GARCH density and functional forecasts

This paper derives the analytic form of multi-step ahead prediction density a Gaussian GARCH(1,1) process with possibly asymmetric news impact curve in GJR class. These results can be applied when single-period returns are modeled as and interest lies at some future forecast horizon. The has been used applications an approximation to this yet unknown density; derived here shows that density, while symmetric, far from Gaussian. explicit compute exact tail probabilities functionals, such Value Risk Expected Shortfall, quantify expected required risk capital for returns. Finally, how estimation uncertainty mapped onto regions any functional distribution.