Put a number on omitted variable bias

Question

The true model is y = 2 x1 + 3 x2 + \varepsilon with \varepsilon exogenous. You omit x2 and regress y on x1 alone. The auxiliary regression of the omitted x2 on the kept x1 has slope \delta = 0.5. What does your estimated coefficient on x1 converge to, and what is the size and sign of the bias?

Accepted Answer

The bias formula. Leaving out x2, the short-regression slope converges to \hat\beta1 \xrightarrow{p} \beta1 + \beta2 \, \delta, where \beta2 is the true effect of the omitted variable and \delta is the slope of the omitted variable on the kept one (how much x2 moves with x1). In words: your kept coefficient absorbs the omitted effect in proportion to how the two regressors overlap. Plug in. \hat\beta1 \xrightarrow{p} 2 + 3 	imes 0.5 = 2 + 1.5 = 3.5. Size and sign of the bias. 	ext{bias} = \beta2 \delta = 3 	imes 0.5 = +1.5. The bias is positive because \beta2 0 and \delta 0 share a sign. Y…

Put a number on omitted variable bias

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