For this question and others that might involve derivation of a formula, after you formulate an algebraic solution, check its validity on some numerical regression examples with small data sets. If you match numerically in some instances, your answer is likely correct. If your theoretical answer doesn't match the numerical cases, go back to review your "derivation".

The raw data are $y_i,x_{i1},x_{i2}$ for $i=1,\ldots,n$. Let ${\hat\beta}_0, {\hat\beta}_1, {\hat\beta}_2$ be the least squares estimates when fitting a plane.

Part a) Suppose the $y$ variable has been rescaled to $y^*$ where $y^*=0.1 y$. Let ${\hat\beta}^*_0, {\hat\beta}^*_1, {\hat\beta}^*_2$ be the least squares estimates for the data $y_i^*,x_{i1},x_{i2}, i=1,\ldots,n$.
Which of the following are correct? Possibly more than one item is correct.

A. ${\hat\beta}^*_0=10{\hat\beta}_0$
B. ${\hat\beta}^*_2={\hat\beta}_2/10$
C. ${\hat\beta}^*_2=10{\hat\beta}_2$
D. ${\hat\beta}^*_0={\hat\beta}_0/10$
E. ${\hat\beta}^*_1={\hat\beta}_1/10$
F. ${\hat\beta}^*_1=10{\hat\beta}_1$
G. None of the above


Part b)
Suppose the $x_1$ variable has been rescaled to $x_1^*$ where $x_1^*=10 x_1$. Let ${\hat\beta}^*_0, {\hat\beta}^*_1, {\hat\beta}^*_2$ be the least squares estimates for the data $y_i,x_{i1}^*,x_{i2}, i=1,\ldots,n$.
Which of the following are correct? Possibly more than one item is correct.

A. ${\hat\beta}^*_1={\hat\beta}_1/10$
B. ${\hat\beta}^*_2=10{\hat\beta}_2$
C. ${\hat\beta}^*_0=10{\hat\beta}_0$
D. ${\hat\beta}^*_1=10{\hat\beta}_1$
E. ${\hat\beta}^*_0={\hat\beta}_0/10$
F. ${\hat\beta}^*_2={\hat\beta}_2/10$
G. None of the above


Part c) Suppose the $x_1$ variable has been shifted to $x_1^*$ where $x_1^*=x_1-2$. Let ${\hat\beta}^*_0, {\hat\beta}^*_1, {\hat\beta}^*_2$ be the least squares estimates for the data $y_i,x_{i1}^*,x_{i2}, i=1,\ldots,n$.
Which of the following are correct? There might be more than one correct answer.

A. ${\hat\beta}^*_2\ne {\hat\beta}_2$
B. ${\hat\beta}^*_2= {\hat\beta}_2$
C. ${\hat\beta}^*_0\ne {\hat\beta}_0$
D. ${\hat\beta}^*_1= {\hat\beta}_1$
E. ${\hat\beta}^*_0= {\hat\beta}_0$
F. ${\hat\beta}^*_1\ne {\hat\beta}_1$
G. None of the above

Hint: