Groupe d’études et de recherche en analyse des décisions

Estimating and Testing GARCH Processes When the Parameter is on a Boundary

Christian Francq Université Lille 3, France

We establish the asymptotic distribution of the quasi-maximum likelihood (QML) estimator for generalized autoregressive conditional heteroskedastic (GARCH) processes, when the true parameter may have zero coefficients. This asymptotic distribution is the projection of a normal vector distribution onto a convex cone. The results are derived under mild conditions which, for important subclasses of the general GARCH, coincide with those made in the recent literature when the true parameter is in the interior of the parameter space. Furthermore, the QML estimator is shown to converge to its asymptotic distribution locally uniformly. Using these results, we consider the problem of testing that one or several GARCH coefficients are equal to zero. The null distribution and the local asymptotic powers of the Wald, score and quasi-likelihood ratio tests are derived. The one-sided nature of the problem is exploited and asymptotic optimality issues are addressed. (Travail conjoint avec Jean-Michel Zakoian)