Density forecast transformations

The common choice of using a direct forecasting scheme implies that the individual predictions ignore information on their cross-horizon dependence. However, this dependence is needed if the forecaster has to construct, based on direct density forecasts, predictive objects that are functions of several horizons (e.g. when constructing annual-average growth rates from quarter-on-quarter growth rates). To address this issue we propose using copulas to combine the individual h-step-ahead predictive distributions into one joint predictive distribution. Our method is particularly appealing to those for whom changing the direct forecasting specification is too costly. We use a Monte Carlo study to demonstrate that our approach leads to a better approximation of the true density than an approach that ignores the potential dependence. We show the superior performance of our method using several empirical examples, where we construct (i) quarterly forecasts using month-on-month direct forecasts, (ii) annual-average forecasts using monthly year-on-year direct forecasts, and (iii) annual-average forecasts using quarter-on-quarter direct forecasts.