help with the parameters

Hello,

Following is a code im using to estimate the parameters using the least absolute deviation using a program I found online.

However, my estimates are not accurate . they should both be equal to 0.1. Also is there a way to add a constraint that the parameters are positive?

Thanks.

clc;
clear;

p=1;
T = 300;

a0 = 0.1; a1 = 0.1; 

seed=123;

ra = randn(T+2000,1); 

epsi=zeros(T+2000,1);
simsig=zeros(T+2000,1);

unvar = a0/(1-a1);

for i = 1:T+2000
                 if (i==1) 
                     simsig(i) = unvar;
                     s=(simsig(i))^0.5;
                     epsi(i) = ra(i) * s;

                   else                     
                       simsig(i) = a0+ a1*(epsi(i-1))^2;
                       s=(simsig(i))^0.5;
                       epsi(i) = ra(i)* s;
                   end
  end

epsi2 = epsi.^2;
y = epsi2(2001:T+2000);

%{
Minimizing the sums of squares of errors is appropriate when
the noise in your model is normally distributed. Its not
uncommon to expect a normal error structure. But sometimes
we choose instead to minimize the sum of absolute errors.

How do we do this? Its a linear programming trick this time.
For each data point, we add a pair of unknowns called slack
variables. Thus

    y(i) = a + b*x(i) + u(i) - v(i)

Here the scalars a and b, and the vectors u and v are all unknowns.
We will constrain both u(i) and v(i) to be non-negative. Solve
the linear programming system with equality constraints as
above, and the objective will be to minimize sum(u) + sum(v).

The total number of unknowns will be 2+2*n, where n is the
number of data points in our "regression" problem.
%}

len = length(y);

x = zeros(len,p);

for i = 1:p
       x(1+i:len,i) = y(1:len-i,1);    
end

% formulate the linear programming problem.
n = length(x);
% our objective sums both u and v, ignores the regression
% coefficients themselves.
f = [0 0 ones(1,2*n)]';

% a and b are unconstrained, u and v vectors must be positive.
LB = [-inf -inf , zeros(1,2*n)];
% no upper bounds at all.
UB = [];

% Build the regression problem as EQUALITY constraints, when
% the slack variables are included in the problem.
Aeq = [ones(n,1), x, eye(n,n), -eye(n,n)];
beq = y;

% estimation using linprog
params = linprog(f,[],[],Aeq,beq,LB,UB);

% we can now drop the slack variables
coef = params(1:2)