# American Institute of Mathematical Sciences

September  2019, 9(3): 383-391. doi: 10.3934/naco.2019025

## Singular arma systems: A structure theory

 Institute for Statistics and Mathematical Methods in Economics, Vienna University of Technology, Wiedner Hauptstrasse 8-10, 1040 Vienna, Austria

Received  July 2018 Revised  April 2019 Published  May 2019

Singular vector ARMA systems are vector ARMA (VARMA) systems with singular innovation variance or equivalently with singular spectral density of the corresponding VARMA process. Such systems occur in linear dynamic factor models, e.g. if the dimension of the static factors is strictly larger than the dimension of the dynamic factors or in linear dynamic stochastic equilibrium models, if the number of outputs is strictly larger than the number of shocks. We describe the relation of factor models and singular ARMA systems and a realization procedure for singular ARMA systems. Finally we discuss kernel systems.

Citation: Manfred Deistler. Singular arma systems: A structure theory. Numerical Algebra, Control and Optimization, 2019, 9 (3) : 383-391. doi: 10.3934/naco.2019025
##### References:
 [1] B. D. O. Anderson and M. Deistler, Properties of zero-free transfer function matrices, SICE Journal of Control, Measurement and System Integration, 1 (2008), 284-292.  doi: 10.1109/TAC.2009.2028976. [2] H. Akaike, Stochastic theory of minimal realization, IEEE-TAC, 19 (1974), 667-674.  doi: 10.1109/tac.1974.1100707. [3] M. Deistler and W. Scherrer, Modelle der Zeitreihenanalyse, Birkhuser-Springer, Cham, 2018. [4] M. Deistler, B. D. O. Anderson, A. Filler, C. Zinner and W. Chen, Generalized dynamic factor models - An approach via singular autoregressions, European Journal of Control, 16 (2010), 1-14.  doi: 10.3166/ejc.16.211-224. [5] M. Forni, M. Hallin, M. Lippi and L. Reichlin, The generalized dynamic factor model: identification and estimation, The Review of Economics and Statistics, 82 (2000), 540-554. [6] M. Forni, M. Hallin, M. Lippi and P. Zaffaroni, Dynamic factor models with infinite dimensional factor spaces: one-sided representations, Journal of Econometrics, 185 (2015), 359-371.  doi: 10.1016/j.jeconom.2013.10.017. [7] E. J. Hannan, M. Deistler, The Statistical Theory of Linear Systems, SIAM Classics in Applied Mathematics, Philadelphia, 2012. doi: 10.1137/1.9781611972191.ch1. [8] B. L. Ho and R. E. Kalman, Effective construction of linear state-variable models from input/output functions, Regelungstechnik, 14 (1966), 545-592. [9] I. Komunjer and S. Ng, Dynamic identification of dynamic stochastic general equilibrium models, Econometrica, 79 (2011), 1995-2032.  doi: 10.3982/ECTA8916. [10] Y. A. Rozanov, Stationary Random Processes, Holden Day, San Francisco, 1967. [11] W. Scherrer and M. Deistler, A structure theory for linear dynamic errors-in-variables models, SIAM J. Cont. Opt, 36 (1998), 2148-2175.  doi: 10.1137/S0363012994262464.

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##### References:
 [1] B. D. O. Anderson and M. Deistler, Properties of zero-free transfer function matrices, SICE Journal of Control, Measurement and System Integration, 1 (2008), 284-292.  doi: 10.1109/TAC.2009.2028976. [2] H. Akaike, Stochastic theory of minimal realization, IEEE-TAC, 19 (1974), 667-674.  doi: 10.1109/tac.1974.1100707. [3] M. Deistler and W. Scherrer, Modelle der Zeitreihenanalyse, Birkhuser-Springer, Cham, 2018. [4] M. Deistler, B. D. O. Anderson, A. Filler, C. Zinner and W. Chen, Generalized dynamic factor models - An approach via singular autoregressions, European Journal of Control, 16 (2010), 1-14.  doi: 10.3166/ejc.16.211-224. [5] M. Forni, M. Hallin, M. Lippi and L. Reichlin, The generalized dynamic factor model: identification and estimation, The Review of Economics and Statistics, 82 (2000), 540-554. [6] M. Forni, M. Hallin, M. Lippi and P. Zaffaroni, Dynamic factor models with infinite dimensional factor spaces: one-sided representations, Journal of Econometrics, 185 (2015), 359-371.  doi: 10.1016/j.jeconom.2013.10.017. [7] E. J. Hannan, M. Deistler, The Statistical Theory of Linear Systems, SIAM Classics in Applied Mathematics, Philadelphia, 2012. doi: 10.1137/1.9781611972191.ch1. [8] B. L. Ho and R. E. Kalman, Effective construction of linear state-variable models from input/output functions, Regelungstechnik, 14 (1966), 545-592. [9] I. Komunjer and S. Ng, Dynamic identification of dynamic stochastic general equilibrium models, Econometrica, 79 (2011), 1995-2032.  doi: 10.3982/ECTA8916. [10] Y. A. Rozanov, Stationary Random Processes, Holden Day, San Francisco, 1967. [11] W. Scherrer and M. Deistler, A structure theory for linear dynamic errors-in-variables models, SIAM J. Cont. Opt, 36 (1998), 2148-2175.  doi: 10.1137/S0363012994262464.
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