% This is a simple illustration of the post model selection sampling
% distribution of the OLS estimator.

clear all
S=1000;  % the number of iterations for the monte carlo simulation
n=500;    % the sample size

for s=1:S,
    X=mvnrnd(zeros(2,1),[1 .8; .8 1],n);
    Y=X*[1;0.2]+normrnd(0,1,n,1);
    W1=[ones(n,1) X];
    W2=[ones(n,1) X(:,1)];
    bhat1(:,s)=inv(W1'*W1)*W1'*Y;
    Vbhat1=inv(W1'*W1)*((Y-W1*bhat1(:,s))'*(Y-W1*bhat1(:,s))/n);
    reject(s)=1*(abs(bhat1(3,s)./sqrt(Vbhat1(3,3)))>1.96);
    bhat2(:,s)=inv(W2'*W2)*W2'*Y;
    bhat(:,s)=(1-reject(s))*bhat2(2,s)+reject(s)*bhat1(2,s);
end  

hist(bhat)