Before revision:   Simulations show that the proposed models have good selection and forecasting performance  , even when the design matrix presents high cross-correlation.
After revision:  We establish good frequentist asymptotic properties of the posterior of the in-sample and out-of-sample prediction error, we recover the optimal posterior contraction rate, and we show optimality of the posterior predictive density. Simulations show that the proposed models have good selection and forecasting performance , even when the design matrix presents high cross-correlation.