clear;clc;
nobs=300;
teste1=randn(nobs,2);
epsilon=randn(nobs,1);
X=[ones(nobs,1) teste1];
y=X*[2;1;3]+epsilon;
tic
beta_1=inv(X'*X)*X'*y;
toc
tic
beta_2=X\y;
toc
tic
beta_3=pinv(X)*y;
toc

results=zeros(500,3);

for i=1:500
    teste1=randn(nobs,2);
    epsilon=randn(nobs,1);
    X=[ones(nobs,1) teste1];
    y=X*[2;1;3]+epsilon;
    results(i,1:3)=(X\y)';
end

hist(results(:,1),100)
hist(results(:,2),100)
hist(results(:,3),100)
%Penúltimo exercício
nobs2=30000;
teste1=randn(nobs2,2);
epsilon=randn(nobs2,1);
X=[ones(nobs2,1) teste1];
y=X*[2;1;3]+epsilon;
results_ex5=zeros(29950,3);

for i=50:30000
    Xtemp=X(1:i,:);
    ytemp=y(1:i,1);
    results_ex5(i,:)=(Xtemp\ytemp)';

end
line(1:29951,results_ex5(50:end,1))
line(1:29951,results_ex5(50:end,2))
line(1:29951,results_ex5(50:end,3))
    
% Último Exercício
resultsRU=zeros(500,1);

for i=1:500
    epsilon=randn(nobs,1);
    y0=1;
    yt=y0+cumsum(epsilon);
    resultsRU(i,1)=yt(1:end-1)\yt(2:end);
end

hist(resultsRU,100)