Analytic Moments for GARCH Processes

TL;DR

This paper derives explicit formulas for the first four moments of GARCH process returns and variances, enabling efficient prediction without simulations.

Contribution

It provides analytic expressions for moments of GARCH models with general innovations, including their limits and convergence properties.

Findings

Analytic moments accurately predict return distributions

Moments converge to normal distribution over time

Reduces reliance on simulation for GARCH predictions

Abstract

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.