Minimization of Objective Function

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Testing for jumps in conditionally Gaussian ARMA–GARCH models, a robust approach.pdf

Dear Member,

I want to find the estimates of my model by minimizing an objective function using "fmincon"command of Matlab. I write the following program.

function[arg]=argmin(x)
  c0=x(1,1);          % constant of equation 2.9
  c1=x(2,1);          % Coefficient associated with first term in equation 2.9
  w=x(3,1);           % constant of eqaution no 2.11
  alpha=x(4,1);       % Coefficient associated with first term in equation 2.9 
  beta=x(5,1);        % Coefficient associated with second term in equaiton 2.9
  load r;             % data series whome we wants to correct for outliers 
  delta=.975;         % percentage of observation to me downweighted 
    %%************************************************************
    %
    t=size(r,1);
    j=zeros(t,1);
    meau=zeros(t,1);
    var=zeros(t,1);
    mpy=zeros(t,1);
    %%This section of program define the jump factor 
    if delta==0.99
    c_delta=1.0185;
    end
    if delta==0.975
    c_delta=1.0465;
    end
    if delta==0.95
    c_delta=1.0953;
    end
    %%This section of program estiamte the equation 2.6, 2.9, 2.10, 2.11 simulatenously
    for i=1:t
        if i==1
            meau0=mean(r);                                                 % initial value of eqation no 2.9
            sd=std(r);                                                     
            j0=(0-meau0)/sd;                                               % initial value of J equation no 2.6
            var0=std(r)^2;                                                 % initial value of eqation no 2.11
            mpy0=sign(j0)*min(abs(j0),norminv(delta));                   % initial value of eqation no 2.10
    %%*******************************************************************************************************
            meau(i,1)=c0+c1*sqrt(var0)*mpy0;                               % equation no 2.9  of the paper
            var(i,1)=w+alpha*var0*c_delta*mpy0^2+beta*var0;                % equation no 2.11 of the paper
            j(i,1)=(r(i,1)-meau(i,1))/sqrt(var(i,1));                      % equation no 2.6  of the paper
            mpy(i,1)=sign(j(i,1))*min(abs(j(i,1)),norminv(delta));       % equaiton no 2.10 of the paper
        else
            meau(i,1)=c0+c1*sqrt(var(i-1,1))*mpy(i-1,1);
            var(i,1)=w+alpha*var(i-1,1)*c_delta*mpy(i-1,1)^2+beta*var(i-1,1);
            j(i,1)=(r(i,1)-meau(i,1))/sqrt(var(i,1));
            mpy(i,1)=sign(j(i,1))*min(abs(j(i,1)),norminv(delta));  
       end
    %%************************************m******************************************
    end
    abs1=abs(j);
    z =2*log10(abs1);
    %%The estimation of Rho, that is used in equation no 2.13 (function which is to be minimize)
     v=4;
    rho=-z+.82660*(1+v)*log10(1+exp(z)/(v-2));
    arg=mean(rho);

These are the codes of mathematics of proposed test in the paper attached. Then I have an other program in which I use the minimization function "fmincon". The program is given below.

function [mn, fv]=minimization()
  x0=[.51; .08; .4; 4.18; .88];
  A=[  0, 0, 0, 1, 1];
  b=[.99];
  lb=[-5,-5,0,0,0];
  lu=[5,5,5,1,1];
    %options = struct('MaxFunEvals', 2000);
    options=optimset('Algorithm','sqp','MaxFunEvals', 2000,'MaxIter',500);
    %opt = optimset('Display','iter');
    [mn, fv]=fmincon(@argmin,x0,A,b,[],[],lb,lu,[],options);

When I run this program it does not work. I have simulated series "r" of AR(1)-GARCH(1,1) process. When there is no outliers in the series the estimated values should be close to the original ones.