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Suppose we have $X_{1}, X_{2}, ..., X_{n}$ sequence of $iid$ random variables with mean $\mu$ and standard deviation $\sigma$. By definition, the time series $x_{1}, x_{2}, ..., x_{n}$ is a stationary process with mean $\mu$ and standard deviation $\sigma$. If so, is it legit to compute a moving average series for the process as follows:

$$s_k=\frac{x_{1}+x_{2}+...+x_{k}}{k}?$$

As the process is stationary, we assume that the mean should be time invariant, i.e., $\mu$. Then if it is legit to compute moving average series, how we interpret it in the context of stationarity? (I assume if it makes sense, then interpretation should relate to long- and short-run dynamics of the series. )

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    $\begingroup$ $s_k$ obviously is "legitimately" computed (there's no obstacle at all to forming linear combinations of finitely many random variables)--but $(s_k)$ is not stationary, as you can tell by calculating that its standard deviation is $\sigma/\sqrt k,$ which varies with $k.$ $\endgroup$
    – whuber ♦
    Commented Jan 9 at 15:33

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Yes it is legit to compute a moving average on a stationary series, and it can be convenient too, for the most various reasons. Stationarity in itself means the process has an unconditional constant expected value, but a stationary process can still have different expected values for different time points, conditioned to the previous values. For instance a process can have a tendency of giving new observations that are close in value to the previous ones, and still be stationary as long as these values tend to go back to the unconditional mean over time.

This answer wouldn't be complete without citing the notorious MA models, which make a weighted moving average of the process in order to make a forecast of new observations. I'm bringing them up because estimating MA weights requires stationarity.

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  • $\begingroup$ To me a weighted moving average of the process sounds more like an AR model than an MA model. The latter works with residuals, not the original process. $\endgroup$
    – Richard Hardy
    Commented Jan 9 at 9:30
  • $\begingroup$ I admit I used the MA process as an example because of the name, and that was probably an error, because the name is quite misleading (the fact that MA processes can have negative parameters is particularly problematic), but an AR model is not a better example. Also MA processes can be refactored as infinite AR processes (and viceversa). $\endgroup$
    – carlo
    Commented Jan 9 at 15:48
  • $\begingroup$ AR model is the general case of what you describe. $\endgroup$
    – Richard Hardy
    Commented Jan 9 at 20:09

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