Omitted Variable Bias

Omitted variable bias arises when a variable that affects $Y$ and is correlated with the included regressor $X$ is left out of the model. The omitted variable lands in the error term $u$, makes $\text{cov}(X, u) \neq 0$, and biases the OLS coefficient on $X$. The sign of the bias is predictable: if the omitted variable has a positive effect on $Y$ and is positively correlated with $X$, OLS overstates $X$'s effect; flip either sign to flip the bias direction.

When to use

Reach for OVB framing whenever a DAG shows a confounder (a variable with arrows into both treatment and outcome) that you can't observe. PS_4's pooled OLS overstates the seatbelt-law effect by ~2.5× because early-adopting states were already safer (state-level unobserved heritage drives both early adoption and lower fatalities). PS_5's union-recall regression is biased because firms with bad management both unionise more and recall more. Fixes: include the omitted variable as a control (if observable), use Fixed Effects to absorb time-invariant confounders, or use Instrumental Variables when no observable proxy exists.