How to add constraints to the constant in GARCH model?

Hi guys! I wish to create a GARCH(1,1) model with a constraint on the constant, take the constraint w = (1-alpha-belta) * 0.8 as an example. I tried to use

Mdl = garch('GARCHLags',1,'ARCHLags',1,'Offset',NaN); 

but I do not know how to specify the constraint on the constant.

Could anybody help me? Thank you!

Answer 1

[Karanjot]

Hi Wenjing,

I understand that you want to learn more about constraints for GARCH (Generalized Autoregressive Conditional Heteroskedasticity) models.

The ‘estimate’ function maximizes the loglikelihood function using ‘fmincon’ from the Optimization Toolbox. ‘fmincon’ has many optimization options, such as choice of optimization algorithm and constraint violation tolerance.

The numeric optimization function ‘fmincon’ further supports inequality constraints. Since the constraints are linear, they can be represented in a matrix form, such as A * x < b. If the parameters are stacked in the [‘constant’, ‘garch’, ‘arch’] format, then the constraints can be formulated as follows:

A = [0 1 1];
b = 1;
lb = [0 0 0];

The constrained optimization can be executed using the below syntax:

fmincon(fun,x0,A,b,[],[],lb,[],[],options);

where the empty matrix ‘[]’ indicates non-existing constraints of other types.

To learn more about constraints for GARCH models, please refer to the below documentation:

https://www.mathworks.com/help/econ/optimization-settings-for-conditional-variance-model-estimation.html#btq0e10

I hope this helps!