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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!

Vincent
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1 Answers1

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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.

Greg
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