Regression and econometrics

Omitted-variable bias

Signs the bias from a variable you left out, using only the two relationships it is involved in.

β^1→β1+β2 δ,δ=Cov(x1,x2)Var(x1)\hat{\beta}_1 \to \beta_1 + \beta_2\,\delta, \qquad \delta = \frac{\mathrm{Cov}(x_1, x_2)}{\mathrm{Var}(x_1)}

Where

β2\beta_2
True effect of the omitted variable on the outcome.
δ\delta
How the omitted variable co-moves with the one you kept.

Assumptions

  • The bias is the product of two signs, so it can go either way — compute it rather than guessing.

Sanity check. Noise in a regressor attenuates toward zero; noise in the outcome does not bias at all.

Where this is taught