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I am currently conducting some GARCH modelling and I am wondering about the robust standard errors, which I can obtain from ugarchfit() in rugarch package in R. I have found a presentation and on page 25 the author says that the robust standard errors are obtained from QMLE estimation, but there is no further explanation.

My question is what is the interpretation of these robust standard errors, that is, what are they robust to? I suspect that they are robust to heteroskedasticity, but I would be grateful for some confirmation. Also, what is more common in practice, reporting the non-robust or robust version of the standard errors?

EDIT: I have found additional information on the topic here. Basically, it confirms what those errors are robust to. Thus, the question whether their use in case of GARCH modeling (on stock index returns) are justifiable?

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There is a mention of robust standard errors in "rugarch" vignette on p. 25. The robust standard errors are due to quasi maximum likelihood estimation (QMLE) as opposed to (the regular) maximum likelihood estimation (MLE). They are robust against violations of the distributional assumption, e.g. when the assumed distribution is Normal while the true distribution is Student-$t$. The source cites White "Maximum likelihood estimation of misspecified models" (1982), the (famous) paper introducing QMLE.

Now, are they justifiable? Roughly speaking, if the true distribution is not particularly ill-behaved, QMLE will work. If the true ditribution coincides with the assumed distributions, QMLE will still work, so there is not much to lose (although MLE would give narrower confidence intervals than QMLE, which could be useful). For a rigorous treatment, see White's (1982) paper or an econometrics textbook.

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