Residual constraints from a bounded response ($L-{\hat{y}}_i\le e_i\le U-{\hat{y}}_i$)

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.

For a finite sample, let $L={\min }_i{y}_i$, $U={\max }_i{y}_i$, $m={\min }_i{\hat{y}}_i$ and $M={\max }_i{\hat{y}}_i$. The regression residuals are $e_i=y_i-{\hat{y}}_i$. Every residual is at least $L-M$ and at most $U-m$. At an observation attaining $y_i=L$, its residual is at most $L-m$, so $e_{\min }\le L-m$. At an observation attaining $y_i=U$, its residual is at least $U-M$, so $e_{\max }\ge U-M$. These observations prove all four bounds. Merely knowing theoretical response limits, without attainment, would not justify the two inner bounds.