Enforcing solvability of a nonlinear matrix equation and estimation of multivariate ARMA time series

Please always quote using this URN:urn:nbn:de:0296-matheon-12409
  • The matrix equation $X+AX^{-1}A^T=B$, arising in parameter estimation of certain time series models, is solvable only for certain values of the matrices $A,B$. We present a numerical method to modify $A,B$ in order to make the matrix equation solvable. Since solvability depends on the location of the eigenvalues of the palindromic matrix polynomial $\lambda^2 A+\lambda B+A^T$, our method works by moving those eigenvalues to specified locations using first order spectral perturbation theory. The method is heuristic but works in practice, as is supported by several compelling numerical examples. These examples arise from parameter estimation of a common time series model, the multivariate ARMA(1,1).

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Author:Christian Schroeder, Tobias Bruell, Federico Poloni, Giacomo Sbrana
Referee:Benjamin Kunz
Document Type:Preprint, Research Center Matheon
Date of first Publication:2013/08/07
Release Date:2013/08/07
Tag:VARMA(1,1) model; first order eigenvalue perturbation theory; palindromic eigenvalue problem; parameter estimation; regularization
Institute:Technische Universität Berlin
MSC-Classification:15-XX LINEAR AND MULTILINEAR ALGEBRA; MATRIX THEORY / 15Axx Basic linear algebra / 15A24 Matrix equations and identities
62-XX STATISTICS / 62Mxx Inference from stochastic processes / 62M10 Time series, auto-correlation, regression, etc. [See also 91B84]
65-XX NUMERICAL ANALYSIS / 65Fxx Numerical linear algebra / 65F30 Other matrix algorithms
91-XX GAME THEORY, ECONOMICS, SOCIAL AND BEHAVIORAL SCIENCES / 91Bxx Mathematical economics (For econometrics, see 62P20) / 91B84 Economic time series analysis [See also 62M10]
93-XX SYSTEMS THEORY; CONTROL (For optimal control, see 49-XX) / 93Exx Stochastic systems and control / 93E10 Estimation and detection [See also 60G35]
Preprint Number:1027
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