Stationary time series having unusual ACF and PACF plots

Question

I'm analysing a highly stationary time series and while plotting ACF and PACF I noticed a strange bump at a later lag very close to the 0.5 threshold level. Does it affect the AR degree=2 and AM degree =1 in the ARIMA model?

Results of Dickey-Fuller test
Test Statistic -6.218402e+00
p-val 5.296241e-08
#Lags Used 2.000000e+00
Number of Observations Used 4.380000e+02
Critical Value 1% -3.445368e+00
Critical Value 5% -2.868161e+00
Critical Value 10% -2.570297e+00

Answer 1

Does it affect the AR degree=2 and AM degree =1 in the ARIMA model?

If it's daily data, then you should look up for what those lags around 34 could mean (economically, financially, etc), which depends on the nature of your variable. Statistically, it could be a concern, but the type of variable could overcome that. Therefore that bump could be spurious (not a concern).

Besides that, a "true" stationary data shouldn't be almost equal to 1 on the first lag at partial correlation. There may be other issues, like structural breaks, that can cause false positive of a unit root test.