Single-state generalized autoregressive conditional heteroscedasticity (GARCH) models identify only one mechanism governing the response of volatility to market shocks, and the conditional higher moments are constant, unless modelled explicitly. So they neither capture state-dependent behaviour of volatility nor explain why the equity index skew persists into long-dated options. Markov switching (MS) GARCH models specify several volatility states with endogenous conditional skewness and kurtosis; of these the simplest to estimate is normal mixture (NM) GARCH, which has constant state probabilities. We introduce a state-dependent leverage effect to NM-GARCH and thereby explain the observed characteristics of equity index returns and implied volatility skews, without resorting to time-varying volatility risk premia. An empirical study on European equity indices identifies two-state asymmetric NM-GARCH as the best fit of the 15 models considered. During stable markets volatility behaviour is broadly similar across all indices, but the crash probability and the behaviour of returns and volatility during a crash depends on the index. The volatility mean-reversion and leverage effects during crash markets are quite different from those in the stable regime. Copyright (c) Blackwell Publishing Ltd and the Department of Economics, University of Oxford, 2009.
