How to check the residuals of a ARIMA-ARCH model?

Question

Lets say we have a AR(1)-ARCH(1) time series model and we want to check the residuals for Ljung-Box. If the residuals of the model is Fit@residuals (using R for modeling) and the variance of the model is Fit@Sigma Is it true that the real residuals that we should check is Fit@residuals/Fit@Sigma ? since from Box et al. 2008 we have:

Answer 1

In an ARMA-GARCH model it is the standardized residuals that should be i.i.d., non-autocorrelated and conditionally homoskedastic, and have the distribution that was assumed when forming the likelihood function of the model. That is not the raw residuals from the ARMA model $a_t$ (following the notation above) but the standardized residuals $e_t$ (as implicitly defined in the equation 10.1.3).

It seems you may be using "fGarch" package in R. Then @residuals will yield $a_t$ as the "fGarch" package pdf says on p. 13: "a numeric vector with the (raw, unstandardized) residual values". Then $e_t$ could indeed be obtained as Fit@residuals/[email protected]; it is $e_t$ that should be the input for tests for presence of autocorrelation, remaining ARCH effects etc.