{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:24:54Z","timestamp":1760149494090,"version":"build-2065373602"},"reference-count":48,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2023,7,28]],"date-time":"2023-07-28T00:00:00Z","timestamp":1690502400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>In this paper, an algorithm for Mathematica is proposed for the computation of the asymptotic Fisher information matrix for a multivariate time series, more precisely for a controlled vector autoregressive moving average stationary process, or VARMAX process. Meanwhile, we present briefly several algorithms published in the literature and discuss the sufficient condition of invertibility of that matrix based on the eigenvalues of the process operators. The results are illustrated by numerical computations.<\/jats:p>","DOI":"10.3390\/a16080364","type":"journal-article","created":{"date-parts":[[2023,7,31]],"date-time":"2023-07-31T01:48:50Z","timestamp":1690768130000},"page":"364","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["An Algorithm for the Fisher Information Matrix of a VARMAX Process"],"prefix":"10.3390","volume":"16","author":[{"given":"Andr\u00e9","family":"Klein","sequence":"first","affiliation":[{"name":"Department of Quantitative Economics, Universiteit van Amsterdam, Roetersstraat 11, 1018WB Amsterdam, The Netherlands"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9575-4793","authenticated-orcid":false,"given":"Guy","family":"M\u00e9lard","sequence":"additional","affiliation":[{"name":"Solvay Brussels School of Economics and Management and ECARES, Universit\u00e9 libre de Bruxelles, CP 114\/04, Avenue Franklin Roosevelt, 50, B-1050 Brussels, Belgium"}]}],"member":"1968","published-online":{"date-parts":[[2023,7,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"L\u00fctkepohl, H. (1991). Introduction to Multiple Time Series Analysis, Springer.","DOI":"10.1007\/978-3-662-02691-5"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1109\/TASSP.1986.1164786","article-title":"Computation of the exact information matrix of Gaussian time series with stationary random components","volume":"14","author":"Porat","year":"1986","journal-title":"IEEE Trans. Acoust. Speech Signal Process."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"539","DOI":"10.1016\/0898-1221(89)90106-5","article-title":"Analytical derivatives for estimation of linear dynamic models","volume":"18","author":"Zadrozny","year":"1989","journal-title":"Comput. Math. Appl."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Terceiro Lomba, J. (1990). Estimation of Dynamic Econometric Models with Errors in Variables, Springer.","DOI":"10.1007\/978-3-642-48810-8"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1016\/S0165-1765(97)00180-8","article-title":"Exact initial conditions for maximum likelihood estimation of state space models with stochastic inputs","volume":"57","author":"Casals","year":"1997","journal-title":"Econ. Lett."},{"key":"ref_6","unstructured":"Jerez, M., Casals, J., and Sotoca, S. (2011). Signal Extraction for Linear State-Space Models, Lambert Academic Publishing."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.laa.2013.12.031","article-title":"An algorithm for the exact Fisher information matrix of vector ARMAX time series","volume":"446","author":"Klein","year":"2014","journal-title":"Linear Algebra Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1016\/j.econlet.2014.01.019","article-title":"On the Fisher information matrix of a vector ARMA process","volume":"123","author":"Bao","year":"2014","journal-title":"Econ. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1080\/00949658508810811","article-title":"A priori sample size evaluation and information matrix computation for time series models","volume":"21","author":"Dharan","year":"1985","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_10","unstructured":"Box, G.E.P., Jenkins, G.M., Reinsel, G.C., and Ljung, G.M. (2015). Time Series Analysis Forecasting and Control, Wiley. [5th ed.]."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1093\/biomet\/70.1.279","article-title":"Evaluation of the covariance matrix for the maximum likelihood estimator of a Gaussian autoregressive-moving average process","volume":"70","author":"Godolphin","year":"1983","journal-title":"Biometrika"},{"key":"ref_12","first-page":"1","article-title":"On algorithms for computing the covariance matrix of estimates in autoregressive-moving average models","volume":"5","author":"Klein","year":"1989","journal-title":"Comput. Stat. Q."