I am new to ARIMA, and I am trying to understand these lag plots.
Are the following ACF and PACF suggesting that the lag of my time series is 4? If I am wrong, please help me understand these plots.
Solved – ACF and PACF plot analysis
acf-pacfarimaautocorrelationinterpretationtime series
Related Solutions
In R acf starts with lag 0, that is the correlation of a value with itself. pacf starts at lag 1.
Just a peculiarity of her R implementation. You can use the Acf function of the package forecast which does not show the lag 0 if that bothers you.
The ACF and PACF are descriptive statistics showing simple correlation and conditional correlation. They are sometimes useful in identifying ARIMA models that 1) have no Pulses (your series does) and 2) have no deterministic time trends or level shifts ( your series seems to have no trend followed by a period that has trend) and whose parameters and model error variances are constant over time. If you actually post your data in an excel format I will try and help you understand more BUT my initial visual assessment is that this is a time series that may have complications/opportunities that will challenge simple (1960 type !) ARIMA analysis or simple model selection (list-based) schemes that use an AIC/BIC criteria to select/identify a model. We should aspire to keeping things/models simple but not too simple so much so that they are of little value !
EDIT AFTER RECEIPT OF DATA:
I took the 269 monthly values and analyzed it with the automatic option in AUTOBOX. After identifying a global model taking into account various pulse effects a significant difference was found between the first 169 values and the most recent 99 values 
. This is visually obvious.The best model for the most recent 99 values was 
and here 
. The residuals from the model appear well behaved [![enter image description here] . [4]](https://i.stack.imgur.com/Kz01u.png)
The ACF of the residuals suggest a slight possibility of a minor seasonal effect but most likely not important .
. The plot of the actual , fit and forecast using standard ARIMA procedures tp compute uncertainty limits is here 
. Note however from http://www.autobox.com/cms/index.php/blog/entry/you-should-have-50-confidence-in-your-confidence-limits that the forecast limits are flawed on two accounts. Taking into account the possibility of future values being effected by pulses i.e.unusual values and that the estimated parameters are not necessarily the poulation parameters we get a more realistic picture 
Best Answer
The threshold statistical significance of the autocorrelations has been noted in the comments and in another answer. What looks interesting is that the autocorrelations in Lag 4 and Lag 8 persist also in the Partial ACF.
Reality should come into play at this point: what are these data? By whatever knowledge you have on the process they measure, is it reasonable to expect that the current level should depend on Lags 4 and 8? If yes, the low estimated strength of autocorrelation is not necessarily an artifact, but an indication that said autocorrelation exists but it is low in strength.