  For a GJR-GARCH specification with a generic innovation distribution we
derive analytic expressions for the first four conditional moments of the
forward and aggregated returns and variances. Moment for the most commonly used
GARCH models are stated as special cases. We also the limits of these moments
as the time horizon increases, establishing regularity conditions for the
moments of aggregated returns to converge to normal moments. Our empirical
study yields excellent approximate predictive distributions from these analytic
moments, thus precluding the need for time-consuming simulations.
