$e^+ e^- \to e^- {\bar \nu_e} u {\bar d}$ from LEP to linear collider energies

The complete tree level cross section for the process $e^+e^- \to e^- \bar\nu_e u \bar{d}$ is computed using the GRACE system, a program package for automatic amplitude calculation. Special attention is brought to the gauge violation problem induced by the finite width of the $W$-boson. The {\it preserved gauge scheme} is introduced and an event generator which includes double-resonant, single-resonant and non-resonant diagrams with no need for a cut on the electron polar angle is presented. Since no cut needs to be applied to the electron, the generator can be used to estimate background for searches involving jets and missing energy. A monojet event rate estimation based on this process at LEP-I energy is discussed.