We propose a new object-centric framework for learning-based stereo 3D object detection. Previous studies build scene-centric representations that do not consider the significant variation among outdoor instances and thus lack the flexibility and functionalities that an instance-level model can offer. We build such an instance-level model by formulating and tackling a local update problem, i.e., how to predict a refined update given an initial 3D cuboid guess. We demonstrate how solving this problem can complement scene-centric approaches in (i) building a coarse-to-fine multi-resolution system, (ii) performing model-agnostic object location refinement, and (iii) conducting stereo 3D tracking-by-detection. Extensive experiments demonstrate the effectiveness of our approach, which achieves state-of-the-art performance on the KITTI benchmark. Code and pre-trained models are available at https://github.com/Nicholasli1995/SNVC.