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Consider the following "true" multiple regression model (observation index omitted for simplicity):

$$ Y = \alpha_{1} + \alpha_{2}X + \alpha_{3}Z + u $$

where $u$ is white noise, normally distributed. Importantly, assume $X$ and $Z$ are correlated.

Say I estimate the following "first-step" regression via OLS:

$$ Y = \beta_{1} + \beta_{2}X + e $$

Note that due to the correlation assumption, $\beta_{2}$ is biased. From here, I construct the residuals $\hat{e}$, which can be understood as the "component of $Y$ not explained by $X$". Then, I estimate the "second-step" regression via OLS:

$$\hat{e} = \gamma_{1} + \gamma_{2}Z + v$$

Will $\gamma_{2}$ be an unbiased/consistent estimator of $\alpha_{3}$?

luchonacho
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  • Why should $\gamma_2$ (a coefficient of $Z$) have anything at all to do with $\alpha_2$ (a coefficient of $X$)? – whuber Jun 06 '16 at 16:17
  • I suspect my answer at http://stats.stackexchange.com/a/46508/919 might shed some light on what you are doing. – whuber Jun 06 '16 at 16:53
  • To be clear, as the link shows it is possible to partial out $X$ from $Z$ (first-stage), and use the uncorrelated part of $Z$ against $Y$ (second-stage) to correctly estimate $\alpha_{3}$. I am aware of that, actually. My question is different. The first stage subtracts from $Y$ the $X$ component (which will be biased due to correlation with $Z$; this is, part of what is taken out of $Y$ will be explained also by $Z$). My intuition is that the answer to my question is no, but I want a more formal proof, if possible. – luchonacho Jun 06 '16 at 18:15
  • The material I directed you to shows that if you also partial $X$ out of $Z$, then in the last step you will obtain the correct multiple regression coefficient estimate. But, since by construction $\hat e$ is orthogonal to $X$, that's not actually necessary. – whuber Jun 06 '16 at 18:20
  • I see. Maybe I should be more precise. I found this two-step method in a paper, and I want to evaluate it's robustness. So my question is really about whether the estimation is consistent or not. – luchonacho Jun 06 '16 at 20:18

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