Trouble with obtaining constant matrix to find variance-covariance matrix of regression parameters

Question

I have been working on an exercise from Applied Linear Statistical Models - 5th edition- by Kutner.

The question is asking me to obtain the variance-covariance matrix for a polynomial regression of one predictor variable in terms of the original predictor variables before centering. I know the mechanics of what needs to be done and how to do everything, but there is one step I could not complete. Fortunately I have a solution for the problem. That follows below:

My only question is how to obtain the constant matrix $A$ that they did in the solution? I know that the vector I'm using in finding the covariance matrix for the regression parameters will be in this particular situation $A\mathbf{b'}$. Where $\mathbf{b'}$ is just the vector of parameter estimates from the polynomial regression. And from that it will follow that to get the needed covariance matrix I will use the relationship $$\Sigma_{bb} = A \Sigma_{b'b'}A^{t}$$.

This I understand. I'm just stuck in understanding how the matrix $A$ was obtained and the motivation for this matrix in particular.

Would someone be able to provide some insight?

Answer 1

We have \begin{align}b_0' &= b_0- b_1\bar{X}+b_{11}\bar{X}^2\\ b_1' &= b_1 - 2b_{11}\bar{X}\\ b_{11}' &=b_{11} \end{align}

Let's rewrite them in matrix multiplication form.

$$\begin{bmatrix} b_0' \\ b_1'\\ b_{11}'\end{bmatrix} = \begin{bmatrix} 1 & -\bar{X} & \bar{X}^2\\ 0 & 1 & -2\bar{X}\\ 0 & 0 & 1\end{bmatrix} \begin{bmatrix} b_0 \\ b_1 \\ b_{11}\end{bmatrix}$$

That is how we obtain the matrix $A$.