From the ACF and PACF, what are the non-seasonal and seasonal part of the model here to find an appropriate ARIMA model

acf-pacfarimaforecastingrtime series

I am working on a monthly average dataset and would like to do forecasting.
I had used the codes ndiffs() and nsdiffs() to check how many differences are required to make the series stationary, which gave an output of

ndiffs(y)
[1] 0
nsdiffs(y)
[1] 1

then the seasonal difference at lag 12 was taken.

diff(lag = 12) %>% # seasonal difference
ggtsdisplay()

The ACF and PACF that was displayed are attached here.
enter image description here

Can you explain how to interpret the ACF and PACF and what are the P,p,Q and q values here?

Best Answer

For the analysis, first, as you made a seasonnal difference and no simple difference, you have $D=1$ and $d=0$.

Now, you should analyze the ACF and PACF at the lags $12, 24, 36, \dots$. You should notice a kind of decreasing PACF and 2 pics (lag $12$ and $24$) for the ACF. Then you will propose that $P=0$ and $Q=2$.

After that, you should analyze the $12$ first lag of ACF and PACF to find $p$ and $q$. Here, you can see that there is no pic at any lag. Then $p=0$ and $q=0$.

After all, the model should be an ARIMA$(p=0,d=0,q=0)\times(P=0,D=1,Q=2)$. It is a pure seasonal process.

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