# confused about acf/pacf of this stock index

all

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. 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) after taking the first difference: plot(diff(x)) and finally a descomposition using stl(x) • While the third lag is significant... it does look like it could be just chance. Jun 14, 2022 at 7:35
• take the log before differencing Jun 14, 2022 at 19:45
• I did it. The ACF/PACF are the same with a different scale. Jun 14, 2022 at 20:46
• 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.
– whuber
Jun 15, 2022 at 20:54

• 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. Jun 14, 2022 at 10:44