Correct fmincon() constraints for GARCH?

Hello, all. So, I am trying to manually to MLE estimate a GARCH(1,1) model using the optimization toolbox's fmincon(). I know that Matlab's built-in garch() has constraints in its optimization, such as briefly mentioned in this mathworks doc, but I do not know how to manually by myself express these by myself as fmincon arguments.

My code so far is this, which works for unconstrained optimization using fminunc/ fminsearch:

lh = @(x,data) -sum(log(fun(x,data))); %Transformation of my input function for optimization.
options = optimset('Display', 'off', 'MaxIter', 1000000, 'TolX', 10^-20, 'TolFun', 10^-20);
[theta, max1] = fmincon(@(x) lh(x,data), guesses, options, constraints???);

The guide for fmincon() is not clearing things up, how I code these constraints myself?

Answer 1

[Hang Qian]

Hi Michael,

Typically we add some inequality constraints to ensure a positive conditional variance in the GARCH(1,1) model, like constant > 0, arch > 0, garch > 0 and arch + garch < 1.

The numeric optimization function fmincon supports inequality constraints. Since the constraints are linear, we can put them in a matrix form like A * x < b. If the parameters are stacked like [constant, garch, arch], then the constraints can be formulated as

A = [0 1 1]

b = 1

lb = [0 0 0]

Then we can run the constrained optimization using the syntax

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

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

By the way, we may replace lb by an augmented A and b with more rows, but it is not as efficient as the explicit lower bounds.

Best,

Hang Qian