Regularizing Bayesian Predictive Regressions

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.