IDEAS home Printed from
   My bibliography  Save this paper

Invertibility Condition of the Fisher Information Matrix of a VARMAX Process and the Tensor Sylvester Matrix


  • André Klein
  • Guy Melard


In this paper the invertibility condition of the asymptotic Fisher information matrix of a controlled vector autoregressive moving average stationary process, VARMAX, is displayed in a theorem. It is shown that the Fisher information matrix of a VARMAX process becomes invertible if the VARMAX matrix polynomials have no common eigenvalue. Contrarily to what was mentioned previously in a VARMA framework, the reciprocal property is untrue. We make use of tensor Sylvester matrices since checking equality of the eigenvalues of matrix polynomials is most easily done in that way. A tensor Sylvester matrix is a block Sylvester matrix with blocks obtained by Kronecker products of the polynomial coefficients by an identity matrix, on the left for one polynomial and on the right for the other one. The results are illustrated by numerical computations.

Suggested Citation

  • André Klein & Guy Melard, 2020. "Invertibility Condition of the Fisher Information Matrix of a VARMAX Process and the Tensor Sylvester Matrix," Working Papers ECARES 2020-11, ULB -- Universite Libre de Bruxelles.
  • Handle: RePEc:eca:wpaper:2013/304274

    Download full text from publisher

    File URL:
    File Function: Full text for the whole work, or for a work part
    Download Restriction: no

    References listed on IDEAS

    1. Hannan, E J, 1971. "The Identification Problem for Multiple Equation Systems with Moving Average Errors," Econometrica, Econometric Society, vol. 39(5), pages 751-765, September.
    2. Leon Wegge, 2012. "ARMAX(p,r,q) Parameter Identifiability Without Coprimeness," Working Papers 1217, University of California, Davis, Department of Economics.
    3. Athanasopoulos, George & Vahid, Farshid, 2008. "VARMA versus VAR for Macroeconomic Forecasting," Journal of Business & Economic Statistics, American Statistical Association, vol. 26, pages 237-252, April.
    4. Peter Brockwell & Alexander Lindner & Bernd Vollenbröker, 2012. "Strictly stationary solutions of multivariate ARMA equations with i.i.d. noise," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 64(6), pages 1089-1119, December.
    Full references (including those not matched with items on IDEAS)


    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.

    Cited by:

    1. Guy Melard, 2020. "An Indirect Proof for the Asymptotic Properties of VARMA Model Estimators," Working Papers ECARES 2020-10, ULB -- Universite Libre de Bruxelles.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Guy Melard, 2020. "An Indirect Proof for the Asymptotic Properties of VARMA Model Estimators," Working Papers ECARES 2020-10, ULB -- Universite Libre de Bruxelles.
    2. Jean-Marie Dufour & Tarek Jouini, 2011. "Asymptotic Distributions for Some Quasi-Efficient Estimators in Echelon VARMA Models," CIRANO Working Papers 2011s-25, CIRANO.
    3. Mélard, Guy, 2022. "An indirect proof for the asymptotic properties of VARMA model estimators," Econometrics and Statistics, Elsevier, vol. 21(C), pages 96-111.
    4. Athanasopoulos, George & de Carvalho Guillén, Osmani Teixeira & Issler, João Victor & Vahid, Farshid, 2011. "Model selection, estimation and forecasting in VAR models with short-run and long-run restrictions," Journal of Econometrics, Elsevier, vol. 164(1), pages 116-129, September.
    5. Chan, Joshua C.C. & Eisenstat, Eric & Koop, Gary, 2016. "Large Bayesian VARMAs," Journal of Econometrics, Elsevier, vol. 192(2), pages 374-390.
    6. van der Genugten, B.B., 1976. "A general approach to identification," Other publications TiSEM fc91ed21-63b4-43d1-b480-6, Tilburg University, School of Economics and Management.
    7. M. Deistler & B. Pötscher & J. Schrader, 1984. "The uniqueness of the transfer function of linear systems from input-output observations," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 31(1), pages 157-181, December.
    8. Cotti, Chad & Courtemanche, Charles & Maclean, Joanna Catherine & Nesson, Erik & Pesko, Michael F. & Tefft, Nathan W., 2022. "The effects of e-cigarette taxes on e-cigarette prices and tobacco product sales: Evidence from retail panel data," Journal of Health Economics, Elsevier, vol. 86(C).
    9. Chan, Joshua C.C., 2013. "Moving average stochastic volatility models with application to inflation forecast," Journal of Econometrics, Elsevier, vol. 176(2), pages 162-172.
    10. Luisa Corrado & Sean Holly, 2006. "The Linearisation and Optimal Control of Large Non-Linear Rational Expectations Models by Persistent Excitation," Computational Economics, Springer;Society for Computational Economics, vol. 28(2), pages 139-153, September.
    11. Giuseppina Guagnano & Silvia Terzi, 1997. "Identifiability conditions for Generalised STARMA models," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 6(3), pages 245-255, December.
    12. Peña, Daniel & Poncela, Pilar, 1996. "Pooling information and forecasting with dynamic factor analysis," DES - Working Papers. Statistics and Econometrics. WS 10709, Universidad Carlos III de Madrid. Departamento de Estadística.
    13. Nyberg, Henri & Saikkonen, Pentti, 2014. "Forecasting with a noncausal VAR model," Computational Statistics & Data Analysis, Elsevier, vol. 76(C), pages 536-555.
    14. Raghavan, Mala, 2020. "An analysis of the global oil market using SVARMA models," Energy Economics, Elsevier, vol. 86(C).
    15. Xu Xiaojie, 2018. "Using Local Information to Improve Short-Run Corn Price Forecasts," Journal of Agricultural & Food Industrial Organization, De Gruyter, vol. 16(1), pages 1-15, January.
    16. Athanasopoulos, George & Hyndman, Rob J. & Song, Haiyan & Wu, Doris C., 2011. "The tourism forecasting competition," International Journal of Forecasting, Elsevier, vol. 27(3), pages 822-844.
    17. Martin Fukac & Adrian Pagan, 2009. "Structural Macro-Econometric Modelling in a Policy Environment," NCER Working Paper Series 50, National Centre for Econometric Research.
    18. Dias, Gustavo Fruet & Kapetanios, George, 2018. "Estimation and forecasting in vector autoregressive moving average models for rich datasets," Journal of Econometrics, Elsevier, vol. 202(1), pages 75-91.
    19. Massimiliano Marcellino & Oscar Jorda, "undated". "Stochastic Processes Subject to Time-Scale Transformations: An Application to High-Frequency FX Data," Working Papers 164, IGIER (Innocenzo Gasparini Institute for Economic Research), Bocconi University.
    20. Tian, Guoqiang, 1982. "Studies on the identification problem of the simultaneous economic models from viewpoint of unique determination of parameters (I)," MPRA Paper 41303, University Library of Munich, Germany.

    More about this item


    Tensor Sylvester matrix; Matrix polynomial; Common eigenvalues; Fisher in- formation matrix; Stationary VARMAX process;
    All these keywords.

    NEP fields

    This paper has been announced in the following NEP Reports:


    Access and download statistics


    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eca:wpaper:2013/304274. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: . General contact details of provider: .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Benoit Pauwels (email available below). General contact details of provider: .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.