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% We evaluate the maximum likelihood estimator of a GARCH(1,1) process
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% Let us clean the environment
close all; clear all; clc;
% We upload the data
data=xlsread(‘SP500daily_level.xlsx’);
% We go from prices to log (or continuously-compounded) returns
prices=data(2:end,2);
r = log(prices(2:end)./prices(1:end-1));
T = length(r);
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% Always plot the data (when you can): why do we need a GARCH model? Let’s see …
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title(‘SP 500 returns in percentage terms’)
ylabel(‘returns’)
xlabel(‘time’)
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% Let us now maximize the log-likelihood, find the parameter estimates, and do inference
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%Estimation, first
parameter_names = [“mu” “delta” “phi”]’;
theta_initial=[0,0.5,0.5]; %Initial conditions (I am just choosing reasonable values, but you can of course change them and see what happens)
options = optimset(‘Display’,’none’,’TolFun’,1e-20,’TolX’,1e-20);
%Now, inference
% Below, we compute the gradient and, with the gradient, the asymptotic
% variance/covariance matrix directly (by taking derivatives “by hand”).
sum4 = zeros(3,3);
h = var(r)*ones(T,1);
grad1_h = zeros(T,1);
grad2_h = zeros(T,1);
grad3_h = zeros(T,1);
h(t) = theta(1)+theta(2)*h(t-1)+theta(3)*r(t-1)*r(t-1);
grad1_h(t) = 1 + theta(2)*grad1_h(t-1);
grad2_h(t) = h(t-1) + theta(2)*grad2_h(t-1);
grad3_h(t) = r(t-1)*r(t-1) + theta(2)*grad3_h(t-1);
grad_h = [grad1_h(t); grad2_h(t); grad3_h(t)];
grad = -(1/2)*1/h(t)+(1/2)*r(t)*r(t)/(h(t)^2);
grad = grad.*grad_h;
sum4=sum4+grad*grad’;
asy_var = (1/T)*inv((1/T)*sum4);
t_stats = theta’./sqrt(diag(asy_var));
outcome = table(parameter_names,theta’,t_stats,…
‘VariableNames’,{‘parameters’ ‘Estimates_MLE’ ‘t_statistics’});
disp(outcome);
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% Value-at-Risk
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h_next = theta(1)+theta(2)*h(end)+theta(3)*r(end)*r(end);
VaR = 1000000*2.33*sqrt(h_next);
fprintf(‘The 1%% Value-at-Risk for an investor holding a “market-like” portfolio is %5.2f \n’,VaR)
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% This is the function which obtains the log-likelihood given an assumption
% of normality on the error terms
% As always, the function is defined at the bottom of the code and then called
% in the fminsearch routine
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function y=garchloglik(theta,data)
h1=var(data); % Recall h needs to be initiated (we loop over h)
% Initial value of h: we use the unconditional
% empirical variance from the data
T = length(data);
mu=theta(1);
delta=theta(2);
phi=theta(3);
sum1=-T/2*log(2*pi);
sum2=-0.5*log(h1);
sum3=-data(1)*data(1)/(2*h1);
h = mu+delta*h+phi*data(t-1)*data(t-1);
sum2=sum2-0.5*log(h);
stand_res=data(t)*data(t)/(2*h);
sum3=sum3-stand_res;
y= -(1/T)*(sum1+sum2+sum3); % Note that I changed the sign of the likelihood. We are maximizing, not minimizing.
% The function garchloglik can now be fed into the
% fminsearch function
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