GARCH density and functional forecasts

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