# OLS basic doubt

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?

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