Grad's moment models for Boltzmann equation were recently regularized to globally hyperbolic systems and thus the regularized models attain local well-posedness for Cauchy data. The hyperbolic regularization is only related to the convection term in Boltzmann equation. We in this paper studied the regularized models with the presentation of collision terms. It is proved that the regularized models are linearly stable at the local equilibrium and satisfy Yong's first stability condition with commonly used approximate collision terms, and particularly with Boltzmann's binary collision model.