Martingale Transforms with Mixed Stable Limits and the QMLE for Conditionally Heteroskedastic Models

We derive a limit theorem for appropriately centered and scaled martingale transforms \sum_{i=1}^{n}\xi_{i}V_{i} to mixed-stable limits when \left(\xi_{i}\right) is an iid sequence in the domain of attraction of an \alpha-stable distribution where \alpha\in(0,2]. Using the Principle of Conditioning we recover and extend known results in the literature while imposing weaker conditions. The results are particularly useful in determining the limit theory of the Gaussian QMLE in conditionally heteroskedastic models when the squared innovations are heavy-tailed. We provide the framework for the QMLE limit theory which in the ergodic case is based on the stochastic recurrence approach used in the relevant literature and we furthermore allow for the parameter vector to lie on the boundary of the parameter space. Then we show that the QMLE weakly converges to an \alpha-stable distribution when \alpha\in[1,2] and is inconsistent when \alpha<1. We relax the assumption on ergodicity and provide analogous results for the QMLE in the non-stationary GARCH(1,1) case. We investigate the limit theory of the usual Wald statistic and provide with the asymptotic exactness and consistency of the relevant testing procedure based on subsampling. In the context of the stationary GARCH(1,1) we construct a testing procedure for weak stationarity and derive its asymptotic properties and numerically evaluate its performance