Taking selfies has become a photographic trend nowadays. We envision the emergence of the video selfie capturing a short continuous video clip (or burst photography) of the user, themselves. A selfie stick is usually used, whereby a camera is mounted on a stick for taking selfie photos. In this scenario, we observe that the camera typically goes through a special trajectory along a sphere surface. Motivated by this observation, in this work, we propose an efficient and globally optimal relative camera pose estimation between a pair of two images captured by a camera mounted on a selfie stick. We exploit the special geometric structure of the camera motion constrained by a selfie stick and define its motion as spherical joint motion. By the new parametrization and calibration scheme, we show that the pose estimation problem can be reduced to a 3-DoF (degrees of freedom) search problem, instead of a generic 6-DoF problem. This allows us to derive a fast branch-and-bound global optimization, which guarantees a global optimum. Thereby, we achieve efficient and robust estimation even in the presence of outliers. By experiments on both synthetic and real-world data, we validate the performance as well as the guaranteed optimality of the proposed method.