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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.

In a linear regression model $Y=X\beta +\epsilon$, the regression error $\epsilon$ is the unobserved discrepancy between the response and the model's systematic component. The assumption $\mathbb{E}[\epsilon \mid X]=0$ makes $X\beta$ the conditional expectation of $Y$ given $X$; it does not follow merely from writing the equation.

For a scalar response $Y=2+3x+\epsilon$, an observation $x=4,Y=15$ has error one if the stated coefficients are the true model coefficients. A regression residual instead uses fitted coefficients: $e=Y-X\hat{\beta }$. Errors and residuals therefore answer different questions. In ordinary least squares, $X^{T}e=0$ holds algebraically when the fit exists, even though $X^T\epsilon$ need not be zero for a realized statistical sample. Assuming independence or constant variance of the errors is a further modeling choice.