Interpreting Ljung–Box Tests on GARCH Residuals and Squared Returns
Summary
The document asks why Ljung–Box tests on squared returns can report very high p-values even when their autocorrelation plots appear to show dependence. It presents test results for returns and squared returns at several lag lengths, then asks whether the high p-values imply independence. The response focuses on interpreting diagnostics after fitting a model rather than treating the test as if it were applied to raw observations.
Fitted residuals are constrained by the estimation process, so the usual reference distribution for the Ljung–Box statistic may not apply. The response further cautions that standardized residuals from a GARCH model can have a nonstandard null distribution, making p-values based on a standard distribution unreliable. Thus, high reported p-values should not automatically be read as proof that squared returns are independent. The document does not identify the software or provide a corrected test procedure, so readers need an inference method suited to the fitted model and their implementation.
Key ideas
- Ljung–Box reference distributions may change when the test is applied after fitting a model.
- Residual constraints can reduce the effective information represented by observations.
- Standardized residuals from GARCH models may require nonstandard null distributions for diagnostic tests.
- High p-values from an unsuitable reference distribution do not establish independence.
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Full text
# Ljung_Box Statistic of R and R^2 values in Return analysis # 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? ## Answer by Richard Hardy (score 1) https://quant.stackexchange.com/a/34769 Two things to consider. - 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. - 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).
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