# Ljung_Box Statistic of R and R^2 values in Return analysis

I have found a result that I find truly puzzling. Here is an extract from a GARCH-Analysis I have performed:

Test______________Statistic_______p-Value

Ljung-Box Test_____R Q(10)_____0.4047773
Ljung-Box Test_____R Q(15)_____0.3371581
Ljung-Box Test_____R Q(20)_____0.4098038
Ljung-Box Test_____R^2 Q(10)___0.9935475
Ljung-Box Test_____R^2 Q(15)___0.9978561
Ljung-Box Test_____R^2 Q(20)___0.9984385

Pardon the formatting by the way. R stands for the Returns and R^2 for the squared returns. In the ACF Plots I have seen, that there is significant autocorrelation for the R^2 values. Nevertheless, this dependence would show in a significantly low p-Value (for example one below 0.05). Nonetheless I here see the exact opposite. All R^2 Values show an extremely high p-Value. How can this be? Shouldn't this mean that the Values are very independent of one another?

• I'm voting to close this question as off-topic because it belongs on Cross Validated as it is purely about statistical properties and interpretations of statistical tests. – Richard Hardy Jun 19 '17 at 20:20
• dw^2 = dt springs to mind – Kian Jun 19 '17 at 20:20
• Have you considered accepting the answer? See how this is supposed to work. – Richard Hardy Dec 13 '20 at 18:06

## 1 Answer

Two things to consider.

1. Once a model has been fit to the data, the Ljung-Box test statistic follows a different null distribution than that applicable to raw data. This also holds for other diagnostic tests and the confidence bounds on ACF and PACF. This is because model residuals are in some way restricted. E.g. residuals from a linear regression always sum to one, so each data point does not carry as much information as raw data: you can reconstruct the i-th residual as zero minus the sum of all other residuals, while you cannot reconstruct raw data that way.
2. The Ljung-Box test has really nonstandard null distribution when applied on standardized residuals from a GARCH model, and the $p$-values you have there are most likely wrong, because they are most likely computed from some standard distribution. (Yes, popular software packages continue making this mistake despite notifications from users.) There have been some discussions on that on Cross Validated (search "Ljung-Box" and "GARCH"; e.g. this).