Endogeneity without correlation?

Question

Is it, in general, possible that two variables are endogenous even if they are not correlated with each other at all?

Answer 1

Your question is bit confusing. The answer is Yes. It is possible that two variables are endogenous even if they are not correlated with each other at all.

Consider the model $Y = a +bX_1+cX_2 +e$. We can construct variables where $Cov(X_1,X_2)=0$, but $Cov(X_1, e)$ and $Cov(X_2,e)$ are non zero.

Here is the example in R:

x1= rnorm(1000); 
x2 =rnorm(1000); 
x3= 0.2*x1+ 0.3*x2 + 0.4*rnorm(1000); 
e=0.1*rnorm(1000); 
y=3+2*x1+3*x2+x3+e; 

Here $x_3$ is the omitted variable, which makes $x_1$ and $x_2$ endogenous. $Cov(x_1,x_2) =0$ by construction and $Cov(x_1,x_3+e)$ and $Cov(x_2,x_3+e)$ are nonzero.