The shared response model (SRM) provides a simple but effective framework to analyze fMRI data of subjects exposed to naturalistic stimuli. However, when the number of subjects or runs is large, fitting the model requires a large amount of memory and computational power, which limits its use in practice. Furthermore, SRM is not identifiable, which makes the shared response difficult to interpret. In this work, we implement an identifiable version of SRM and show on real data that it improves the stability of the recovered shared response. We then introduce FastSRM, which relies on a dimension reduction step and yields the same solution as the original algorithm. We show experimentally using synthetic and real fMRI data that FastSRM is considerably faster and more memory efficient than current implementations.