I have used NIST data to calculate ACF in excel which worked fine and coded in our programming language (NOT R).

Here is the plot of ACF:

enter image description here

Now my questions are:

1) From this ACF series how can I we find out this is stationary or not (By looking at values programatically without plotting)? Is there any statistical approach like if X number of values fall out of some MAX_VALUE, then this series fails ACF stationary check or something like that?

2) I have observed that sometimes people using "differenced" series to calculate ACF instead of using original series. Can you help us with when to use "differenced" series and when to use "original" series to calculate ACF?

Thanks for your time!


1 Answer 1


1) For a stationary series, you would ultimately expect to see autocorrelations to decay to zero at higher lags (although that is not enough to indicate stationarity). This does not seem to be the case here (yet at the number of lags you plot?). A statistical approach you might want to look at is a "unit root test" - there are many posts on that topic on SE.

2) If a series is nonstationary in the sense that $Y_t=Y_{t-1}+\epsilon_t$ (a "random walk") for (in the simplest case) iid $\epsilon_t$, its first difference $\Delta Y_t:=Y_t-Y_{t-1}$ will be stationary, because $\Delta Y_t=\epsilon_t$. Such stationary series have "more classical" statistical properties. E.g., estimated coefficients satisfy central limit theorems under suitable technical conditions and hence are normally distributed, which is not generally the case for nonstationary series. So many people prefer to model stationary series.

  • $\begingroup$ Thanks for answer! Will check unit root tests, I couldn't understand your answer for 2, you mean if a series is Y = Yt-1 +E then use differenced series, otherwise use original series? $\endgroup$
    – kosa
    Mar 12, 2015 at 17:18
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    $\begingroup$ Yes, my answer was a bit convoluted. The reason for this is that many people draw your conclusion, although it is in principle also possible (and, in the case of for instance cointegration, even strongly advisable!) to work with undifferenced series. $\endgroup$ Mar 13, 2015 at 7:51
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    $\begingroup$ Second, "nonstationary in the sense that" refers to the fact that stationarity refers to the condition that moments are stable over time, and this is violated in the above random walk case. But it would also be violated in other cases, say when a variance of a series jumps from $\sigma_1^2$ to $\sigma_2^2$ after a fraction $\tau$ of observations. Differencing the series is not going to deal with that type of nonstationarity. $\endgroup$ Mar 13, 2015 at 7:51

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