The extratropical circulation response to cooling of the polar-winter stratosphere in a simple AGCM is investigated. The AGCM is a dry hydrostatic primitive equation model with zonally symmetric boundary conditions and analytically specified physics. It is found that, as the polar-winter stratosphere is cooled, the tropospheric jet shifts poleward. This response projects almost entirely and positively (by convention) onto the AGCM’s annular mode. At the same time, the vertical flux of wave activity from the troposphere to the stratosphere is reduced and the meridional flux of wave activity from high to low latitudes is increased. Thus, as the stratosphere is cooled, the stratospheric wave drag is reduced. In order to understand this response, the transient adjustment of the stratosphere‐troposphere system is investigated using an ensemble of ‘‘switch on’’ stratospheric cooling runs of the AGCM. The response to the switch-on stratospheric cooling descends from the upper stratosphere into the troposphere on a time scale that matches simple downward-control theory estimates. The downward-control analysis is pursued with a zonally symmetric model that uses as input the thermal and eddy-driving terms from the eddying AGCM. With this model, the contributions to the response from the thermal and eddy-driving perturbations can be investigated separately, in the absence of eddy feedbacks. It is found that the stratospheric thermal perturbation, in the absence of such feedbacks, induces a response that is confined to the stratosphere. The stratospheric eddy-driving perturbation, on the other hand, induces a response that penetrates into the midtroposphere but does not account, in the zonally symmetric model, for the tropospheric jet shift. Furthermore, the tropospheric eddy-driving perturbation, in the zonally symmetric model, induces a strong upward response in the stratospheric winds. From these experiments and from additional experiments with the eddying AGCM, it is concluded that the stratospheric eddy-driving response induces a tropospheric response, but that the full circulation response results from a two-way coupling between the stratosphere and the troposphere.