How should I determine the data generating process from the correlogram below? This is non-seasonally adjusted monthly data that has been 1st differenced. I am trying to conduct univariate time series analysis for a homework assignment and was instructed to use the Box-Jenkins methodology.

I tried to specify the model by first including AR and MA terms for each spike in the correlogram. I dropped the AR/MA terms that were not significant and continued until I ended up including AR(12) and MA(13) but my inverted roots were .99. I know this is undesirable.

How should I go about specifying my univariate model from this correlogram?

1st Differenced

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Seasonally Differenced and 1st Differenced

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  • $\begingroup$ You don't include AR and MA terms for every spike. You have spikes at lag 6 and 12 in both, but you probably won't need to fit a SARMA(2,2), since either a pure seasonal MA or AR might look like that. The critical information which could differentiate those three possibilities would be the behavior of the correlation at lag 18. $\endgroup$
    – Glen_b
    Feb 22, 2015 at 10:05
  • $\begingroup$ @Glen_b Having the advantage of being a highly focused specialist in time series, I have seen data sets presenting this kind of correlogram reflecting the need for ar(12) AND no ar(6) structure. After incorporating an appropriate seasonal component perhaps seasonal dummies or seasonal ARIMA and any anomalies (deterministic structure) that would incorrectly downwards bias the sample acf it might be possible to identify additional non-seasonal AR/MA structure. In cases like this the original data is always useful to eliminate guesswork. $\endgroup$
    – IrishStat
    Feb 22, 2015 at 12:31
  • $\begingroup$ @Glen_b I added the correlogram of the 1st difference up to 36 lags. I also included the correlogram after 1st differencing and seasonally differencing the data. Using these correlograms, how should I determine if seasonal differencing was necessary? $\endgroup$
    – Amaziah
    Feb 22, 2015 at 16:59
  • $\begingroup$ Also, it may be helpful to let us know the number of observations (T). As a general rule, one inspects T/4 of the autocorrelations. $\endgroup$ Feb 22, 2015 at 21:36

1 Answer 1


If applying the Box-Jenkins methodology, I'd consider an ARIMA(0,0,0)(2,0,0) to be among my set of tentative models.

Justification: the ACF decays at seasonal lags and there are two (possibly three?) significant spikes in the PACF at seasonal lags; namely, six and twelve (and eighteen?). The ARIMA(0,0,0)(3,0,0) model would also be lined up for consideration, but by appeal to the principle of parsimony, I'd begin with the ARIMA(0,0,0)(2,0,0) model and go from there.

Ultimately, if you follow the Box-Jenkins methodology, you'll need to perform an iterative modelling procedure to decide upon a final model.

Additional Information

There are books dedicated solely to showing how to implement a Box-Jenkins ARIMA model selection strategy. For example, Forecasting with Univariate Box-Jenkins Models: Concepts and Cases by Alan Pankratz does this, so you may find it to be a useful guide if you want a longer term solution to developing ARIMA modelling skills.

Alternatively, you could avoid the subjectivity (and difficulty!) of trying to select models based on interpreting ACF and PACFs. There are a various other methodologies that remove this subjective element of interpreting ACFs and PACFs, which include sequential testing and minimizing some information criteria. For more details, you could refer to Chapter 7 of Judge, Griffiths, Hill, and Lee (1980), The Theory and Practice of Econometrics.

Lastly, the output shown in the question appears to be from the EViews software(?). If you had chosen to do your analysis using the R software, you could use handy tools (directly, sans external interfaces) like the forecast package, which allows automatic selection of an ARIMA model given a time-series via its auto.arima() function.

  • 1
    $\begingroup$ Thank you very much. Your answer and justification have helped me better understand the thought process of specifying the ARIMA model. I will look into both of the resources you mentioned. Additionally, thank you for enlightening me on alternatives to Eviews. That is what we are using for my program but I will seriously consider expanding my experience with different softwares. $\endgroup$
    – Amaziah
    Feb 24, 2015 at 4:28

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