TAIL RISK MONOTONICITY 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. 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 normally distributed innovations.