A simple linear regression model as follows,
\begin{align} Y=\beta_0+\beta_1 X+\epsilon \end{align}
Now I would like to replace $X$ with another variable $Z$. I only know $X$ and $Z$ are correlated but do not know the true functional form of the relationship between these two variables.
The new regression model with $Z$ is
\begin{align} Y=\beta^*_0+\beta^*_1 Z+\epsilon^* \end{align}
How can I show that (asymptotically)
1) $\beta^*_0=\beta_0$
2) $\beta^*_1=\beta_1*\rho$, where $\rho$ is the coefficient from least square model $X=\rho Z+ \nu$
3) $\epsilon^*=\beta_1 \nu + \epsilon$
Basically why it must be least square projection of $X$ on $Z$ when we replace $X$ with $Z$?
Any suggestion or references on how to approach this question will be appreciated!
Take the original equation:
$Y = \beta_0 + \beta_1X+\epsilon $
Sub in $X=\rho Z + v$ and re-arrange terms
$Y = \beta_0 + \beta_1(\rho Z + v)+\epsilon = \beta_0 + (\beta_1\rho) Z + (\beta_1 v + \epsilon)$
This is equivalent to regressing Y on Z so the coefficients must be identical.