% EXAMPLE TO DISPLAY LEAST SQUARES ESTIMATION TECHNIQUE 

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

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

% Calculate Values for Normal Equations 
y = undergrads;
x = oilprice;
meanx = mean(x);
meany = mean(y);

% Calculating a & b -- Method #1:  Using Univariate Formulae 
demeanxy = (y - meany) .* (x - meanx);
demeanx = x - meanx;
b2 = sum(demeanxy)/sum(demeanx.^2);
a2 = meany - b2*meanx;  
disp(['a = ' num2str(a2)]);
disp(['b = ' num2str(b2)]);

% Calculating a & b -- Method #2:  Using Matrix Formula 
constant = ones(n,1);
xmat = [constant,x];
b = inv(xmat' * xmat)*(xmat' * y);
disp(' ');
disp('b vector  = '); disp(b); 

% Predicted Values 
yhat = xmat*b;

% Graphs 
scatter(x,y);
hold on;
line(x,yhat);
title('Regression of UW Resident Enrollment on the Price of Oil');
ylabel('UW resident undergraduate enrollment');
xlabel('Price of oil');