Thread Subject:

GARCH(1,1) MLE Question

Hi all just some questions:

We have a process:
y(t) = c + d*y(t-1) + e(t),
h(t) = w+a*e(t)^2 + b*h(t)^2,

We got the log-likelihood function with is:

sum(-0.5*ln(2pi)) -sum(0.5*ln(h(t)^2))-sum(0.5*e(t)^2/h(t)^2))
where e(t) = y(t)-c-d*y(t-1)

How do we estimate the coefficients c,d,w,a,b? Is it that we first run an MLE estimator for the c and d and then continue with the GARCH process or something else? If possible add the derivatives as well. Thank you very much !!