I am analysing the stock index returns data for few countries. From observation of the ACF and PACF there seem to be no significant peaks at any lags.

However, applying the auto.arima function from the forecast package yields an ARIMA(4,0,4) model. I am confused with such results as the ACF/PACF do not even show significance. The model seems to pass the portmanteau test.

I am using the Box-Jenkins approach for my project and do not know how to explain this. I wonder if any analysis is possible with no significant acf/pacf lags, and whether if I can state that ARIMA(4,0,4) is the best fitted model only considering AIC SBC etc. (for instance stating that 'although acf/pacf does not show any significant lags, but considering the possible models' AIC/SBC values of p,q<=5, ARIMA(4,0,4) model was found to be best fitted')

There does not seem to be seasonality present in my data as there was no pattern in ACF/PACF.


1 Answer 1


It is one of the properties of ARIMA models, that the AR (autoregressive) and the MA (moving average) terms tend to "eliminate each other out" or "they work in opposite direction" (if you want to think about this in this way). Therefore a model (4,0,4) might be actually very close to (0,0,0) as the 4 AR terms "eliminate" the effect of 4 MA terms (depends on the coefficients).

Although I am not able to give theoretical foundation for this, it is for practical purposes recommended to fit (if possible) either AR models or MA models, because mixing both parts may lead to the above described behavior. It is also a practical rule to not go (if possible) above order of 2 for AR or MA terms.

I have faced similar problems and I also witnessed that if you allow ARIMA models with higher orders (>2) and both (AR and MA) terms, it is often the case that models of higher and higher orders will be fitted with better AIC criterion. It seems here that the AIC is not fully able to penalize the kind of "overfitting" that emerges here.

  • $\begingroup$ I have seen and wondered about this same phenomenon with increasing AIC with increasing lags in fit, but seemingly no important differences elsewhere! $\endgroup$ Commented Feb 20, 2021 at 0:40

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