Matrix inequality constraints for vector (asymmetric power) GARCH/HEAVY models and MEM with spillovers : some new (mixture) formulations

In this paper we review and generalize results on the derivation of tractable non-negativity (necessary and sufficient) conditions for N-dimensional asymmetric power GARCH/HEAVY models and MEM. We show that these non-negativity constraints are translated into simple matrix inequalities, which are easily handled. One main concern is that the existence of such conditions is often ignored by researchers. We hope that our paper will create more awareness of the presence of these non-negativity conditions and increase their usage. In practice these constraints may not be fulfilled. To handle these cases we propose a new mixture formulation in order to eliminate some of these constraints. By using the exponential specification for some (but not all) of the conditional variables in the system we considerably reduce the dimensions of them. We also obtain new theoretical results about the second moment structure and the optimal forecasts of such multivariate processes. Four empirical examples are included to show the effectiveness of the proposed method.