I fitted a GARCH(1,1), GARCH-M and EGARCH of first order (using maximum likelihood) to my return dataset using both, Gaussian normal and Student-t distribution assumption for the error term.
When using the rugarch package one output is the "Adjusted Pearson Goodness-of-Fit Test". According to the manual published by Alexios Ghalanos ("Introduction to the rugarch package"), this diagnostic test
"..calculates the chi-squared goodness of fit test, which compares the empirical distribution of the standardized residuals with the theoretical ones from the chosen density. The implementation is based on the test of Palm (1996) which adjusts the tests in the presence on non-i.i.d. observations by reclassifying the standardized residuals not according to their value (as in the standard test), but instead on their magnitude, calculating the probability of observing a value smaller than the standardized residual, which should be identically standard uniform distributed. The function must take 2 arguments, the fitted object as well as the number of bins to classify the values."
Question 1: Could someone explain this descprition in an easier way?
Question 2: Using student-t I dont get a normal distribution in standardized residuals (using Kolmogorov test for normality). But Pearson's goodness of fit test tells me, that my distribution assumption was right.. Can I still use this model? As Everyone is always saying that residuals have to be Gaussian i.i.d. (remark: my standardized residuals are idependently distributes according to ljung-box and mcleod li tests)