I am trying to choose the correct ARIMA model. To get a stationary series on which to plot the ACF and PACF on I've done the following transformations on my original series:

  1. natural log
  2. 1st non-seasonal differences
  3. 1st seasonal differences of the non-seasonal differences

Judging from my time series plot(included below) the final series seems stationary but unfortunately I can't interpret the ACF plot. It looks to me like it would need further or different transformations.

This is my data set with the transformations I've done.

I included my plots below. I am using minitab 18 to generate these.

Time Series Plot of Original Series

Time Series Og

Time Series Plot of Transformed Series

Time Series Transformed

ACF of Transformed Series (Lags: 48)


PACF of Transformed Series (Lags: 48)


ACF of 1st Non-seasonal Diff:

ACF 1st Diff

PACF of 1st Non-seasonal Diff:

PACF 1st Diff

  • $\begingroup$ What does the original series look like? $\endgroup$ Nov 17, 2017 at 1:52
  • $\begingroup$ Hi @MichaelChernick! I've just updated my question with a time series plot of the original series. Thanks for your help! $\endgroup$ Nov 17, 2017 at 1:58
  • $\begingroup$ the series are clearly non-stationary judging by the original series $\endgroup$
    – Aksakal
    Nov 17, 2017 at 1:58
  • $\begingroup$ @Aksakal, yes I agree which is why I transformed the series with the 3 steps described in my question and the time series plot of my transformed series looks stationary to me. Am I missing something? $\endgroup$ Nov 17, 2017 at 2:00
  • $\begingroup$ I would run ACF/PACF on a simple first difference of log of series and see what comes. You have annual seasonality. You have less than 200 observations, I wouldn't worry about lags beyond 12, there's not too many observations to calculate them robustly $\endgroup$
    – Aksakal
    Nov 17, 2017 at 2:03

1 Answer 1


enter image description here is a useful model for the most recent 78 values. There is a significant change in parameters over time (approx period 111) which is clearly visual from the time series plot. This conclusion was based on the CHOW test for constant parameters. Seasonal differencing (order 12) is needed without any power transform. Note the need for a quarterly auto-regressive effect often overlooked in the rush to just using 12 period effects suggesting tw0 seasonal effects. Finally a single anomaly is identified at the penultimate point in time (period 187)


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