## 1 Introduction  
In time series models, developing statistically efficient and computationally quick methods for identifying and estimating VAR or VARMA models with single-frequency data (SFD) or mixed-frequency data (MFD) is an important task. In this paper, \"SFD means that all the variables of a model are observed at the same discrete-time frequency at which the model operates, and MFD means that some of the variables are observed at the same discrete-time frequency at which the model operates, and others are observed at one or more lower frequencies\" [1].  
Linear algebra tools are extensively used to study transfer functions, Yule-Walker equations and Hankel matrices associated with identifying and estimating VAR or VARMA models (see, for instance, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]). In particular, the extended Yule-Walker method (XYW) was proposed by [5], and it is considered in several later papers, in particular in [2, 3, 8] for estimating VAR models from available covariance matrices of MFD. Two of the principal and parallel strands of this literature which extend the method to the case of VARMA models are [1, 2]. The first considers both exact and generic identification and the second considers only exact identification. They make both common and differing assumptions involving the parameters of a VARMA model and prove that their assumptions are, as a whole, sufficient to identify all of the parameters of a VARMA model with MFD. In particular, [1] questions whether its \"conditions as a whole are necessary for identification\". This question has motivated our work, and our aim is to expand the methodology to more identifiable models.  
This paper is focused on identification. In econometrics and statistics, identification means the coefficients (parameters) of a model are determined uniquely from data covariances (and higher moments, depending on the data distribution) under assumed conditions for the coefficients of a model.  
Our aim is to provide conditions for the identifiability of VARMA models that cannot be identified by following the procedure in [1] and, in this regard, to complement it. We use a subset of conditions in [1] and some more, and we prove with two counterexamples that the whole set of conditions in [1] is not necessary. The main results are obtained from subsystems of extended Yule-Walker equations. Therefore, we expand the set of models identified with extended Yule-Walker methods.  
Because with MFD some autocovariance blocks are not available from observed variables, the main idea of our work is to provide conditions that, in addition to ensuring that the model with SFD is identifiable, allow rebuilding the unknown blocks from available covariance blocks. This thus yields all the complete autocovariances. As is well known, there is a bijection between the covariance and the corresponding spectral density of the process (see, for instance, [6]), meaning we can ensure the identifiability of the model.  
Section 2 summarizes and comments on the sufficient conditions used in [1] to identify VARMA models with SFD and MFD. In Section 3, we prove our main theoretical results. With a suitable change, our paper expands the applicability of XYW methods to more identifiable models. Section 4 tests and illustrates our main insight with two counterexamples. We close this paper with Conclusions, References and an Appendix with a MATLAB subroutine used in a counterexample.