# Error variable correlated with explanatory variables

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.