# How to interpret autocorrelation of residuals and what to do with it?

I was wondering what does it mean when time series residuals have autocorrelation? How should I deal with it?

Since this is a very comprehensive topic, you will have to do research on model checking within the Box-Jenkins framework by yourself, but I try to give a short answer to your short question:

In general, after fitting a time series model (at least you are using a standard model and not assuming a special individual model, where the residuals may behave differently) the residuals should be white noise. So they should have no autocorrelation. If you have significant autocorrelation (you have to test this) and not just one spike at high lag order in your acf/pacf this maybe an indicate that you specified your model wrong. It depends, you could for example plot the residuals along with your other variable in a scatter plot and look, e.g. if there is still a seasonal behaviour, but you should also definitely also test this. If the autocorrelation is significant, yes, this is a problem, since this implies, you missed to include some information. So e.g. your lag order is not sufficient. If the autocorrelation turns out to be not significant, this is not a problem.

So the answer: Test it. See: How to test the autocorrelation of the residuals?

If this answer satisfies you, it would be nice, if you click on the left hook to my post and accept my post as the answer.

• The key part is that if a Box-Jenkins model is correct, the only difference between the model's output and actual data should be random (white) noise. If the difference has a pattern, you've left something out of your model or otherwise mis-specified it. Statistically-significant autocorrelation of the residuals is a pattern -- your model's output differs from reality in a systematic way -- so your model needs work. Mar 4, 2014 at 18:51
• Thank yoy for your answer. I want to know whether the autocorrelated residuals indicate the presence of autocorrelation of independent variables or of the dependent variables? Or, both is possible? Aug 16, 2021 at 7:04

Following Stat Tistician's suggestion, the solution is to use Regression Model with ARIMA error .

library(nlme)

#create some fake data with AR4 noise
n=500
set.seed(42)
v = arima.sim(list(ar=c(0.2,0.2,0.2,0.2)), n)
x= seq(-3,3,length.out = n)
y = 0.5*x + 2*v + 0.5*rnorm(n)

#fit
summary(gls(y~x, correlation = corARMA(form = ~1, p=4,q=0)))