% PARTIAL REGRESSION PLOTS 

% 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;
x2 = oilprice;
constant = ones(n,1);
x1 = [constant seniors];

% Auxillary Regressions
M1 = eye(n) - x1*inv(x1'*x1)*x1';
e21 = M1*x2;
ey1 = M1*y;

% Calculating OLS Estimates 
xmat = [x1,x2];
b = inv(xmat' * xmat)*(xmat' * y);
disp(' ');
disp('b vector  = '); disp(b); 
b2 = inv(e21'*e21)*(e21'*ey1);

% Calculating R^2
M0 = eye(n)-(1/n)*ones(n,n);
SST = y'*M0*y;
SSR = b'*xmat'*M0*xmat*b;
SSE = (y-xmat*b)'*(y-xmat*b);
R2a = SSR/SST;
R2b = 1 - SSE/SST;

% Predicted Values 
yhat = xmat*b;
ey1hat = e21*b(3,1);

% Graphs 
scatter(mean(x2)+e21,mean(y)+ey1);
hold on;
line(mean(x2)+e21,mean(y)+ey1hat);
title('Conditional Regression of UW Enrollment on the Price of Oil');
ylabel('UW resident undergraduate enrollment');
xlabel('Price of oil');