non-normal residuals in ARIMA

I am trying to fit an ARIMA model and I have already evaluated a few variations which I finally selected ARIMA(1,1,3) model. The residuals seems to be uncorrelated and all the lags are significant. However, in this model and even in all the others I tried, the normality condition for residuals is always violated and they look like this when plotted against normal distribution.

Should I transform my data somehow? I have already used natural log and first differencing in order to make the data stationary or can I ignore the assumption when I have a lot of observations (1,5M) ?

• Why do you expect that errors should be normally distributed, whether or not the other checks you mention look good? There is no particular reason they should be. – Christoph Hanck Mar 24 '15 at 13:41
• I have read in some works, that the normality of the residuals is imortant so that the t-statistics of the AR and MA terms are valid. As I understood, if the residuals wont be normally distributed I may mistakenly exclude some significant term or include some insignificant one. But I am not sure and therefore I came here to ask :) – m3d1v0 Mar 24 '15 at 13:49
• Under very strict conditions, normality of the errors is important to get that t-statistics are indeed t-distributed. These are not met here anyhow, so that you need to rely on asymptotics to get a distribution for your t-statistics. Now, if you rely on asymptotics, normality of the errors is not (that) important anymore. – Christoph Hanck Mar 24 '15 at 14:15
• I found two conflicting opinions here on Cross Validated: Aksakal says that MLE is not reliable if the normality assumption is violated. Rob J. Hyndman says that non-normality is not that big of a problem. I am sure Rob J. Hyndman is very experienced and authoritative, I just wonder what the explanation for his observation is. – Richard Hardy Mar 24 '15 at 17:16
• @ChristophHanck, your statements about asymptotics make sense in a linear regression setting. However, ARIMA models may be more sensitive to non-normality since the normality assumption is used in the maximum likelihood estimation (but not in OLS) and the ARIMA likelihood is quite complicated. Could you give an explanation or a reference supporting your opinion? I share the doubts of m3d1v0... – Richard Hardy Mar 24 '15 at 17:18

Your QQplots could indicate $t$-distributed error terms might fit better. You could try to fit an ARIMA-model with $t$-distributed innovation terms, and see if the fit is very different from the fit you have now. I have done such things with the bugs software, there are certainly other options.