving Average Process is first-order, second order, third order, etc. MA(1), MA(2), MA(3)? How to simple understand it, without any complicated formulas, etc?

Kind regards, thank you for help!

kathy kinga
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2 Answers2

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A moving average process is a weighted "moving average" of a stationary white noise process. Usually it is not an actual average but rather a linear combination of previous noise, or error terms, originating from an unknown process we cannot measure. These noise points are assumed to be independent and identically distributed (and usually Normal).

The order of this process is the number of previous points that are taken into account. A MA(1) process will use only the last time point, MA(2) uses two, etc.

Even if you ask to avoid formulas it is usually easier to read this in a formula instead of in words. I will try to explain what I write in detail.

  • MA(1): $ x_{i} = \varepsilon_{i} + a_1 \cdot \varepsilon_{i-1}$
  • MA(2): $ x_{i} = \varepsilon_{i} + a_1 \cdot \varepsilon_{i-1} + a_2 \cdot \varepsilon_{i-2}$
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This means that at a certain time point (here called $i$) the process will take the value $x_{i}$ which is modelled as the current point in the driving noise $\varepsilon_{i}$ and some factor $a_1$ times the previous noise value $\varepsilon_{i-1}$. In the MA(2) we also have a term for yet another point, two steps back in time.

When fitting a model of this type you generally try to infer the noise variance and regression parameters (i.e find the multipliers called $a$ in the formula) so that you can predict future values or in some other way understand your process better. For a higher order model you therefore need to find more multipliers than you would need to do for a lower order model.

while
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Just take the ma polynomial (psi weights )and compute it's inverse. This will be the auto-projective polynomial ( pi weights) reflecting how previous values are weighted. In this way the model's memory in terms of the Y's will be better understood and accepted.

IrishStat
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