Multicollinearity and endogeneity (omitted variable bias)

[Jack Grover]

Dear Stata users, I had a doubt while running some regressions on Stata, which is nonetheless theoretical rather than operational.

My doubt concerns the relationship between multicollinearity and endogeneity provoked by omitted variable bias.

Now; if the true model is Y = a X1 + b X2 + e, but we estimate instead Y = a X1 + u, if the correlation of X1 and X2 is not 0, then X1 is correlated with the error term u = b X2 + e, and we have hence endogeneity provoked by omitted variable bias.

Nonetheless, if I estimate correctly Y = a X1 + b X2 + e, I do not have endogeneity anymore (since I do not have omitted variable bias), however I have multicollinearity (since the correlation of X1 and X2 is not 0). Therefore, in light of this, does this imply that solving for omitted variable bias always leads to multicollinearity?

Thank you.

Jack

[FernandoRios]

Hi Jack
Broadly speaking, multicollinearity is not a problem for model estimation unless its perfect multicolinearity, where certain parameters cannot be estimated.
So technically speaking, your statement is correct, except for the fact that there is almost always multicolinearity among variables, because in finite samples, the correlation between any to variables is almost never zero (unless constructed in such a way).
HTH
Fernando

[Jack Grover]

That is exactly what I was thinking.
Thank you Fernando.
Jack