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We use multivariate regression in econometrics to (hopefully) get around the problem of the omitted variable. Look at the most basic case of a simple linear regression:

$$Y_i = β_0 + β_1 x_i + u_i $$

Here an omitted variable would be included in the error term. It must be both: explaining the dependent variable and correlated with the explanatory variable $x_i$.

Now I am asking myself the simple question, why we would not measure in practice the correlation between the residual $û_i$ and the explanatory variable. I am now in some intermediate econometrics courses, but never heard of this idea before. Why would we not want to do that? Is it because there might be omitted variables that are negatively and positively correlated and offset each other?

kjetil b halvorsen
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Marlon Brando
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    What happens when you estimate the correlation between residuals and explanatory variable? Have you checked that? – T.E.G. Oct 15 '23 at 08:16
  • Thanks for the hint! I see the problem, correlation is always getting zero. $Xû = 0$ is orthogonal, if we think of the geometry of OLS and thus uncorrelated. So there is no point in doing that. But how to think of the situation correctly? We believe that in the true model, u contains the omitted variable and therefore u and x must be correlated. But in practice û and x are never correlated. Is it because the information is just not implied in the residual, but only in the real error term? – Marlon Brando Oct 15 '23 at 10:45
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    I think the information is implied but in a different way. For example, this answer (last two paragraphs) explains how omitted variables might bias coefficient estimators which in turn affect residual distribution. – T.E.G. Oct 16 '23 at 20:52

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