Given a time series, one can estimate the autocorrelation-function and plot it, for example as seen below:

The time series


What is it then possible to read about the time series, from this autocorrelation-function? Is it for example possible to reason about the stationarity of the time series?

Edited: Here I have included the ACF of the differenced series with more lags

ACF after differencing

  • 1
    $\begingroup$ Might it be help to plot the ACF up to larger lags, perhaps a few hundred? $\endgroup$
    – onestop
    Commented Nov 12, 2011 at 13:57
  • $\begingroup$ How do you define stability of the time series? $\endgroup$
    – mpiktas
    Commented Nov 12, 2011 at 14:19
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    $\begingroup$ Did you mean, perhaps, stationarity? $\endgroup$
    – cardinal
    Commented Nov 12, 2011 at 14:33
  • $\begingroup$ Yes, I did mean stationarity. $\endgroup$
    – utdiscant
    Commented Nov 12, 2011 at 17:04

1 Answer 1


this acf suggests non-stationarity which might be remedied by incorporating a daily effect as it appears to evidence structure at lag 24. The daily effect could be either auto-regressive of order 24 or it might be deterministic where 23 hourly dummies might be needed. You could try either of these and assess the results. Further structure appears to be needed. This could be either the need to include level shifts or some form of short-term auto-regressive structure like a differncing operator of lag 1. After identifying and estimating a useful mode, the residuals might suggest further action (model augmentation)to ensure that the signal has fully extracted all information and rendered a noise process that is normal or Gaussian. This will then answer your vague question regarding "stability". Hope this helps !

A slight addition !

The word "suggests" is used as the acf is not the final word on this while the actual data is. In the absence of the actual data the acf is sometimes useful in characterizing the process.

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    $\begingroup$ I think the time-series plot makes it pretty clear that the nonstationarity is not going to be remedied by anything on the order of 24 lags. I suspect the "structure" you see at around 24 lags is actually the high frequency oscillations also very apparent in the first plot. Indeed, as a coarse estimate, I counted the visible troughs between index 3500 and 4000 and I see 20 of them. If a simple lag-1 difference were to take care of it, you'd probably see a fairly pronounced 1/f like decay in the ACF coefficients. It doesn't immediately look like that to me, but there are very few lags plotted. $\endgroup$
    – cardinal
    Commented Nov 12, 2011 at 13:53
  • $\begingroup$ :cardinal What you say might be correct. The actual data would help assess the underlying signal. I don't have access to a data scrubbing program although I have seen some other posters refer to that. Perhaps the actual data could be posted or a reference to a data/screen scrubbing program that performed tha. $\endgroup$
    – IrishStat
    Commented Nov 12, 2011 at 15:20
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    $\begingroup$ Why analyze the ACF before differencing the series? Isn't that nearly universal practice when there's a clear trend? $\endgroup$
    – rolando2
    Commented Nov 12, 2011 at 15:55
  • $\begingroup$ :Rolando The reason I analyzed or commented on the acf is that is what the OP wanted. I agree with your comment that you might want to deal with the "persistency of the acf" by remedying the apparent nonstationarity. The correct remedy may not necessarily be differencing, please see insead.edu/facultyresearch/research/doc.cfm?did=46900 . You might simply simulate a time series that has one or more "drastic" changes in a mean but otherwise is random. Study the acf and will find that it is false evidence that one needs to difference the series to obtain a stationary series. $\endgroup$
    – IrishStat
    Commented Nov 12, 2011 at 20:25
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    $\begingroup$ @IrishStat: thank you for your comment. The paper you referenced certainly seems to be at odds with the vast majority of the time series literature. It seems to be from 1995; how has it been received? It is labeled a "working paper"; did it ever get peer-reviewed? $\endgroup$
    – rolando2
    Commented Nov 12, 2011 at 21:34

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