The coupling between the stratosphere and troposphere following Stratospheric Sudden Warming (SSW) events is investigated in an idealized atmospheric General Circulation Model, with focus on the influence of stratospheric memory on the troposphere. Ensemble forecasts are performed to confirm the role of the stratosphere in the observed equatorward shift of the tropospheric midlatitude jet following an SSW. It is demonstrated that the tropospheric response to the weakening of the lower stratospheric vortex is robust, but weak in amplitude and thus easily masked by tropospheric variability. The amplitude of the response in the troposphere is crucially sensitive to the depth of the SSW. The persistence of the response in the troposphere is attributed to both the increased predictability of the stratosphere following an SSW, and the dynamical coupling between the tropospheric jet and lower stratosphere. These results suggest value in resolving the stratosphere and assimilating upper atmospheric data in forecast models.