Analytic Moments for GARCH Processes
ICMA Centre Discussion Papers in Finance DP 2011-07
59 Pages Posted: 4 Nov 2010 Last revised: 14 Apr 2011
Date Written: April 12, 2011
Conditional returns distributions generated by a GARCH process, which are important for many problems in market risk assessment and portfolio optimization, are typically generated via simulation. This paper extends previous research on analytic moments of GARCH returns distributions in several ways: we consider a general GARCH model -- the GJR specification with a generic innovation distribution; we derive analytic expressions for the first four conditional moments of the forward return, of the forward variance, of the aggregated return and of the aggregated variance -- corresponding moments for some specific GARCH models largely used in practice are recovered as special cases; we derive the limits of these moments as the time horizon increases, establishing regularity conditions for the moments of aggregated returns to converge to normal moments; and we demonstrate empirically that some excellent approximate predictive distributions can be obtained from these analytic moments, thus precluding the need for time-consuming simulations.
Keywords: approximate predictive distributions, conditional and unconditional moments, GARCH, kurtosis, skewness, simulation
JEL Classification: C53
Suggested Citation: Suggested Citation