Suppose I have a linear model;

$Y=X\beta+\epsilon$

Where $X$ is $(n \times p)$, with the first column of $X$ being an intercept column (consisting only of ones). Now suppose I construct $\tilde{X}$ by demeaning and rescaling (i.e. dividing by the column norm) all columns apart from the intercept column and now use the model:

$Y=\tilde{X}\tilde{\beta}+\epsilon$

My questions are:

• Will $\hat{\beta}=\hat{\tilde{\beta}}$ when I use standard OLS? Will any of the $\beta$ values be the same and why?
• If they are not the same, is there some way for me to convert $\hat{\beta}$ to $\hat{\tilde{\beta}}$ and vice versa?
• Do I still need the intercept column in $\tilde{X}$ if all my other columns are demeaned?
• Does the intercept still have the same interpretation for both $X$ and $\tilde{X}$, why/why not?

Thank you in advance!

Suppose I have a linear model;

$Y=X\beta+\epsilon$

Where $X$ is $(n \times p)$, with the first column of $X$ being an intercept column (consisting only of ones). Now suppose I construct $\tilde{X}$ by demeaning (i.e. removing the mean from each column) and rescaling (i.e. dividing by the column norm) all columns apart from the intercept column and now use the model:

$Y=\tilde{X}\tilde{\beta}+\epsilon$

My questions are:

• Will $\hat{\beta}=\hat{\tilde{\beta}}$ when I use standard OLS? Will any of the $\beta$ values be the same and why?
• If they are not the same, is there some way for me to convert $\hat{\beta}$ to $\hat{\tilde{\beta}}$ and vice versa?
• Do I still need the intercept column in $\tilde{X}$ if all my other columns are demeaned?
• Does the intercept still have the same interpretation for both $X$ and $\tilde{X}$, why/why not?

Thank you in advance!