Construction and visualization of optimal confidence sets for frequentist distributional forecasts

The focus of this paper is on the quantification of sampling variation in frequentist probabilistic forecasts. We propose a method of constructing confidence sets that respects the functional nature of the forecast distribution, and use animated graphics to visualize the impact of parameter uncertainty on the location, dispersion and shape of the distribution. The confidence sets are derived via the inversion of a Wald test and are asymptotically uniformly most accurate and, hence, optimal in this sense. A wide range of linear and non-linear time series models - encompassing long memory, state space and mixture specifications - is used to demonstrate the procedure, based on artificially generated data. An empirical example in which distributional forecasts of both financial returns and its stochastic volatility are produced is then used to illustrate the practical importance of accommodating sampling variation in the manner proposed.