Consider the following models fitted to the same time series:

  • ARIMA(0,1,1)

  • ARIMA(1,0,0) (that is, AR(1)) with an external regressor

Can I use the AIC (or any other information criteria) to decide which one is better?

I know that in order to use the AIC to compare models these have to be fitted to the same dataset. Does this hold when I = 1? Does this hold when I use an external regressor in my model?

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    $\begingroup$ No, the AICs on differenced model cant be compared to AR $\endgroup$ – Aksakal Jul 21 '15 at 15:20
  • $\begingroup$ Thanks! Is there any other way I can compare these models then? $\endgroup$ – arroba Jul 21 '15 at 15:25

You can't compare AIC of IMAX(1,1) and ARX(1), because in the former there's a differencing and in the latter there is none.

You could go for $R^2$ like measure called FVU. Make sure you apply it on the dependent variable itself, not the differences. This is a "mechanical" approach. I'd hold off on this one though.

The real problem is that you're comparing very different specifications: non-stationary (unit root) and stationary. They're fundamentally different. For instance, if the constant is significant in IMAX(1,1) then your process will wander away from where it starts, while ARX(1) will be jumping around the unconditional mean, i.e. mean-reverting.

You have to make up your mind as to which approach to use using other means than in-sample fit statistics such as AIC. For instance, you may examine the autoregressive coefficient in ARX(1), whether it's close to 1 or not. If it's close then maybe ARX(1) is not a good fit at all. On the other hand, if the residuals from IMAX(1) have very strong negative autocorrelation at lag 1 this may indicate over-differencing etc.

  • $\begingroup$ Thank you for your answer, if I understand you think that $R^2$ like measures wouldn't be good on this case as well, is that right? I'll take a look at the autoregressive coefficients and residuals... I'm aware that comparing $I = 1$ with $I = 0$ isn't ideal, but in my specific case I'm getting different results depending on which unit root test I use, so I can't quite accept / reject the presence of an unit root... $\endgroup$ – arroba Jul 23 '15 at 13:41
  • $\begingroup$ You don't have $R^2$ for MA models, generally, they can't be estimated by linear regression function. $\endgroup$ – Aksakal Jul 23 '15 at 14:16

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