ARIMA - What is the proper ARIMA model for these data?

Question

I am doing my project on forecasting and due to I have limited knowledge in ARIMA, I would like to ask what is the appropriate ARIMA model for these two data. Both data are monthly.

Figure 1

The first data is shown above, i differenced the data once due to stationary problems (as it cuts off slowly, CMIIW), thus the result of differencing is shown below

Figure 2

I have found that the data is under 0.05 on lag 1 and 2. Does it mean that there are 4 available ARIMA model for this data [(1,1,0), (0,1,1), (2,1,0) and (0,1,2)]? I also tried to use ARIMA(1,1,1) but failed since the solution do not converge.

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Another one is another data below without differencing (different training data), i found that both ACF and PACF cuts off at lag 3 (with 95% confidence level). Thus are ARIMA(2,0,0) and ARIMA (0,0,2) applicable for this or do i need to do differencing first? The reason i am putting MA(2) and/or AR (2) is that some resources found that the sum of p & q should not exceed 2 to prevent overfitting the model.

Figure 3

Thank you very much for your help :D

Answer 1

The first series could be a (0,0,0)(1,0,0)3 or (0,0,0(0,0,1)3 OR a hybrid deterministic model with with three seasonal pulses reflecting a quarterly effect and possible short term arima structure. Only your data knows for sure as the acf and pacf can be heavily influenced/clouded by pulses , level shifts ,local time trends and other "infections" or "opportunities" . If you wish you can post your data and I will try and help further.

EDITED AFTER RECEIPT OF DATA:

Neither series requires differncing : see the acf of KL1 kl2 . Unnecessary differencing can inject structure much like rolling in the mud and then washing your shirt.

I took your two data sets into AUTOBOX which sorts out the best approach/model to your data .. the model for series kl1 is and series kl2 here

The Actual/Fit and Forecast graph for kl1 is here and for kl2 is here

Both series had trend , additive seasonal factors while kl1 had an arima component (1,0,0) and kl2 had three pulses and an evidented error variance change suggesting that weighted least squares be employed ( see http://docplayer.net/12080848-Outliers-level-shifts-and-variance-changes-in-time-series.html ) for a discussion of both Intervention DEtection and Error Variance Deterministic Changes.

SARIMA approaches though usually useful when integrated with deterministic structure is often the preferred route but you had too few observations(24) to pursue this approach.

When following other writers , make sure that they were employing best practices .. which unfortunately is not always the case due to either lack of subject knowledge or lack of software or both.