I am interested in the correct way to estimate a GARCH/ARMA-GARCH model.
I will refer to the coefficients as:
ARMA-GARCH: ret = const + ar1 + ma1 residuals = y - ret sigma_2 = alpha + arch1 + garch1 GARCH: residuals = y sigma_2 = alpha + arch1 + garch1
There is a range of information on how to initialise variables in the GARCH framework. One common way is to set sigma_2 at time 0 to the long run variance: alpha / (1 - arch1 - garch1). Is this preferred? What is the preferred initialisation procedure for returns and residuals?
For the optimisation procedure, is there a 'correct' way to choose the starting coefficients? I am currently pulling from a uniform distribution as follows:
U(0.1) * 0.5. This seems to work OK.
I imagine the only constraint is that
arch1 + garch1 <= 1? Are there any other constraints to be aware of?
I have set the bounds for arch1 and garch1 as
(0.0001,1), in the format of
(lower_bound,upper_bound), giving alpha a boundary of
(0.0001,None). In the ARMA-GARCH specification, are there any bounds for the constant, AR term, or MA term?