GARCH density and functional forecasts

Abstract

This paper derives the analytic form of the multi-step ahead prediction density for single-period returns, when the latter follow a Gaussian GARCH(1,1) process with a possibly asymmetric news impact curve. The Gaussian density has been used in applications as an approximation of the multi-step ahead prediction density; the analytic form derived here shows that the prediction density, while symmetric, can be far from Gaussian. The explicit form of the prediction density 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 the stated prediction distribution.