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1111\/j.1467-9892.1990.tb00054.x","article-title":"Fisher\u2019s information matrix for seasonal autoregressive-moving average models","volume":"11","author":"Klein","year":"1990","journal-title":"J. Time Ser. Anal."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"579","DOI":"10.1016\/0024-3795(95)00552-8","article-title":"On Fisher\u2019s information matrix of an ARMAX process and Sylvester\u2019s resultant matrices","volume":"237\u2013238","author":"Klein","year":"1996","journal-title":"Linear Algebra Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"684","DOI":"10.1109\/78.277866","article-title":"Computation of the Fisher information matrix for SISO models","volume":"42","author":"Klein","year":"1994","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1016\/0047-259X(78)90082-9","article-title":"The information matrices of the parameters of multiple mixed time series","volume":"8","author":"Newton","year":"1978","journal-title":"J. Multivar. Anal."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1016\/j.laa.2005.02.006","article-title":"On the resultant property of the Fisher information matrix of a vector ARMA process","volume":"403","author":"Klein","year":"2005","journal-title":"Linear Algebra Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"674","DOI":"10.1016\/j.laa.2008.09.019","article-title":"Matrix differential calculus applied to multiple stationary time series and an extended Whittle formula for information matrices","volume":"430","author":"Klein","year":"2009","journal-title":"Linear Algebra Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1975","DOI":"10.1016\/j.laa.2009.06.027","article-title":"Tensor Sylvester matrices and the Fisher information matrix of VARMAX processes","volume":"432","author":"Klein","year":"2010","journal-title":"Linear Algebra Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1016\/0898-1221(92)90207-X","article-title":"Errata to \u2018Analytical derivatives for estimation of linear dynamic models\u2019","volume":"24","author":"Zadrozny","year":"1992","journal-title":"Comput. Math. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"857","DOI":"10.2307\/2951765","article-title":"Asymptotic distributions of impulse responses, step responses, and variance decompositions of estimated linear dynamic model","volume":"61","author":"Mittnik","year":"1993","journal-title":"Econometrica"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1007\/BF02590998","article-title":"Estimation and information in time series","volume":"2","author":"Whittle","year":"1953","journal-title":"Ark. Mat."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"565","DOI":"10.1090\/S0002-9904-1976-14103-1","article-title":"Resultants of matrix polynomials","volume":"82","author":"Gohberg","year":"1976","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_24","first-page":"96","article-title":"An indirect proof for the asymptotic properties of VARMA model estimators","volume":"21","year":"2022","journal-title":"Econom. Stat."},{"key":"ref_25","unstructured":"Hannan, E.J., and Deistler, M. (1988). The Statistical Theory of Linear Systems, Wiley."},{"key":"ref_26","unstructured":"Caines, P. (1988). Linear Stochastic Systems, Wiley."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Brockwell, P.J., and Davis, R.A. (1991). Time Series: Theory and Methods, Springer. [2nd ed.].","DOI":"10.1007\/978-1-4419-0320-4"},{"key":"ref_28","unstructured":"Gohberg, I., Lancaster, P., and Rodman, L. (1982). Matrix Polynomials, Academic Press."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1089","DOI":"10.1007\/s10463-012-0357-x","article-title":"Strictly stationary solutions of multivariate ARMA equations with i.i.d. noise","volume":"64","author":"Brockwell","year":"2012","journal-title":"Ann. Inst. Stat. Math."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Magnus, J.R., and Neudecker, H. (1988). Matrix Differential Calculus with Applications in Statistics and Econometrics, Wiley.","DOI":"10.2307\/2531754"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1016\/0005-1098(86)90064-6","article-title":"Optimal experiment designs with respect to the intended model application","volume":"22","author":"Gevers","year":"1986","journal-title":"Automatica"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"627","DOI":"10.1111\/j.1467-9892.2004.01863.x","article-title":"An algorithm for computing the asymptotic Fisher information matrix for seasonal SISO models","volume":"25","author":"Klein","year":"2004","journal-title":"J. Time Ser. Anal."