Unizor - Linear Regression - Problem 1

Notes to a video lecture on http://www.unizor.com


Linear Regression - Problem 1

Consider a linear regression model described in the previous lecture:
Y = a·X + b + ε
where independent variable X is represented by sample data
x₁, x₂ ...x_n
and observed values of dependent variable Y are
y₁, y₂ ...y_n.

We have introduced two averages of the sample data:
Ave(x)=(x₁+x₂+...+x_n)/n = U
Ave(y)=(y₁+y₂+...+y_n)/n = V

Using new variables
X_k = x_k − U and Y_k = y_k − V
(where index k is from 1 to n)
we came up with the best possible value for a coefficienta in the formula for linear regression as
a = ΣX_k·Y_k / ΣX_k²

Problem

Using the sample averaging function Ave() applied as
Ave(x)=(x₁+x₂+...+x_n)/n
Ave(y)=(y₁+y₂+...+y_n)/n
Ave(xy)=(x₁y₁+...+x_ny_n)/n
Ave(x²)=(x₁²+...+x_n²)/n
prove that the expression for a coefficient a in the formula for linear regression in terms of original sample data x_k and y_klooks like
a = [Ave(xy)−Ave(x)Ave(y)] / [Ave(x²)−Ave²(x)]