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After differentiating a monthly climatic data with a lag of 12, and being sure that, at least, one more differentiation will turn my series into white noise (ndiffs only results in 1)...I used auto.arima to model my series and I got a result like $ARIMA(p,1,q)(0,0,Q)_{12}$.

I am hesitating why $d=1$ if data has been previously differentiated.

Answer: My data was differentiated without taking logarithms. Variance got better with that!

How can I make sure I choose a good lag for seasonal data?

Answer: try with a subseries plot ggsubseriesplot().

fina
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    https://stats.stackexchange.com/questions/368157/determine-ar-term-from-pacf-plot/368159#368159 discuses how summary statistics (acf/pacf) can often be vague about the underlying model. Post your observed data and I will try and help further. – IrishStat May 07 '19 at 09:02

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I took your 456 monthly values into AUTOBOX which examines series by ITERATIVELY considering anomalies , power transformations , transience in EITHER model parameters or model error variance over time .... https://autobox.com/pdfs/ARIMA%20FLOW%20CHART.pdf

It developed a model which broadly speaking which does not use a regular differencing as yours did BUT it did introduce the need for a logarithmic transformation.

Following is the identified model enter image description here and here enter image description here ... (2,0,0)(1,1,0)12 in logs

The ambiguity that you encountered was to apply regular differencing or not . The acf of the seasonally differenced data is hereenter image description here suggesting an ar(2) for short term memory NOT regular differencing . The message here is that the conclusion not to regularly difference and ar/ma structure should be made AFTER the seasonal differencing is in place.

The two pulses are of minor importance but might be of interest as they reflect values that were identified as extraordinary. AUTOBOX detected an ar(12) factor while you used an ma(12) .

The recent actuals and forecasts are here enter image description herewith the residual acf here suggesting model sufficiency enter image description here

IrishStat
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  • Thank you very much for your analysis. However, what do you mean with "not to regularly difference and ar/ma structure should be made AFTER the seasonal differencing is in place." ? It is necessary to differencing with lag 12 and then another time? – fina May 07 '19 at 11:44
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    given that seasonal differencing is in effect ... it is not necessary to difference any more. That having been said the polynomial ( 1-.618B -.284B**2) suggests a near equivalence to a first differencing operator as it has a root close to 1. Note that the sim of the coefficients fro an AR(2) must be less than 1.0 ( .618 + .284 =.902 which is less than 1.0) – IrishStat May 07 '19 at 11:50
  • Please, let me a final question. If I try `auto.arima(diff(log(x), lag=12), ...)` I should get the same results as you? $SARIMA(2,0,0)(1,1,0)_{12}$ in logs. – fina May 07 '19 at 11:55
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    not so sure . probably not... as logs are needed to be identified and the two readings adjusted for the estimated anomalies . AUTOBOX conducts extensive :investigations" in order to detect significant structure. It found a simple ar(2) versus a "complicated ar(1) and ma(1) " . All things having been said I don't think there would be a large difference in the forecasts between the tour de force of AUTOBOX and the simple fitting a set of models from a pre-specified/determined list (for this series with such a strong signal) – IrishStat May 07 '19 at 12:36
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    probably not ....it would depend on the "whiteness" of the error term and the nature of the forecasts . Of course your first model is flawed by the two anomalies which you did not treat AND the lack of a power transform (logs) causing you to select the inferior auto.arima model. – IrishStat May 07 '19 at 13:21