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This video's second half formulates the GARCH autoregressive model combined with the heavy-tailed t-distribution (t-GARCH) and implies its log-likelihood function based on the first half's derivation for the Normal distribution. Although not written out in full for the t-GARCH, could someone provide the source article where the log-likelihood function is fully derived for the t-GARCH model?

(especially the more general case where the assumption of standard-t ($\mu=0,\sigma=1$) is relieved). Please don't just say "any GARCH textbook", I have looked, and they seldom count as the originator. just want to narrow the search faster

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    $\begingroup$ The assumption does not have to be relaxed, because in a GARCH model, the standardized residuals must have zero mean and unit variance by definition. Also, why the quasi-maximum-likelihood tag (and no maximum-likelihood tag) if you are specifically interested in maximum likelihood estimation for a given distribution (Student-$t$) rather than with a normal distribution (that would be quasi)? Also, when you say autoregressive, do you mean an AR-GARCH model, i.e. one where the conditional mean is modelled using an autoregression? $\endgroup$
    – Richard Hardy
    Commented Dec 11, 2020 at 19:17
  • $\begingroup$ GARCH stands for generalized autoregressive conditional heteroskedasticity, so I wasn't implying the further case of AR-GARCH, no, just drawing out the terminology that GARCH is a type of autoregressive model that's all $\endgroup$
    – develarist
    Commented Dec 11, 2020 at 19:21
  • $\begingroup$ OK, thanks. I was confused because the GARCH autoregressive model spells out as the generalized autoregressive conditional heteroskedasticity autoregressive model and as such contains the term autoregressive twice. $\endgroup$
    – Richard Hardy
    Commented Dec 11, 2020 at 19:23
  • $\begingroup$ It's a good reminder that GARCH itself assumes $\mu=0,\sigma=1$ regardless of parametric distribution it is combined with. still waiting for that source though, even for the Gaussian-GARCH derivation $\endgroup$
    – develarist
    Commented Dec 11, 2020 at 19:24
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    $\begingroup$ See "Derivation of GARCH Student-$t$ log-likelihood". $\endgroup$
    – Richard Hardy
    Commented Dec 11, 2020 at 19:44

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For GARCH-Student-$t$ model, the likelihood is available in MathWorks page "Maximum Likelihood Estimation for Conditional Variance Models" which references several sources. The relevant one is probably Bollerslev (1987).
For ARCH(m)-Student-$t$ model, the likelihood is available in Hamilton (1994) Chapter 21 Time Series Models of Heteroskedasticity, p. 662.
For GARCH(p,q)-Normal model, the likelihood is available in Francq & Zakoian (2010) Chapter 7 Estimating GARCH Models by Quasi-Maximum Likelihood, pp. 142.
For ARCH(p)-Normal model, the likelihood is available in Tsay (2010) Chapter 3 Conditional Heteroskedastic Models, pp. 120.

References

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    $\begingroup$ Since the OP said still waiting for that source though, even for the Gaussian-GARCH derivation, I am providing what I found for the Gaussian case. I may follow it up with the Student-$t$ case later on. $\endgroup$
    – Richard Hardy
    Commented Dec 11, 2020 at 19:38

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