We consider inference and testing in extended constant conditional correlation GARCH models in the case where the true parameter vector is a boundary point of the parameter space. This is of particular importance when testing for volatility spillovers in the model. The large-sample properties of the QMLE are derived together with the limiting distributions of the related LR, Wald, and score statistics. Due to the boundary problem, these large-sample properties become nonstandard. The size and power properties of the tests are investigated in a simulation study. As an empirical illustration we test for (no) volatility spillovers between foreign exchange rates.