We predict the pseudoelasticity of the $⟨100⟩/{100}$ copper nanowire using atomistic simulations with the embedded atom method potential under uniaxial compressive loading. The $⟨100⟩/{100}$ copper nanowire exhibits pseudoelasticity which depends on the reorientation of the crystalline structure of the nanowire due to twinning. The twinning that governs the pseudoelasticity of the $⟨100⟩/{100}$ nanowire results from an external compressive loading whereas the twinning that governs the pseudoelasticity of the $⟨110⟩/{111}$ nanowire results from a tensile loading. Therefore, the pseudoelasticity of the $⟨100⟩/{100}$ nanowire is referred to as ``compressive pseudoelasticity.'' This difference in the twinning phenomena distinguishes the $⟨100⟩/{100}$ nanowire from the $⟨110⟩/{111}$ nanowire. This study also shows that the pseudoelasticity of the nanowire is related to the stacking-fault energy of the nanowire material and the Schmid factor which depends on the structural orientation of the nanowire. The $⟨100⟩/{100}$ nanowire shows the maximum recoverable strain of 22% which is a remarkable amount compared with $5--10\text{ }\mathrm{%}$ for bulk shape memory alloys. In addition, the $⟨100⟩/{100}$ nanowire does not need to reach the critical temperature to exhibit pseudoelasticity because the lateral free surfaces of the twin region have lower energy than the surface of the nanowire.