  We show that regularizing Bayesian predictive regressions provides a
framework for prior sensitivity analysis. We develop a procedure that jointly
regularizes expectations and variance-covariance matrices using a pair of
shrinkage priors. Our methodology applies directly to vector autoregressions
(VAR) and seemingly unrelated regressions (SUR). The regularization path
provides a prior sensitivity diagnostic. By exploiting a duality between
regularization penalties and predictive prior distributions, we reinterpret two
classic Bayesian analyses of macro-finance studies: equity premium
predictability and forecasting macroeconomic growth rates. We find there exist
plausible prior specifications for predictability in excess S&P 500 index
returns using book-to-market ratios, CAY (consumption, wealth, income ratio),
and T-bill rates. We evaluate the forecasts using a market-timing strategy, and
we show the optimally regularized solution outperforms a buy-and-hold approach.
A second empirical application involves forecasting industrial production,
inflation, and consumption growth rates, and demonstrates the feasibility of
our approach.
