Bayesian Forecasting Methods

Bayesian forecasting methods typically involve calculating the predictive density Bfor the outcome variable, Y, given both observables and assumptions on so-called unobservables. The observables are the data we condition on, i.e., z, which are outcomes of the random variables generating the data, Z. The unobservables are a given model, M(θ ) ∈ M and associated parameters θ , for the joint density of the data and outcome along with a prior distribution over the parameters of this joint density, π (θ ). Using Bayes’ rule, these can all be combined to construct the forecast distribution of Y given z and the choices on the unobservables, PY (Y|Z, M).

The forecaster’s loss function is generally not involved in the construction of the predictive density, although robustness to variations in either the proposed model or the prior can (and should) be viewed from the perspective of the loss function. Point forecasts can be formed by taking summary statistics of this predictive density with respect to the loss function in a manner analogous to the “known distribution” results presented in chapter 3. In practice, the known distribution is replaced with the predictive density and the point forecast is chosen to minimize the expected loss given the predictive density. In turn, predictive densities can themselves be used as density forecasts for the outcome of interest.

Several practical difficulties arise in implementing Bayesian forecasting methods. First on the list is how to specify the joint density of the data which is typically unknown. Properties of the forecast will depend on the choices made for this joint density. To address uncertainty about the form of the correct density, one possibility is to consider multiple models and use Bayesian model averaging—see chapter 14. In this chapter we assume that there is a single model, M, and although the results depend on the model choice, we suppress this dependence in the notation. Second, priors must be elicited for both the parameters and the unobservables. A third problem for forecasters is the construction of point forecasts, a process that involves summary statistics of the predictive density.

Section 5.1 introduces the basic theory behind Bayes risk as used in Bayesian forecasting analysis. Section 5.2 covers Ridge and shrinkage estimators while Section 5.3 provides a brief review of computational methods and discusses Bayesian modeling of predictive densities. Section 5.4 discusses economic applications of Bayesian forecasting methods and goes through an illustrative portfolio allocation example. Section 5.5 concludes.