Omitted variable bias, derive the formula

Question

The true model is y = \beta1 x1 + \beta2 x2 + \varepsilon (variables centered, \varepsilon exogenous), but you regress y on x1 alone. Derive what your estimated slope converges to, state the sign rule for the bias, and give a finance example.

Accepted Answer

The derivation. The short-regression slope estimator satisfies \hat{\gamma}1 \xrightarrow{p} \frac{\operatorname{Cov}(x1, y)}{\operatorname{Var}(x1)} = \frac{\operatorname{Cov}(x1,\ \beta1 x1 + \beta2 x2 + \varepsilon)}{\operatorname{Var}(x1)} = \beta1 + \beta2\,\underbrace{\frac{\operatorname{Cov}(x1, x2)}{\operatorname{Var}(x1)}}{\delta}, using \operatorname{Cov}(x1, \varepsilon) = 0. Here \delta is exactly the slope of the auxiliary regression of the omitted x2 on the included x1. So 	ext{OVB} = \beta2 	imes \delta = (	ext{effect of omitted variable}) 	imes (	ext{its loading on the in…

Omitted variable bias, derive the formula

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