In this paper, we consider MCMC algorithms for Student-$t$ regression models. We investigate the efficiency of Markov chains based on the algorithms in terms of whether trace-class results hold or not. We first consider the case where the parameters follow a matrix-normal-inverse-Wishart distribution and show that the Markov operator associated with a standard data augmentation algorithm is trace-class. We next consider the case of an improper prior and univariate outcomes. In this case, the standard Markov operator is not trace-class but the Markov operator associated with a collpased Gibbs algorithm is trace-class. Finally, we consider the case of an improper prior and multivariate outcomes. We obtain a trace-class result for a parameter expanded data augmentation algorithm which is based on a univariate working parameter.