Are excess returns predictable, and if so, what does this mean for investors? In classic studies of rational valuation (e.g. Samuelson (1965, 1973), Shiller (1981)), risk premia are constant over time and thus excess returns are unpredictable. However, an extensive empirical literature has found evidence for predictability in returns on stocks and bonds by scaled-price ratios and interest rates.
Confronted with this theory and evidence, the literature has focused on two polar viewpoints. On the one hand, if models such as Samuelson (1965) are correct, investors should maintain constant weights rather than form portfolios based on possibly spurious evidence of predictability. On the other hand, if the empirical estimates capture population values, then investors should time their allocations to a large extent, even in the presence of transaction costs and parameter uncertainty. Between these extremes, however, lies an interesting intermediate view: that both data and theory can be helpful in forming portfolio allocations.
This paper models this intermediate view in a Bayesian setting. We consider an investor who has a prior belief about the R2 of the predictive regression. We implement this prior by specifying a normal distribution for the regression coefficient on the predictor variable. As the variance of this normal distribution approaches zero, the prior belief becomes dogmatic that there is no predictability. As the variance approaches infinity, the prior is diffuse: all levels of predictability are equally likely.
In between, the distribution implies that the investor is skeptical about predictability: predictability is possible, but it is more likely that predictability is "small" rather than "large". By conditioning this normal distribution on both the unexplained variance of returns and on the variance of the predictor variable, we create a direct mapping from the investor's prior beliefs on model parameters to a well-defined prior over the R2.