Does absence of endogeneity problem means absence of heteroscedasticity?

For example, if an estimator is designed to correct for endogeneity does this mean that heteroscedaticity will be corrected for, too?

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#### Best Answer

No, not at all. Endogeneity is a first-moment problem, while heteroskedasticity is a second-moment problem.

For example, consider a linear model

$$ y_i=x_ibeta+u_i $$ An assumption like $$E(u_i|x_i)=0$$ implies no correlation of error term and regressor, so no endogeneity. No (conditional) heteroskedasticity in turn can be written as $$Var(u_i|x_i)=sigma^2,$$ where $sigma^2$ is a constant number.

There is no reason whatsoever that $$E(u_i|x_i)=0$$ would imply $$Var(u_i|x_i)=sigma^2.$$

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