# How does one estimate parameters in a GARCH-M(1,1) model?

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?

• You can use M.L.E. to estimate parameter vector. – Neeraj Dec 10 '18 at 3:35
• Thanks @Neeraj but I need more detail. What is the form of the conditional log-likelihood? – Vykta Wakandigara Dec 10 '18 at 6:58
• Have you tried looking this up in the research paper that proposed the model the first time? Or in some textbook? – Richard Hardy Dec 10 '18 at 8:55
• @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. – Vykta Wakandigara Dec 10 '18 at 15:00
• You could mention the sources you found helpful explicitly by editing your post. Perhaps they will be helpful for other users in the future. – Richard Hardy Dec 10 '18 at 15:28