Evaluation of joint density forecasts of stock and bond returns: predictability and parameter uncertainty

One of the most important findings in empirical finance has been the fact that returns are not i.i.d. Predictability, or time variation in the conditional distribution of returns, is one of the basic ingredients of asset pricing and portfolio choice models nowadays. Under the current renewed interest in its implications for portfolio management, there is a growing literature on the computation of optimal portfolios for a given utility function and a given estimation of a particular model of returns. But there is an obvious problem with that approach. If the estimated model is not an accurate description of the distribution of returns the conclusions may be misleading. The approach in this paper is to study the properties of a particular model by means of the evaluation of its joint density forecasts of stock and bond returns. The focus is not on model testing, the procedure is based instead on out-of-sample checks of real-time density forecasting rules against realizations of returns. That is the relevant context for portfolio management. In addition, two important and controversial questions are addressed for each model: time-variation in risk premia and parameter uncertainty.