Exogeneity

In linear regression, a common mean-exogeneity condition is $\mathbb{E}[\epsilon \mid X]=0$: after conditioning on the design matrix, the regression error has zero conditional expectation. For a full-column-rank design, ordinary least squares satisfies $\hat{\beta }=\beta +(X^{\top }X{)}^{-1}{X}^{\top }\epsilon$, so this condition gives $\mathbb{E}[\hat{\beta }\mid X]=\beta$ without requiring constant error variance.

A weaker condition, $\mathbb{E}[X_i{\epsilon }_i]=0$, states an unconditional moment restriction and is not identical to zero conditional mean. If an omitted variable contributes to the regression error and is correlated with a predictor, either condition can fail. Exogeneity must therefore be stated relative to a specified model and information set; it does not by itself turn an associational regression into a causal effect.