GARCH models, tail indexes and error distributions: An empirical investigation

• We examine whether generalized autoregressive conditional heteroskedasticity models capture extreme events.
• We estimate Hill's tail indexes for individual S&P 500 stock market returns.
• We compare actual tail indexes to the indexes produced by simulating GARCH models.
• GARCH models with normal conditional distributions underestimate tail risk.
• GARCH models with Student's t conditional distributions perform better.

We perform a large simulation study to examine the extent to which various generalized autoregressive conditional heteroskedasticity (GARCH) models capture extreme events in stock market returns. We estimate Hill's tail indexes for individual S&P 500 stock market returns and compare these to the tail indexes produced by simulating GARCH models. Our results suggest that actual and simulated values differ greatly for GARCH models with normal conditional distributions, which underestimate the tail risk. By contrast, the GARCH models with Student's t conditional distributions capture the tail shape more accurately, with GARCH and GJR-GARCH being the top performers.