% EXAMPLE TO SHOW THE OLS ESTIMATOR IS ASYMPTOTICALLY NORMAL

% Load in Data 
clc;
n = 51;
M = importdata('matlabex10data.txt','\t',1);
data = M.data;
n = size(data,1);
loops = 50000;

% Name Variables 
year = data(:,1);
undergrads = data(:,2);
oilprice = data(:,3);
seniors = data(:,4);

% Redefine Variables 
y = undergrads;
x = [ones(n,1) oilprice seniors];
[n,k] = size(x);

% Initial OLS Regression
b = inv(x'*x)*(x'*y);
e = y - x*b;
s2 = (e'*e)/(n-k);

% Loop to Simulate Artifical Data and Calculate OLS 
theta = 1;
bhat = zeros(k,loops);
n2 = 40;
for (i=1:1:loops);
    x2 = x(1:n2,:);
    y = x2*b + sqrt(s2)*((rand(n2,1)-0.5)/sqrt(1/12));
    bhat(:,i) = inv(x2'*x2)*(x2'*y);
end;

% Sampling Distribution
hold on;
sx = inv(x2'*x2);
bz = (bhat(2,:) - b(2))/(sqrt(s2)*sqrt(sx(2,2)));
histogram(bz,100,'Normalization','pdf');
xvalues = linspace(-3,3,100);
f = (1/sqrt(2*pi))*exp(-xvalues.^2/2);
plot(xvalues,f);