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I'm trying to determine the AR and MA of a time series by looking at the ACF/PACF plots but they doesn't look like the classic examples of the textbooks.

it has 264 monthly values of a stock index from 2000 to 2022. The series is not stationary so I take a diff and then plot acf/pacf. As you can see the first and second lags of either plots are not significant but the third lag is. I don't know how to interpret this.

enter image description here

Thanks for your help! Edited just to add some extra info (commands are from R)

Regarding UNIT Roots:

forecast::ndiffs(x) gives me "1", so it should be safe that taking a first difference is ok

Regarding seasonality:

forecast::nsdiffs(x) gives "0"

A plot of the time series: plot(x)

enter image description here

after taking the first difference: plot(diff(x))

enter image description here

and finally a descomposition using stl(x)

enter image description here

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    $\begingroup$ While the third lag is significant... it does look like it could be just chance. $\endgroup$
    – user2974951
    Jun 14, 2022 at 7:35
  • $\begingroup$ take the log before differencing $\endgroup$
    – Taylor
    Jun 14, 2022 at 19:45
  • $\begingroup$ I did it. The ACF/PACF are the same with a different scale. $\endgroup$
    – ulises
    Jun 14, 2022 at 20:46
  • $\begingroup$ A hint of quarterly seasonality tends to show up in certain markets where managers need to report their holdings on a quarterly basis. Often, a close look at the data shows unusual activity right at the end of each quarter. Thus, the spike at lag 3 is no surprise and suggests exactly what to investigate for understanding the data further. $\endgroup$
    – whuber ♦
    Jun 15, 2022 at 20:54

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This could indicate the presence of seasonality in your data. The initial impression looks like your series could be a 3-period seasonal-AR or seasonal-MA process because the 3rd lag is significant, followed by a significant 6th lag in ACF. In PACF, 3rd lag is again significant (the 6th lag is high but insignificant).

The plot of the original variable can give more insight into the patterns of behaviour and seasonality, or if seasonality exists in data or not. Check for any continuous repeating patterns of peaks and valleys in the variable over time (in this example, the pattern might repeat every 3 periods/months). If the peaks are of similar height over time, it indicates additive seasonality. If the peaks are increasing in height with an increasing trend, it indicated multiplicative seasonality.

You could also try seasonal differencing with a lag of 3 periods (3 months in your case). Seasonal differencing will be enough if the seasonality is additive. If it's multiplicative seasonality, you can use the SARIMA model for estimation.

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  • $\begingroup$ First, if there were seasonality, you would see signs of it on all integer multiples of 3 (3, 6, 9, ...), but here you do not. Second, differencing a series that does not have a unit root is asking for trouble (keyword: overdifferencing). $\endgroup$
    – Richard Hardy
    Jun 14, 2022 at 7:43
  • $\begingroup$ There is evidence of the 3rd and 6th lag being significant. But that alone is not enough to conclude that seasonality exists. That is why I mentioned that there could be seasonality in the data, but needs further checking. It could indicate seasonal AR or MA terms since both AFC and PACF die out after the 3rd or 6th lag. Again, it needs further checking to make sure this is the case. Only ACF and PACF may not be enough and we might need to check the behaviour of original variables with time or even detrend it. $\endgroup$
    – Spur Economics
    Jun 14, 2022 at 10:30
  • $\begingroup$ Seasonality may not always be visible along with all multiples (3, 6, 9 and so on here). It could die out after a few multiples in both ACF and PACF, or in any one of them. $\endgroup$
    – Spur Economics
    Jun 14, 2022 at 10:36
  • $\begingroup$ OK, SMA(3) could make some sense here if we only looked at the ACF-PACF plots and ignored the fact these are stock returns which should normally be unpredictable (and the plots being completely in line with an assumption of white noise; recall that 5% of bars should stick out purely by chance). Yet there is no way to advocate for seasonal differencing based on the graphs. Also regarding the 6th lag is high: it only seems high on a scale from roughly -0.17 to +0.17 as in the graph. The actual value is about 0.1 which implies an $R^2$ of about 0.01. This is really low. $\endgroup$
    – Richard Hardy
    Jun 14, 2022 at 10:44
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    $\begingroup$ The point I am trying to get across is that additive seasonality absolutely does not warrant seasonal differencing. Looking at the ACF and PACF plots, it is also absolutely clear that seasonal differencing is unwarranted and would most likely be harmful. $\endgroup$
    – Richard Hardy
    Jun 14, 2022 at 11:30

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