I am practicing MA and AR modelling by using autocorrelation and partial autocorrelation values. My data is in the image below; I can see that only at lag 12 there is a value that might be considered but I have a dilemma: what model is this if any?

  1. To be a MA of higher order the autocorr should be 0 After lag 12, but not before.
  2. To be an AR I should see value oscillating and decaying up to lag 12.

Am I correct in my thinking or am I wrong?

I performed the Box-Jenkins test (lbqtest) with Matlab and only at lag 12 I have the rejection of the null hypothesis ( h =1 ) for an Alpha of 0.10.

enter image description here

  • $\begingroup$ What's the sample size? If it's greater than n=80 (so that n/4>20) then you may want to inspect higher lags, in particular lag 24 and possibly lag 36. Have you calculated a t value for the autocorrelation coefficient at lag 12? If so, what is it? Lastly, what do the blue lines in the ACF represent? Since you're trying to identify a tentative model, make sure that these are based on Bartlett's errors. $\endgroup$ Commented May 22, 2013 at 5:20

2 Answers 2


Though I am not an expert, the spike is fairly easy to interpret if you are evaluating monthly data.

One thing you may be interested in is trying to correct your data for seasonality as this may improve your regression results.


First of all, what kind of data are you analyzing? That could give you an idea of whether you should expect a spike at h=12. You are mostly correct in your reasoning about the look of an ACF for different ARMA models. However, if you had an MA(12) process like

$$ Y_t=\theta_{12}\epsilon_{t-12}+\epsilon_t $$

then you could have an ACF plot that is approx. zero for h=1,...,11, nonzero for h=12, and then approx. zero afterwards.

In your case, this doesn't look like anything worth modelling. Remember that if you test a hypothesis of $H_0$: the autocorrelation at lag $h$ is significantly different from zero at significance level $\alpha=0.10$, on average you expect to make a type I error 10% of the time. This is probably one of those times.

If you are curious, try fitting some MA(12) models and look at some model selction criterion (AIC, AICc, BIC, etc.) and see if any model gives a better fit than $Y_t=\epsilon_t$, $\epsilon_t \overset{iid}{\sim} N(0, \sigma^2)$. I am guessing no.


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