Non-identifiability of VMA and VARMA systems in the mixed frequency case

Recently, identifiability results for VAR systems in the context of mixed frequency data have been shown in a number of papers. These results have been extended to VARMA systems, where the MA order is smaller than or equal to the AR order. Here, it is shown that in the VMA case and in the VARMA case, where the MA order exceeds the AR order, results are completely different. Then, for the case, where the innovation covariance matrix is non-singular, “typically” non-identifiability occurs - not even local identifiability. This is due to the fact that, e.g., in the VMA case, as opposed to the VAR case, the not directly observed autocovariances of the output can vary “freely”. In the singular case, i.e., when the innovation covariance matrix is singular, things may be different.