Regression error ($\epsilon$)

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.