In a linear regression problem, we have: $$ y = x\beta_1 + \epsilon $$ For the convenience of discussion, let's just assume this is a univariate regression problem. (i.e. both $x$ and $y$ are vectors and $\beta_1$ is a scalar)

Because of the existence of $\epsilon$, the solution of above equation ($\hat{\beta_1}$) will be different from the solution of this one: $$ R^{\frac{1}{2}}y = R^{\frac{1}{2}}x\beta_2 + \epsilon $$ where $R^{\frac{1}{2}}$ is a square rotation matrix that maps $x$ and $y$ into another space. We know that $R^{\frac{1}{2}}$ is positive semidefinite and we know that $I-R^{\frac{1}{2}}$ is also p.s.d. (I am not sure if this is useful.)

We have: $$ \hat{\beta_1}=(x^Tx)^{-1}x^Ty $$ and $$ \hat{\beta_2}=(x^TRx)^{-1}x^TRy $$ Is there any relasionship regarding $\hat{\beta_1}$ and $\hat{\beta_2}$? It seems we can get something if we assume $\beta_1>0$.

Eventually, I want to compare the test statistics: $$ \dfrac{\beta^2}{var(y-x\beta)} $$ for $\hat{\beta_1}$ and $\hat{\beta_2}$. Will there be any relationship of these two?


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