One of the most important findings in empirical finance has been the fact that returns are not independent neither identically distributed over time. This fact means that there is time variation in the conditional distribution of returns and it is called predictability. Its economic meaning is that the investor faces time-varying investment opportunities. This concept was introduced by Merton (1973), who developed the Intertemporal CAPM to describe the implications of predictability for portfolio choice and asset pricing.
In terms of portfolio management, predictability implies additional terms in the optimal portfolio of an investor with respect to the usual ones of risk-return trade-off. These terms are called hedging demands as they are driven by the correlation between shocks to the returns and shocks to the state variables. This portfolio choice has implications in terms of asset pricing. The market portfolio is no longer the only priced risk factor and a multifactor model should be used for pricing.
Predictability has become one of the basic ingredients of asset pricing and portfolio choice models nowadays. There is a renewed interest in its portfolio management implications. This is shown by a growing literature on the computation of optimal portfolios for a given utility function and the estimation of a particular model of returns. The latter is required to evaluate the expected utility of the agent for any portfolio choice. An investor needs density forecasts, not only first and second moments, unless some restrictions are imposed on her preferences or the distribution of returns. In addition, a joint density forecast of several assets is required to be useful for portfolio management.
Campbell et al. (2003) is an example of this literature using classic inference. On the other hand, Kandel and Stambaugh (1996), Stambaugh (1999) and Barberis (2000) study portfolio selection based on Bayesian inference. Those papers are based on a normal vector autoregression (VAR) of returns and predictors. Although new models are being introduced, for instance the Markov switching VAR as in Ang and Bekaert (2001), and its portfolio implications studied.
But there is an obvious problem with that approach since conclusions may be misof joint density forecasts of stock and bond returns computed from different models. Multiple assets must be included in the analysis and therefore monthly U.S. excess returns of bonds and stocks are jointly studied in the post-war period.
The evaluation is going to be implemented in a relevant context for portfolio management. 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. In addition, my evaluation is not going to use comparisons of models in terms of their performance for a given loss function. It is based on the probability integral transform (PIT), which is defined as the cumulative distribution function from the density forecast evaluated at the final realization. The PIT should be uniform and independent over time if the forecasts are accurate. This methodology is advocated in Diebold, Gunther and Tay (1998).
