Inference in GARCH when some coefficients are equal to zero

The asymptotic distribution of the QML estimator for GARCH processes, with coefficients possibly equal to zero, is established. This distribution is the projection of a normal vector distribution onto a convex cone. The results are derived under mild conditions which, for important subclasses, coincide with those made in the recent literature when the coefficients are positive. 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 null. The null distribution and the local asymptotic powers of the Wald, score and quasi-likelihood ratio tests are derived. Asymptotic optimality issues are addressed. A set of numerical experiments illustrates the practical relevance of our theoretical results