Regress y on residuals of x and z

Asked Sep 24 '23 at 20:12

Active Sep 25 '23 at 00:20

Viewed 42 times

I have the following set up:

$y_i = \beta_0 + \beta_1 x_i + \beta_2 z_i + e_i$, where $e_i$ is extracted from a Normal (0,1) distribution independently of $x$ and $z$. The true values are $\beta_0 = \beta_1 = \beta_2 = 1$.

I want to compare the coefficients of regressing $y$ on $x$ and $z$ with the coefficients of regressing $y$ on the residuals of $x$ (obtained by regressing $x$ on $z$) and the residuals of $z$ (obtained by regressing $z$ on $x$).

I can see that the estimates are now biased but I can't find a way to understand the rational as to why they are biased. Any ideas?

edited Sep 25 '23 at 00:20

Durden

1,171

asked Sep 24 '23 at 20:12

user986483

Your question seems related to the Frisch-Waugh theorem. – Durden Sep 25 '23 at 00:03
In what sense are the estimates "biased"? What do you suppose you are estimating with these residual regressions in the first place? Only then can we know what you might mean by a "bias." – whuber Sep 25 '23 at 00:21

Regress y on residuals of x and z

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