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 of the form sum_{i=1}^{n}\xi_{i}V_{i} to mixed-stable limits when (\xi_{i})_{i\in\mathbb{Z}} 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 and we fully characterize the limits while imposing weaker conditions. The results are particularly useful in determining the limit theory of the Gaussian QMLE in conditionally heteroskedastic models since they allow for the avoidance of difficult to establish mixing conditions present in the literature. We provide the limit theory based on the stochastic recurrence approach used in the relevant literature while allowing for the parameter vector to lie on the boundary of the parameter space. Then we show that the QMLE weakly converges to an appropriate projection of a multivariate alpha-stable distribution when alpha\in[1,2] and is inconsistent when alpha<1 under certain conditions. We investigate the limit behavior of a usual self-normalized test statistic, construct inferential procedures based on the percentile-t bootstrap and conduct a Monte Carlo study which shows significant gains in performance when the additional information provided by our limit theorem is exploited