Limit Theory for the QMLE of the GQARCH (1,1) Model

We examine the asymptotic properties of the QMLE for the GQARCH (1,1) model. Under suitable conditions, we establish that the asymptotic distribution of na-1a QMLE -θ0$n^{\frac{a-1}{a}}\left({\rm QMLE}-\theta _{0}\right)$ for θ0 the true parameter, is the one of the unique minimizer of a quadratic form over a closed and convex subset of R4$\mathbb {R}^{4}$. This form is the squared distance (w.r.t. a p.d. matrix) to a random vector that follows a normal distribution when a = 2 or is a linear transformation of an a-stable random vector when a ∈ (1, 2). This implies that we have distributional convergence to this random vector when θ0 is an interior point. Hence, we describe cases in which non normal limit distributions are obtained either due to convergence of the estimator on the boundary of the parameter space, and/or due to the non existence of high order moments for random elements involved in this framework. Possible extensions concern the establishment of analogous results for indirect estimators based on the QMLE which could have desirable first order asymptotic properties.