  This paper considers the problem of isolating a small number of exchange
traded funds (ETFs) that suffice to capture the fundamental dimensions of
variation in U.S. financial markets. First, the data is fit to a vector-valued
Bayesian regression model, which is a matrix-variate generalization of the well
known stochastic search variable selection (SSVS) of George and McCulloch
(1993). ETF selection is then performed using the decoupled shrinkage and
selection (DSS) procedure described in Hahn and Carvalho (2015), adapted in two
ways: to the vector-response setting and to incorporate stochastic covariates.
The selected set of ETFs is obtained under a number of different penalty and
modeling choices. Optimal portfolios are constructed from selected ETFs by
maximizing the Sharpe ratio posterior mean, and they are compared to the
(unknown) optimal portfolio based on the full Bayesian model. We compare our
selection results to popular ETF advisor Wealthfront.com. Additionally, we
consider selecting ETFs by modeling a large set of mutual funds.
