# Ljung-Box test for squared residuals values interpretation

I ran the Ljung-Box for a single series and find that the statistic is very high.

I am using 20 lags so the critical value is 31.4104 and the statistic is greater than that. So my conclusion is that the data is not independently distributed. I also test the squared residuals for the same series. Again I have high values. However, it is not clear to me how to interpret this last result? I am using R:

LjungBox(obj,lags= 20,order=0,season=1,squared.residuals=FALSE)


And tells me that when squared.residuals = TRUE, then apply the test on the squared values to check for Autoregressive Conditional Heteroscedastic, ARCH, effects.

How can I interpret this test and the results?

• You talk about a series first and about residuals thereafter. Are you dealing with residuals from some model throughout or first with some original series and then residuals, or with an original series throughout? – Richard Hardy Oct 25 '17 at 18:44
• @RichardHardy It is the same series all the way. No model has been applyed yet to the data. But I guess I might find heteroscedasticity like in real data. Thanks – Wilmer Rojas Oct 25 '17 at 23:55

So you have applied the Ljung-Box test on your original data, first with the option squared.residuals = FALSE and then squared.residuals = TRUE. In both cases you found the test statistic exceeds the critical value, hence you reject the null hypothesis. The null hypothesis in the first case is "there is no autocorrelation up to lag 20" and in the second case "there is no autocorrelation in the squares up to lag 20". The first result thus suggests presence of autocorrelation, and the second suggests presence of autoregressive conditional heteroskedasticity. That is how you interpret the test results.