},{"key":"ref_33","unstructured":"Peterka, V., and Vidin\u010dev, P. (1967, January 12\u201317). Rational-fraction approximation of transfer functions. Proceedings of the IFAC Symposium on Identification in Automatic Control Systems, Prague, Czech Republic."},{"key":"ref_34","unstructured":"S\u00f6derstr\u00f6m, T. (1984). Description of a Program for Integrating Rational Functions Around the Unit Circle, Department of Technology, Uppsala University. Technical Report 8467R."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"769","DOI":"10.1109\/29.17573","article-title":"Cram\u00e9r-Rao bounds for AR parameter and reflection coefficient estimators","volume":"37","author":"Pham","year":"1989","journal-title":"IEEE Trans. Acoust. Speech Signal Process."},{"key":"ref_36","first-page":"303","article-title":"Some efficient computational procedures for high order ARMA models","volume":"8","year":"1979","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"545","DOI":"10.1109\/29.17535","article-title":"The Euclid algorithm and the fast computation of cross-covariance and autocovariance sequences","volume":"37","author":"Demeure","year":"1989","journal-title":"IEEE Trans. Acoust. Speech Signal Process."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1111\/j.1467-9892.2005.00461.x","article-title":"On the evaluation of the information matrix for multiplicative seasonal time series models","volume":"27","author":"Godolphin","year":"2005","journal-title":"J. Time Ser. Anal."},{"key":"ref_39","first-page":"223","article-title":"The identification of vector mixed autoregressive-moving average systems","volume":"56","author":"Hannan","year":"1969","journal-title":"Biometrika"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"223","DOI":"10.2307\/1909577","article-title":"The identification problem for multiple equation systems with moving average errors","volume":"39","author":"Hannan","year":"1971","journal-title":"Econometrica"},{"key":"ref_41","unstructured":"Wegge, L.L. (2012). Armax(p,r,q) Parameter Identifiablity without Coprimeness, University of California. Available online: https:\/\/www.researchgate.net\/publication\/254396797_ARMAXprq_Parameter_Identifiability_Without_Coprimeness."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1664","DOI":"10.1016\/j.mcm.2010.12.034","article-title":"Least squares based iterative parameter estimation algorithm for multivariable controlled ARMA system modelling with finite measurement data","volume":"53","author":"Bao","year":"2011","journal-title":"Math. Comput. Model."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1198\/073500107000000313","article-title":"VARMA versus VAR for macroeconomic forecasting","volume":"26","author":"Athanasopoulos","year":"2008","journal-title":"J. Bus. Econ. Stat."},{"key":"ref_44","unstructured":"Terceiro, J., Casals, J.M., Jerez, M., Serano, G.R., and Sotoca, S. (2023, May 19). Time Series Analysis Using MATLAB, Including a Complete MATLAB Toolbox. Available online: http:\/\/www.ucm.es\/info\/icae\/e4."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"891","DOI":"10.1081\/STA-200054428","article-title":"Calculation of the Fisher information matrix for periodic ARMA models","volume":"34","author":"Bentarzi","year":"2005","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"4182","DOI":"10.1080\/03610926.2011.569864","article-title":"Computing the exact Fisher information matrix of periodic state-space models","volume":"41","author":"Hamdi","year":"2012","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"124","DOI":"10.1016\/j.jmva.2017.03.004","article-title":"Asymptotic Fisher information matrix of Markov switching VARMA models","volume":"157","author":"Cavicchioli","year":"2017","journal-title":"J. Multivar. Anal."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"112268","DOI":"10.1016\/j.chaos.2022.112268","article-title":"R\u00e9nyi entropy and divergence for VARFIMA processes based on characteristic and impulse response functions","volume":"160","year":"2022","journal-title":"Chaos Solitons Fractals"}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/16\/8\/364\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:21:56Z","timestamp":1760127716000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/16\/8\/364"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,28]]},"references-count":48,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2023,8]]}},"alternative-id":["a16080364"],"URL":"https:\/\/doi.org\/10.3390\/a16080364","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2023,7,28]]}}}