Omitted variable bias from a covariance and a variance

Question

The true model is y = 1 \cdot x1 + 2 x2 + \varepsilon with \varepsilon exogenous. You omit x2 and regress y on x1. You are told \operatorname{Cov}(x1, x2) = 6 and \operatorname{Var}(x1) = 3. Compute what the x1 coefficient converges to and the bias.

Accepted Answer

Build the loading first. The loading \delta is the slope of the omitted variable on the kept one, which is a covariance over a variance: \delta = \frac{\operatorname{Cov}(x1, x2)}{\operatorname{Var}(x1)} = \frac{6}{3} = 2. Apply the bias formula. \hat\beta1 \xrightarrow{p} \beta1 + \beta2 \delta = 1 + 2 	imes 2 = 1 + 4 = 5. Size and sign. The bias is \beta2 \delta = 2 	imes 2 = +4, upward, because both the omitted effect and the loading are positive. Your slope reads 5 when the true direct effect is only 1, so four-fifths of the estimate is borrowed from the omitted variable. Takeaway. The …

Omitted variable bias from a covariance and a variance

Your answer

Discussion

Learn the concepts

Related questions