How to determine Confidence Interval of each Parameter Estimation results of the GARCH model through fminsearch syntax?

I doing the estimation of the parameters of the GARCH model through this data and syntax below where the rv is realized volatility and then I want to know the standars error and the P-value for each parameter estimation results but I think I must to know the confidence interval of these result parameter estimation first. I hope and I'm so glad if one can help me to solve this problem. Thank You.

Y65 = xlsread("Y65",1,"A2:A89");
n = length(Y65);

rv = [];
moving_window = 5;
for i = 1:n-moving_window+1
    rv(i) = var(Y65(i:i+moving_window-1));
end

Y65_1 = Y65(moving_window:end)';

logL = @(theta) -(sum((log(1./(sqrt(2.*pi.*(theta(1)+theta(2).*...
Y65_1(1:end-1).^2+theta(3).*rv(1:end-1)))))-(1./2).*...
(Y65_1(2:end)./(sqrt(theta(1)+theta(2).*...
Y65_1(1:end-1).^2+theta(3).*rv(1:end-1)))).^2)));

theta_0 = [0.99 0.3 0.3];
[theta_hat NegLL] = fminsearch(logL, theta_0);

beta_0 = theta_hat(1)

beta_0 = 9.0051e-04

beta_1 = theta_hat(2)

beta_1 = 0.3875

beta_2 = theta_hat(3)

beta_2 = 0.3557

-NegLL

ans = 160.0541

Answer 1

[Vanshika Vaishnav]

The 'fminsearch' function is a black box, unconstrained nonlinear solver. This means that no assumptions about the problem are made and 'fminsearch' is unaware that it is solving a curve-fitting problem. As a result, the solver returns only basic information. This means that confidence intervals will need to be calculated by the user. If you would like to use 'fminsearch' for curve fitting, a couple of resources on how to manually compute confidence intervals can be found here:

1. Confidence Intervals-Yale Course

2. Plot confidence interval of a signal