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We know that sometimes the error variable in a regression framework may be correlated with explanatory variables; this happens e.g. when we omit an important predictor from our study, getting the well known "omitted variable bias" issue.

Now, suppose that in our case study $(Y,X_1,X_2)$ are jointly distributed but we're aware only of $X_1$.

Let's assume for simplicity that $E[Y|X_1,X_2]$ and $E[Y|X_1]$ are both linear.

We collect $n$ data , $i=1,...,n$ ; In our Hp. the data generating process is in the $long$ form: $$y_i=\beta_0+\beta_1x_{i1} +\beta_2x_{i2}+\epsilon_i$$

our regression model is in the $short$ form: $$y_i=\Gamma_0 +\Gamma_1x_{i1} +u_i$$

Let us recall $\mathbf{X}=(1,X_1)$ and $\mathbf{\Gamma }=(\Gamma_0,\Gamma_1$), such that $y=\mathbf{X}^t\mathbf{\Gamma}+u$.


If we take $\mathbf{\Gamma}$ as $\mathbf{\Gamma}=\left({E[\mathbf{X}\mathbf{X^t}]}\right )^{-1}E[\mathbf{X}y]$ we end up with $E[\mathbf{X}u]=0$, in fact:

$u=y-\mathbf{X^t}\mathbf{\Gamma} \\ E[\mathbf{X}u]=E[\mathbf{X}y-\mathbf{X}\mathbf{X^t}\mathbf{\Gamma}]=E[\mathbf{X}y-\mathbf{X}\mathbf{X^t}\left({E[\mathbf{X}\mathbf{X^t}]}\right )^{-1}E[\mathbf{X}y]]=E[\mathbf{X}y]-E[\mathbf{X}\mathbf{X^t}]\left({E[\mathbf{X}\mathbf{X^t}]}\right )^{-1}-E[\mathbf{X}y]=0.$

But $E[\mathbf{X}u]=0$ means $cov(\mathbf{X},u)=0.$

This means that with this particular choice of $\mathbf{\Gamma} $ the endogeneity issue fades out,providing we're able to compute the aforementionend expectation. Clearly there's something wrong with my argument but I can't figure out what it is.

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