For the regression model y = + x + , a. Show that the least squares

For the regression model y = α + βx + ε,

a. Show that the least squares normal equations imply Σ_i e_i = 0 and Σ_i x_i e_i = 0.

b. Show that the solution for the constant term is a = y̅ − bx̅.

c. Show that the solution for b is b = [Σⁿ_i=1 (x_i – x̅) (y_i – y̅)]/[Σⁿ_i=1(x_i – x̅)²].

d. Prove that these two values uniquely minimize the sum of squares by showing that the diagonal elements of the second derivatives matrix of the sum of squares with respect to the parameters are both positive and that the determinant is 4n[(Σⁿ_i=1 x²_i) – nx̅²] = 4n[Σⁿ_i=1(x_i – x̅)²], which is positive unless all values of x are the same.