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Say you have a GARCH-M(1,1) model as follows:

$y_t = \beta y_{t-1} + \delta h_t + \epsilon_t, \quad \epsilon_t \sim N(0, h_t) $

$h_t = a_0 + a_1 \epsilon^2_{t-1} + b_1 h_{t-1}.$

How exactly does one estimate the parameter vector: $\Theta = (\delta, \beta, a_0, a_1, b_1)^\top$? This paper, shows the likelihood function for estimating the parameters in the model using the MLE method. For those who have applied this or a similar methodology, are there any nuances to take note of during the estimation process?

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  • $\begingroup$ You can use M.L.E. to estimate parameter vector. $\endgroup$
    – Neeraj
    Dec 10, 2018 at 3:35
  • $\begingroup$ Thanks @Neeraj but I need more detail. What is the form of the conditional log-likelihood? $\endgroup$
    – Vykta Wakandigara
    Dec 10, 2018 at 6:58
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    $\begingroup$ Have you tried looking this up in the research paper that proposed the model the first time? Or in some textbook? $\endgroup$
    – Richard Hardy
    Dec 10, 2018 at 8:55
  • $\begingroup$ @RichardHardy the univariate GARCH-M was not explicitly implemented in the form above but in the multivariate case by Engle, Lilien and Robins (1987). The ARCH-M was the first to be implemented though. I have found a paper, however, that writes down the log-likelihood, just wanted some advice on the nuances. I hope that by the end of the day I would have answered the question. $\endgroup$
    – Vykta Wakandigara
    Dec 10, 2018 at 15:00
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    $\begingroup$ You could mention the sources you found helpful explicitly by editing your post. Perhaps they will be helpful for other users in the future. $\endgroup$
    – Richard Hardy
    Dec 10, 2018 at 15:28

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