Confidence and Prediction Intervals Proofs

Definition 1: Suppose we have a 1 × (k+1) row vector X₀ = [1 x₁ … x_n]. 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 … x_n]

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

References

Howell, D. C. (2010) Statistical methods for psychology (7^th ed.). Wadsworth, Cengage Learning.
https://labs.la.utexas.edu/gilden/files/2016/05/Statistics-Text.pdf

Johnson, R. A., Wichern, D. W. (2007) Applied multivariate statistical analysis. 6th Ed. Pearson
https://mathematics.foi.hr/Applied%20Multivariate%20Statistical%20Analysis%20by%20Johnson%20and%20Wichern.pdf