  We propose a new approach to mixed-frequency regressions in a
high-dimensional environment that resorts to Group Lasso penalization and
Bayesian techniques for estimation and inference. In particular, to improve the
prediction properties of the model and its sparse recovery ability, we consider
a Group Lasso with a spike-and-slab prior. Penalty hyper-parameters governing
the model shrinkage are automatically tuned via an adaptive MCMC algorithm. 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 in small samples, even when the design matrix presents
cross-correlation. When applied to forecasting U.S. GDP, our penalized
regressions can outperform many strong competitors. Results suggest that
financial variables may have some, although very limited, short-term predictive
content.
