Endogeneity without correlation?

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

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 I constructed an index, consisting of x1 and x2. Denote it as x3. People in seminars are now concerned whether the index x3 is endogenous to y - mainly because x1 may drive y. However, neither x1 and the index x3 are strongly correlated (correlation is 0.08), nor is x1 strongly correlated with y. Can we thus say that x1 does not cause endogeneity of x3 with respect to y? – lula Dec 30 '11 at 13:51

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.

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Could you please construct the example? It is easy to create if we consider three variables, which do not have any interpretation, but I think it is less trivial in the case of the regression. – mpiktas Dec 29 '11 at 9:18
Let me know if I have to update this in the above post. I have created the example using 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; ================================ Let x3 is the omitted variable, that make x1 and x2 are endogenous. cov(x1,x2) =0 and cov(x1,x3+e) & cov(x2,x3+e) are nonzero. I hope this do the explanation – vinux Dec 29 '11 at 12:15
I am new to this forum. Please forgive me if I am not following the procedure. – vinux Dec 29 '11 at 12:19
No worries. Your example is ok, include it in answer. I just bugged you, since the answer in its previous form could have been better. Since the goal of SE network is to make internet the better place, I thought I could bug you to improve the question. Welcome to the site! – mpiktas Dec 29 '11 at 12:59
Thank you. Next time I will be careful. – vinux Dec 29 '11 at 13:15