Threshold arch models and asymmetries in volatility

This paper attempts to enlarge the class of Threshold Heteroscedastic Models (TARCH) introduced by Zakoïan (1991a). We show that it is possible to relax the positivity constraints on the parameters of the conditional variance. Unconstrained models provide a greater generality of the paths allowing for nonlinearities in the volatility. Cyclical behaviour is permitted as well as different relative impacts of positive and negative shocks on volatility, depending on their size. We give empirical evidence using French stock returns.