Testing the existence of moments and estimating the tail index of augmented garch processes

We investigate the problem of testing finiteness of moments for a class of semi-parametric augmented GARCH models encompassing most commonly used specifications. The existence of positive-power moments of the strictly stationary solution is characterized through the Moment Generating Function (MGF) of the model, defined as the MGF of the logarithm of the random autoregressive coefficient in the volatility dynamics. We establish the asymptotic distribution of the empirical MGF, from which tests of moments are deduced. Alternative tests relying on the estimation of the Maximal Moment Exponent (MME) are studied. Power comparisons based on local alternatives and the Bahadur approach are proposed. We provide an illustration on real financial data, showing that semi-parametric estimation of the MME offers an interesting alternative to Hill's nonparametric estimator of the tail index.