<p>We extend the generalized information criteria framework for model selection to high-dimensional sparse statistical jump models, a recent class of statistically robust and computationally efficient alternatives to hidden Markov models. Specifically, we derive expressions for the model fit and complexity to construct suitable information criteria for hyperparameter selection. In extensive simulation studies, we demonstrate that our approach selects the correct hyperparameters with high probability. Finally, providing an empirical application, we infer the key features that drive the return dynamics of the world equity market. We find that a three-state model best describes the dynamics of MSCI developed and emerging markets indexes.</p>