I'm reading about correlograms, and how they can be used to detect non-stationarity.

Supposedly, if the autocorrelation constant is significant, and/or declines slowly, we would deem the time series non-stationary.

An example of this can be seen here : http://3.bp.blogspot.com/-4aVTQ9XbMRc/T8lNSFLc8FI/AAAAAAAAAgg/C2otWD3QInw/s1600/corr_inc1.gif (source: davegiles.blogspot.com)

But why is this the case?

The definition of stationarity does not state that there can be no covariance/autocorrelation. It only states that the covariance needs to be the same between say, $Y_t$ and $Y_{t+1}$ as between $Y_{t+1}$ and $Y_{t+2}$)

Shouldn't we need "multiple" correlograms, to check if this were the case?

That is, to check that the covariance at the same number of lags is the same, even with different starting points?

I'm taking my first year of statistics, so please do say if I've gotten something wrong ^^

  • $\begingroup$ Could you quote or direct us to some source that claims correlograms are tools for detecting non-stationarity? A search of related threads on this site uncovers some posts that explicitly controvert that claim, such as stats.stackexchange.com/questions/137967 . $\endgroup$
    – whuber
    Dec 28, 2015 at 20:50
  • $\begingroup$ From "Basic Econometrics" (5 ed) by D. Gujarati and D. Porter p. 751 ch.21.8 ("Tests of stationarity") "The most striking feature of the correlogram is that the autocorrelatoin coefficieents at various lags are very high even up to a lag of 33 quarters. As a matter of fact, if we consider lags of up to 60 quarters, the autocorrelation coefficients are quite high; the coefficient is about 0.7 at lag 60. Figure 21.7 is the typical correlogram of a nonstationary time series. The autocorrelation coefficient starts at a very high value and declenes cery slowly toward zero as the lag lenghtens" $\endgroup$
    – Magnus
    Dec 28, 2015 at 20:57
  • $\begingroup$ On p. 753 Gujarati goes on to describe how to test if the autocorrelation constant is statisticlly significant. This is done (I must assume, since it's part of the subsection) to test wether of not a time series shoud be considered stationary or not $\endgroup$
    – Magnus
    Dec 28, 2015 at 20:59
  • $\begingroup$ How do you infer that? Does he state explicitly that this is a test of stationarity? One would instead think that a test of an autocorrelation coefficient would not make any sense for a nonstationary process, suggesting Gujarati's concern with stationarity might have been as a precondition for this test to be applicable. Your quotation, though, is intriguing, because it points out how a stationarity assumption can be seen as a modeling decision in some cases, rather than as an inherent property of a process: one analyst's non-stationarity is another analyst's correlation. $\endgroup$
    – whuber
    Dec 28, 2015 at 21:13
  • 1
    $\begingroup$ Re what are they used for: I'm sure a search of this site will turn up hundreds, if not thousands, of concrete examples of how people use the ACF and PACF plots to help characterize time series data and choose appropriate models for them. Please note that a correlogram usually has no meaning unless the process is already known (or assumed) to be stationary (possibly after a deterministic function is subtracted in order to produce a constant mean). $\endgroup$
    – whuber
    Dec 29, 2015 at 2:00

1 Answer 1


The quote in your comment claims too much but does relate to something real, and that something can be useful in figuring out suitable models for data.

If you have an $I(1)$ series (a very specific kind of nonstationarity), you should see an ACF that doesn't exhibit the kind of geometric "decay" in the characteristic manner that you see with data generated by typical lowish-order stationary ARMA. [There will still be a tendency to decrease in the ACF of an $I(1)$ series, but it often tends to look more "linear" than geometric]

[You can get some sense of this by actually generating data from stationary low order AR and ARMA models and comparing the ACFs from (say) that of a random walk. It's worth trying for a number of different models]

You will also tend to see it with $I(2)$ (etc) series, but data that's reasonably modelled by $I(1)$ or is at least stationary after differencing tends to be more common.

So if you regard the major possibilities as either the series being I(1) or low-to-moderate order stationary ARMA, then the ACF can sometimes be of some help in distinguishing them (but you'd typically difference and look again before saying too much).

  • 1
    $\begingroup$ So...the slowly declining autocorrelation isn't synonumous with non-stationarity (since it still doesn't rule out that autocorrelations between $Y_t$ and $Y_{t+1}$ and between $Y_{t+1}$ and $Y_{t+2}$ would be the same), but rather a characteristic shared by many non-stationary time series? $\endgroup$
    – Magnus
    Dec 29, 2015 at 13:11
  • 1
    $\begingroup$ @Magnus A characteristic shared by series that are well-described by a common form of nonstationary model, at least. $\endgroup$
    – Glen_b
    Dec 29, 2015 at 14:08

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