Fitting an ARMA-GARCH model, I checked the Weighted Ljung-Box test on standardized residuals and squared residuals to verify if the model is adeguate in describing the linear dependence in the return and volatility series. Combining different orders of the ARCH and GARCH part, for example a GARCH(1,1), GARCH(2,1), GARCH(2,2),
I always get that $p$-value of the test is below 0.05 for the standardized residuals and above 0.05 for the squared standardized residuals. So that seems contrasting to me, and I don't know what type of conclusion I can make. Given the results of the test for the squared standardized residuals, I would say that the model fits well the data, but the test on the standardized residuals suggests me the opposite.
What should I do? Can I priviledge the test result on the squared standardized residuals? Should I try with higher order of the model?
In all the attempts mentioned in the post, changing only the orders of the GARCH model, I always kept fixed the ARMA model. I just tried to change the ARMA order and it looks better. The best choise seems an GARCH(2,2) without the ARMA part. This is happening assuming that innovations follow a Skew Student-$t$ distribution.
Using just a GARCH model without the mean specification seems better in terms of the Ljung-Box test on residuals, and a GARCH(1,1) model fits well the data. At the same time, adding a mean specification improves the AIC and BIC values but requests me to use a GARCH model of higher order. What should I prefer between the two specification?