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

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. The parameter transformation we study results from approximating a GARCH process observed at one frequency with another observed at a lower frequency. We 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 regions where monotonicity fails. These counterexamples include several standard choices for innovation distributions, including the normal case.