Qualitative threshold ARCH models

In this paper we consider a class of dynamic models in which both the conditional mean and the conditional variance are endogenous stepwise functions. We first consider the probabilistic properties of these models: stationarity conditions, leptokurtic effect, linear representation, optimal prediction. In this first part most results are based on Markov chains theory. Then we derive statistical properties of this class of models; pseudo-maximum likelihood estimators, conditional homoscedasticity tests, tests of weak or strong white noise, CAPM test, factors determination, ARCH-M effects. We also discuss the introduction of exogenous variables and the case of multiple lags. Finally, an application to the Paris Stock Index is proposed.