ACF and PACF after differencing time series

Can you please tell me what should be the values of ACF and PACF from the graphs I have attached? I think it should be ($p=0$, $d=1$, $q=3$). I have differenced the data once so $d=1$ and there are three spikes in ACF so $q=3$. But function auto.arima from "forecast" package in R is giving the answer as (0,0,1).

Also, what should be the values for the seasonal $P$, $D$, $Q$ and its period value?

  • $\begingroup$ I believe this is a duplicate question $\endgroup$ – Jon Aug 13 '17 at 16:06

Firstly, inferring from the ACF and PACF plots of the data, I would say your series is already stationary. There is no need for first order differencing.

If the lag-1 autocorrelation is more negative than -0.5 (and theoretically a negative lag-1 autocorrelation should never be greater than 0.5 in magnitude), this may mean the series has been overdifferenced.

Please refer to the following link for a better understanding on selecting the degree of differencing and order of AR and MA for ARIMA models.


  • $\begingroup$ Unfortunately the duke reference is stuck in the 60's and totally ignores major innovations in model identification that have been developed. Modern i.e. more correct/robust model Identification of the order of arima models requires 1) identification and adjustment for pulses, level shifts, seasonal pulses and local time trends 2) validation that the parameters are stable over time 3) validation that the error variance is constant over time i.e. free of deterministic change points and is independent of the observed value of the series $\endgroup$ – IrishStat Jun 2 '17 at 10:05

I'm not going to second guess auto.arima, but try this: get the diagnostics for auto.arima on a test set (after you've fit the model on a training set). Get the same on your model (again using the forecasting package for comparability) See what you get.

The concept of training and test sets, and the use of measures of accuracy, and how to generate them in the forecast package, is covered in Hyndman and Athana­sopou­los's textbook, section 2.5 (it's Hyndman's team that wrote the forecasting package in R). https://www.otexts.org/fpp/2/5


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