# 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

## 3 Answers

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.

Before you do an ARIMA model you have to check if the data is be stationary and if any seasonality should be defined using autocorrelation (ACF) and partial correlation functions(PACF). The auto correlation should follow the 95% confidence bands. Stationary data is detected using a run sequence plot or auto correlation.

If it is not stationary you might have to detrend it. My guess is it was not stationary.

• Yes, that is exactly what I did. First I took a log transform because I am using exchange rate data and I wanted to smoothen a bit the excesive pikes. Then I run a Dickey-Fuller test which suggested that my series is not stationary, hence I differenced it by 1 and then it looked much beter visually and it passed the Dickey-Fuller. I also tried to regres the differenced values on various trends(linear,quadratice...) but all options turned to be insignificant. Therefore I think my data are stationary. Moreover, when I was evaluating the ARIMAs, there was no autorroelation in the residuals. – m3d1v0 Mar 24 '15 at 13:43
• The wording of the answer may be a bit confusing, so let me make a couple of remarks. (1) Statistically significant ACF or PACF values do not mean that the underlying series is nonstationary. (2) There are multiple different forms of nonstationarity, and different methods to detect them. (Augmented) Dickey-Fuller test mentioned in the comment above is one option. Interpreting autocorrelations is less straightforward. – Richard Hardy Mar 24 '15 at 16:56
• @m3d1v0 One doesn't/shouldn't take logs to "smoothen a bit excessive peaks" that is the role of pulse Indicators.. This post speaks to the role of power transforms of which a log is one choice stats.stackexchange.com/questions/18844/… . You have to carefully select the appropriate remedy for the problem at at hand. Transformations are like drugs ... some are good for you and some are not. – IrishStat Sep 27 '15 at 22:44

If the residuals contain pulses or level shifts this can lead to "non-normality" . Try detecting Interventions and add them as necessary. Another way residuals can exhibit non-normality is if there is a deterministic change in error variance suggesting Weighted Least Squares OR if the model's parameters are not constant over time suggesting data segmentation..