Consider regression on $p$ explanatory variables with data $(x_{i1},\ldots,x_{ip},y_i)$ for $i=1,\ldots,n$.
This exercise will lead you to another interpretable equation for the vector $({\hat\beta}_1,\ldots,{\hat\beta}_p)$. It is a partial review of some of the theory (before the final exam).

First consider shifting the $j$th variable $x_j$ to $x^*_j=x_j-a_j$, that is, $x_{ij}^*=x_{ij}-a_j$ for $i=1,\ldots,n$.
Let $({\hat\beta}_0,{\hat\beta}_1,\ldots,{\hat\beta}_p)$ be the least squares vector of coefficients for the original data and let $({\hat\beta}^*_0,{\hat\beta}^*_1,\ldots,{\hat\beta}^*_p)$ be the least squares vector of coefficients for the shifted data.

Part a)
Which of the following are correct statements?


A. ${\hat\beta}^*_0={\bar y}$
B. ${\hat\beta}^*_0={\bar y}-{\hat\beta}_1{\bar x}_1-\cdots-{\hat\beta}_p{\bar x}_p$
C. ${\hat\beta}^*_0={\bar y}-{\hat\beta}_1{\bar x}_1-\cdots-{\hat\beta}_p{\bar x}_p-{\hat\beta}_1a_1-\cdots-{\hat\beta}_pa_p$
D. ${\hat\beta}^*_0={\bar y}-{\hat\beta}_1{\bar x}_1-\cdots-{\hat\beta}_p{\bar x}_p+{\hat\beta}_1a_1+\cdots+{\hat\beta}_pa_p$
E. ${\hat\beta}^*_j={\hat\beta}_j-a_j$ for all $j=1,\ldots,p$
F. ${\hat\beta}^*_j={\hat\beta}_j+a_j$ for all $j=1,\ldots,p$
G. ${\hat\beta}^*_j={\hat\beta}_j$ for all $j=1,\ldots,p$
H. ${\hat\beta}^*_0={\bar y}-{\hat\beta}_1a_1-\cdots-{\hat\beta}_pa_p$
I. None of the above


For the remaining questions below, suppose $a_j={\bar x_j}$ for $j=1,\ldots,p$, that is, $x_j^*$ are now centred variables with sample means of 0.
Part b)
Which of the following are correct statements?


A. ${\hat\beta}^*_0={\bar y}-{\hat\beta}_1{\bar x}_1-\cdots-{\hat\beta}_p{\bar x}_p-{\hat\beta}_1a_1-\cdots-{\hat\beta}_pa_p$
B. ${\hat\beta}^*_0={\bar y}-{\hat\beta}_1{\bar x}_1-\cdots-{\hat\beta}_p{\bar x}_p+{\hat\beta}_1a_1+\cdots+{\hat\beta}_pa_p$
C. ${\hat\beta}^*_0={\bar y}-{\hat\beta}_1{\bar x}_1-\cdots-{\hat\beta}_p{\bar x}_p$
D. ${\hat\beta}^*_j={\hat\beta}_j+a_j$ for all $j=1,\ldots,p$
E. ${\hat\beta}^*_j={\hat\beta}_j-a_j$ for all $j=1,\ldots,p$
F. ${\hat\beta}^*_0={\bar y}$
G. ${\hat\beta}^*_0={\bar y}-{\hat\beta}_1a_1-\cdots-{\hat\beta}_pa_p$
H. ${\hat\beta}^*_j={\hat\beta}_j$ for all $j=1,\ldots,p$
I. None of the above


Part c)
Continuing from (b), let ${\bf X}$ be the $n\times (p+1)$ matrix with the original x variables and let ${\bf X}^*$ be the $n\times p$ matrix with the shifted x variables without a column of 1s at the beginning. and let ${\bf y}$ be the $n\times 1$ vector.
What is ${{\bf X}^*}^T{\bf X}^*$. Which of the following are correct statements? It is:


A. same as ${\bf X}^T{\bf X}$
B. sample correlation matrix of $x_{i1},\ldots,x_{ip}$
C. $n-1$ times the sample correlation matrix of $x_{i1},\ldots,x_{ip}$
D. sample covariance matrix of $x_{i1},\ldots,x_{ip}$
E. $n-1$ times the sample covariance matrix of $x_{i1},\ldots,x_{ip}$
F. None of the above


Part d)
What is ${{\bf X}^*}^T{\bf y}$. Which of the following are correct statements? The $j$th component of ${{\bf X}^*}^T{\bf y}$ is:


A. $n-1$ times the sample covariance of $x_{ij},y_{i}$
B. sample correlation of $x_{ij},y_{i}$
C. $n-1$ times the sample correlation of $x_{ij},y_{i}$
D. sample covariance of $x_{ij},y_{i}$
E. same as $j$th position of ${\bf X}^T{\bf y}$
F. None of the above


Part e)
Now consider the $p\times 1$ vector $({\hat\beta}_1,\ldots,{\hat\beta}_p)^T$. Let ${\tt xmat}$ be an $n\times p$ matrix of the original unshifted explanatory variables, and let ${\tt yvec}$ be an $n\times 1$ vector of the response variables. Then the vector of slopes (without intercept) can be computed in R as solve(mat1,mat2), where
mat1=
and mat2=

(Because WebWork will match on answers that are strings, the exact string is needed; please do not use spaces in the middle of your strings.)

Part f)
The derivation on this page shows that the least squares coefficients ${\hat\beta}_0,{\hat\beta}_1,\ldots,{\hat\beta}_p$ can all be computed as functions of sample means and sample . [Fill in one suitable word.]

Hint: