The present work focuses on the performance of two types of shrinkage priors--the horseshoe prior and the recently developed regularized horseshoe prior--in the context of inducing sparsity in path analysis and growth curve models. Prior research has shown that these horseshoe priors induce sparsity by at least as much as the "gold standard" spike-and-slab prior. The horseshoe priors are compared to the ridge prior and lasso prior, as well as default non-informative priors, in terms of the percent shrinkage in the model parameters and out-of-sample predictive performance. Empirical studies using data from two large-scale educational assessments reveal the clear advantages of the horseshoe priors in terms of both shrinkage and predictive performance. Simulation studies reveal clear advantages in terms of shrinkage, but less obvious advantages in terms of predictive performance, except in the small sample size condition where both horseshoe priors provide noticeably improved predictive performance.