Tail Risk Monotonicity Under Temporal Aggregation in GARCH(1,1) Models

44 Pages Posted: 3 Jan 2020 Last revised: 16 Jun 2021

See all articles by Paul Glasserman

Paul Glasserman

Columbia Business School

Dan Pirjol

Stevens Institute of Technology

Qi Wu

City University of Hong Kong, School of Data Science

Date Written: December 11, 2019

Abstract

The stationary distribution of a GARCH(1,1) process has a power law decay, under broadly applicable conditions. We study the change in the exponent of the tail decay under temporal aggregation of parameters, with the distribution of innovations held fixed. This comparison is motivated by the fact that GARCH models are often fit to the same time series at different frequencies. The resulting models are not strictly compatible so we seek more limited properties we call forecast consistency and tail consistency. Forecast consistency is satisfied through a parameter transformation. Tail consistency leads us to derive conditions under which the tail exponent increases under temporal aggregation, and these conditions cover most relevant combinations of parameters and innovation distributions. But we also prove the existence of counterexamples near the boundary of the admissible parameter region where monotonicity fails. These counterexamples include several standard choices for innovation distributions, including the normal case.

Keywords: GARCH, Tail Risk

JEL Classification: C22, C58

Suggested Citation

Glasserman, Paul and Pirjol, Dan and Wu, Qi, Tail Risk Monotonicity Under Temporal Aggregation in GARCH(1,1) Models (December 11, 2019). Available at SSRN: https://ssrn.com/abstract=3502425 or http://dx.doi.org/10.2139/ssrn.3502425

Paul Glasserman (Contact Author)

Columbia Business School ( email )

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Dan Pirjol

Stevens Institute of Technology ( email )

Hoboken, NJ 07030
United States

Qi Wu

City University of Hong Kong, School of Data Science ( email )

83 Tat Chee Avenue
Kowloon
Hong Kong

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