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In my econometrics professors' slides he states that:

”We said that $\sum_{i=1}^nX_i\hat{u}_i=0$ (where $\hat{u}$ are the residuals) implies zero sample correlation between X and Y“.

My problem is in understanding WHY this is (the statement in bold), or if he made a mistake and actually meant that each regressor always has zero correlation with residuals estimator, due to first order conditions used to grant OLS (even in a multivariate regression, not necessarily single variable regression).


IMPORTANT EDIT: X is the regressor in a single variable linear regression and Y is the outcome of such regression

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That is clearly wrong as stated. As you say, orthogonality between regressors and residuals is true for any regression estimated by OLS. At the same time, the OLS estimator in a bivariate regression is known to be $$ \hat\beta_1=\frac{\widehat{cov}(x,y)}{\widehat{var}(x)} $$ Hence, if the statement were true, we would have a proof that, always, $\hat\beta_1=0$...

So I suppose that is a typo indeed.

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  • $\begingroup$ Thank you very much! $\endgroup$ Commented Aug 5, 2022 at 10:54

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