Omitted variable bias: sign and size

Question

The true model is y = \beta0 + \beta1 x1 + \beta2 x2 + \varepsilon, but you fit only y = \gamma0 + \gamma1 x1 + u, leaving x2 out. What does \hat\gamma1 estimate, and how do you predict the sign and size of the bias?

Accepted Answer

The bias identity. The short regression's slope has probability limit \hat\gamma1 \;	o\; \beta1 + \beta2\,\delta, where \delta is the slope from regressing the omitted x2 on the included x1. So \hat\gamma1 estimates the true partial effect \beta1 plus a contamination term \beta2\delta. The bias is \beta2\delta, and it vanishes only if the omitted variable has no effect (\beta2 = 0) or is uncorrelated with x1 (\delta = 0). Predicting the direction. The sign of the bias is the sign of \beta2 	imes \delta, the product of "how x2 affects y" and "how x2 correlates with x1": \beta2 (effect of x2 …

Omitted variable bias: sign and size

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