I am solving an exercise on multiple linear regression. Near the end I will be asked for the same data as the previous model, it is the maximum likelihood estimates.
The previous model: I I have the matrix $(X'X)^{-1}$ and the matrix $X'y$ and the model is:
$Y_i = B_0 + B_1x_i + B_2x^2_i + e_i$
$i= 1,...,10$
\begin{align}
Y_i &= B_0 + B_1x_i + B_2x^2_i + e_i \\
i &= 1,...,10
\end{align}
Now I have:
$Y_i = g_0 + g_1x^*_i + g_2(x^*_i)^2 + e_i$
$ x^*_i = x_i -10$
$i=1,...,10$\begin{align} Y_i &= g_0 + g_1x^*_i + g_2(x^*_i)^2 + e_i \\ x^*_i &= x_i -10 \\ i &= 1,...,10 \end{align}
To transform the model, can I decrease the Matrixmatrix data for 10?
For example the first row of the matrix $(X'X)^{-1}$ is:
$[4,0,0]$
$4-10;0-10;0-10 ....$
$$ [4,\hspace{12mm} 0,\hspace{12mm} 0]\hspace{20mm} \\ 4-10,\ 0-10,\ 0-10,\ ... $$
The same thing can I make it also for the matrix $X'y$? Thank you.
I have many doubts. Publishing I am publishing the text so it is more understandable.
Consider the regression model linear:
$Y_i = B_0 + B_1x_i + B_2x^2_i + e_i$$$Y_i = B_0 + B_1x_i + B_2x^2_i + e_i$$
with $ e_1, ...., e_n $$e_1, ...., e_n$ independent and identically distributed random variables and x_i, i = 1, ..., 0$x_i,\ i = 1, ..., 0$ constant fix.
The only data I have are these:
https://s32.postimg.org/o20r1i5et/Immagine.png
. I have to rewrite the whole thing ..., so I tried so:
$(X)$ = \begin{bmatrix} 1 & x_1 & x^2_1\\ . & . & . \\ . & . & . \\ . & . & . \\ 1 & x_{10} & x^2_{10}\\ \end{bmatrix}
$(X^*)$ = \begin{bmatrix} 1 & x_1-10 & (x_{1}-10)^2\\ . & . & . \\ . & . & . \\ . & . & . \\ 1 & x_{10}-10 & (x_{10}-10)^2\\ \end{bmatrix}
$X^{*'}X^*$ = \begin{bmatrix} 10 & \sum_{i=1}^{10} x_i-10 & \sum_{i=1}^{10}(x_{i}-10)^2\\ \sum_{i=1}^{10} x_i-10 & \sum_{i=1}^{10} (x_{i}-10)^2 & \sum_{i=1}^{10}(x_{i}-10)^3\\ \sum_{i=1}^{10} (x_{i}-10)^2 & \sum_{i=1}^{10}(x_{i}-10)^3 & \sum_{i=1}^{10}(x_{i}-10)^4\\ \end{bmatrix}\begin{align} (X) &= \begin{bmatrix} 1 & x_1 & x^2_1\\ . & . & . \\ . & . & . \\ . & . & . \\ 1 & x_{10} & x^2_{10}\\ \end{bmatrix} \\[10pt] (X^*) &= \begin{bmatrix} 1 & x_1-10 & (x_{1}-10)^2\\ . & . & . \\ . & . & . \\ . & . & . \\ 1 & x_{10}-10 & (x_{10}-10)^2\\ \end{bmatrix} \\[10pt] X^{*'}X^* &= \begin{bmatrix} 10 & \sum_{i=1}^{10} x_i-10 & \sum_{i=1}^{10}(x_{i}-10)^2\\ \sum_{i=1}^{10} x_i-10 & \sum_{i=1}^{10} (x_{i}-10)^2 & \sum_{i=1}^{10}(x_{i}-10)^3\\ \sum_{i=1}^{10} (x_{i}-10)^2 & \sum_{i=1}^{10}(x_{i}-10)^3 & \sum_{i=1}^{10}(x_{i}-10)^4\\ \end{bmatrix} \\ \end{align}
Now do I have to calculate the inverse? I'mAm I following the proper solution?
Do I have to use the property tofor equivariance?
