Definition 1: Suppose we have a 1 × (k+1) row vector X0 = [1 x1 … xn]. Then the prediction of the multiple regression model based on X0 is ŷ0 where
or in matrix terminology
Property 1: For any 1 × (k+1) row vector X0 = [1 x1 … xn]
Proof: By Property 3 of Multiple Regression using Matrices
Property 2:
Proof: By Property 1 and Property 3 of Multiple Regression using Matrices
where the last equality results from Property 4 of Multiple Regression using Matrices.
Property 3:
Proof: Since y0 = Xβ + ε0 and Xβ is a constant, it follows that the variance of Xβ is zero, and so
Also
But one of the assumptions for linear regression is that ε0 is independent of the coefficients in B, and so cov(ε0,ŷ0) = 0 since ŷ0 = XB. It now follows from Property 2 that
