Making Density Forecasting Models Statistically Consistent

We propose a new approach to density forecast optimisation and apply it to Value-at-Risk estimation. All existing density forecasting models try to optimise the distribution of the returns based solely on the predicted density at the observation. In this paper we argue that probabilistic predictions should be optimised on more than just this accuracy score and suggest that the statistical consistency of the probability estimates should also be optimised during training. Statistical consistency refers to the property that if a predicted density function suggests P percent probability of occurrence, the event truly ought to have probability P of occurring. We describe a quality score that can rank probability density forecasts in terms of statistical consistency based on the probability integral transform (Diebold et al., 1998b). We then describe a framework that can optimise any density forecasting model in terms of any set of objective functions. The framework uses a multi-objective evolutionary algorithm to determine a set of trade-off solutions known as the Pareto front of optimal solutions. Using this framework we develop an algorithm for optimising density forecasting models and implement this algorithm for GARCH (Bollerslev, 1986) and GJR models (Glosten et al., 1993). We call these new models Pareto-GARCH and Pareto-GJR. To determine whether this approach of multi-objective optimisation of density forecasting models produces better results over the standard GARCH and GJR optimisation techniques we compare the models produced empirically on a Value-at-Risk application. Our evaluation shows that our Pareto models produce superior results out-of-sample