# Adjusted Pearson Goodness-of-Fit Test - Rugarch Package

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)

• I see two distinct questions here which could (should) be split into two posts. Q2 has been answered in several related posts, so you can just delete it (you may also look up the existing questions related to Q2). The answer is: you need the empirical distribution to match the theoretically assumed distribution. The theoretically assumed distribution in your case is Student-$t$, not normal, so there is no point in comparing the empirical distribution to a normal distribution. Pearson's test is comparing empirical to Student-$t$, which is correct, and it gives you an answer to that question. – Richard Hardy Mar 27 at 10:02
• Just to clarify further, As Everyone is always saying that residuals have to be Gaussian i.i.d. In this case, "everyone" is pretty often incorrect. As simple as that. – Richard Hardy Mar 27 at 10:39
• Hey, I just discovered you had essentially asked Question 2 here where I answered it. Isn't this the same question? – Richard Hardy Mar 27 at 12:48
• Thank you Richard for your help. Now just in general, if a assume my errors to be student t distributed. Does this somehow impact my coefficients? Or standard errors? I mean, do I face some statistical problems? Thank you for your help! – lilo Mar 28 at 9:01
• If you use maximum likelihood estimation, the coefficient estimates as well as standard errors will depend on the assumed distribution (will differ for different distributions). However, I do not think you would face any serious problem in estimation if you switch from, say, normal to Student-$t$. – Richard Hardy Mar 28 at 9:05