Analytic moments for GJR-GARCH (1, 1) processes

For a GJR-GARCH(1, 1) specification with generic innovation distribution we derive analytic expressions for the first four conditional moments of forward and aggregated returns variances. Moments most commonly used GARCH models are stated as special cases. We also limits these time horizon increases, establishing regularity conditions to converge normal moments. A simulation study using produces approximate predictive distributions which free from bias affecting simulations. An empirical almost 30 years daily equity index, exchange rate interest data applies Johnson SU Edgeworth expansion fitting our closed-form formulae higher returns.