 Residual constraints from a bounded response

If observed responses satisfy $L\le y_i\le U$, their regression residuals $e_i=y_i-{\hat{y}}_i$ necessarily obey the displayed inequalities. Subtracting the corresponding fitted values proves both bounds, without a normal distribution or a fitted-model assumption. On a residual-versus-fitted plot, all points must lie between two parallel lines of slope $-1$. This is also a consistency check when $L,U$ are merely the reported observed minimum and maximum, rather than theoretical response limits. A bounded response does not itself prove that a straight conditional mean is inappropriate.