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In a multivariate OLS model,

$$ Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \epsilon$$

Is my estimator for $\beta_1$ given by:

$$\hat \beta_1 = [X_1'X_1]^{-1} X_1'Y$$

or:

$$\hat \beta_1 = [X_1' M_2X_1]^{-1} X_1'M_2Y$$

where

$$M_2 = I - X_2[X_2'X_2]^{-1}X_2'$$

Which of these estimators is correct?

Also, if it is the latter, how do we obtain unbiasedness?

$$\hat \beta_1 = [X_1' M_2X_1]^{-1} X_1'M_2Y$$

Because, here if I replace $Y$ with $ Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \epsilon$

$\hat \beta_1 = [X_1' M_2X_1]^{-1} X_1'M_2(\beta_0 + \beta_1 X_1 + \beta_2 X_2 + \epsilon)$

It will be unbiased if $\beta_0 = 0 , \beta_2 = 0$ or if $X_1$ and $X_2$ are orthogonal.

Where am I going wrong?

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  • $\begingroup$ I may be wrong, but aren't both estimations of $\hat{\beta_1}$ equal to one another? $\endgroup$ – David May 17 at 23:38
  • $\begingroup$ @David I am not sure $\endgroup$ – Raghav Goyal May 17 at 23:48
  • $\begingroup$ wrong about $\beta_1$ $\endgroup$ – Aksakal May 18 at 0:01
  • $\begingroup$ @Aksakal which $\beta_1$ ? $\endgroup$ – Raghav Goyal May 18 at 0:02
  • $\begingroup$ Because the first estimator ignores $X_2$ altogether, isn't it a fortiori obvious that it has problems? $\endgroup$ – whuber Jun 6 at 13:54

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