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

See all articles by Carol Alexander

Carol Alexander

University of Sussex Business School

Emese Lazar

University of Reading - ICMA Centre

Silvia Stanescu

University of Kent - Kent Business School

Date Written: April 12, 2011

Abstract

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

Alexander, Carol and Lazar, Emese and Stanescu, Silvia, Analytic Moments for GARCH Processes (April 12, 2011). ICMA Centre Discussion Papers in Finance DP 2011-07. Available at SSRN: https://ssrn.com/abstract=1702623 or http://dx.doi.org/10.2139/ssrn.1702623

Carol Alexander (Contact Author)

University of Sussex Business School ( email )

Falmer, Brighton BN1 9SL
United Kingdom

HOME PAGE: http://www.carolalexander.org

Emese Lazar

University of Reading - ICMA Centre ( email )

Whiteknights Park
P.O. Box 242
Reading RG6 6BA
United Kingdom
+44 (0)1183 786675 (Phone)
+44 (0)1189 314741 (Fax)

Silvia Stanescu

University of Kent - Kent Business School ( email )

Canterbury, Kent CT2 7PE
United Kingdom

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