A big omitted effect that causes no bias at all

Question

The true model is y = 4 x1 + 8 x2 + \varepsilon with \varepsilon exogenous. You omit x2 and regress y on x1. Crucially, x2 is uncorrelated with x1, so the loading is \delta = 0. What does the x1 coefficient converge to, and what is the bias?

Accepted Answer

Apply the formula. \hat\beta1 \xrightarrow{p} \beta1 + \beta2 \delta = 4 + 8 	imes 0 = 4. Size and sign of the bias. 	ext{bias} = \beta2 \delta = 8 	imes 0 = 0. There is no bias, even though the omitted variable has a large true effect of 8. The reason is that bias requires two ingredients: the omitted variable must affect y (it does, \beta2 = 8) and it must be correlated with the kept regressor (it is not, \delta = 0). Missing either ingredient kills the bias. The one cost that remains. Omitting an uncorrelated but relevant variable does not bias the slope, but it does inflate the residua…

A big omitted effect that causes no bias at all

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