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Symmetry, 0-1 Matrices and Jacobians: A Review

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  • Magnus, Jan R.
  • Neudecker, H.

Abstract

In this paper we bring together those properties of the Kronecker product, the vec operator, and 0-1 matrices which in our view are of interest to researchers and students in econometrics and statistics. The treatment of Kronecker products and the vec operator is fairly exhaustive; the treatment of 0–1 matrices is selective. In particular we study the “commutation” matrix K (defined implicitly by K vec A = vec A′ for any matrix A of the appropriate order), the idempotent matrix N = ½ (I + K), which plays a central role in normal distribution theory, and the “duplication” matrix D, which arises in the context of symmetry. We present an easy and elegant way (via differentials) to evaluate Jacobian matrices (first derivatives), Hessian matrices (second derivatives), and Jacobian determinants, even if symmetric matrix arguments are involved. Finally we deal with the computation of information matrices in situations where positive definite matrices are arguments of the likelihood function.

Suggested Citation

  • Magnus, Jan R. & Neudecker, H., 1986. "Symmetry, 0-1 Matrices and Jacobians: A Review," Econometric Theory, Cambridge University Press, vol. 2(2), pages 157-190, August.
  • Handle: RePEc:cup:etheor:v:2:y:1986:i:02:p:157-190_01
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    1. Turkington, Darrell A., 1998. "Efficient estimation in the linear simultaneous equations model with vector autoregressive disturbances," Journal of Econometrics, Elsevier, vol. 85(1), pages 51-74, July.
    2. Eduardo Abi Jaber & Bruno Bouchard & Camille Illand & Eduardo Jaber, 2018. "Stochastic invariance of closed sets with non-Lipschitz coefficients," Working Papers hal-01349639, HAL.
    3. Deaton, Angus, 1988. "Household Behavior in Developing Countries," Center Discussion Papers 321313, Yale University, Economic Growth Center.
    4. Martha Misas Arango & Enrique López Enciso & Juana Téllez Corredor & José Fernando Escobar, 2005. "La Inflación Subyacente en Colombia: Un Enfoque de Tendencias Estocásticas Comunes Asociadas a un VEC Estructural," Borradores de Economia 324, Banco de la Republica de Colombia.
    5. Reinaldo B. Arellano-Valle & Adelchi Azzalini, 2022. "Some properties of the unified skew-normal distribution," Statistical Papers, Springer, vol. 63(2), pages 461-487, April.
    6. Christian M. Hafner & Oliver Linton & Haihan Tang, 2016. "Estimation of a Multiplicative Covariance Structure," CeMMAP working papers CWP23/16, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    7. Hafner, Christian M. & Linton, Oliver B. & Tang, Haihan, 2020. "Estimation of a multiplicative correlation structure in the large dimensional case," Journal of Econometrics, Elsevier, vol. 217(2), pages 431-470.
    8. D.A. Turkington, 1997. "Some results in matrix calculus and an example of their application to econometrics," Economics Discussion / Working Papers 97-07, The University of Western Australia, Department of Economics.
    9. Filipiak, Katarzyna & Klein, Daniel & Roy, Anuradha, 2016. "Score test for a separable covariance structure with the first component as compound symmetric correlation matrix," Journal of Multivariate Analysis, Elsevier, vol. 150(C), pages 105-124.
    10. Shaobo Jin & Sebastian Ankargren, 2019. "Frequentist Model Averaging in Structural Equation Modelling," Psychometrika, Springer;The Psychometric Society, vol. 84(1), pages 84-104, March.
    11. Eduardo Abi Jaber & Bruno Bouchard & Camille Illand & Eduardo Abi Jaber, 2018. "Stochastic invariance of closed sets with non-Lipschitz coefficients," Post-Print hal-01349639, HAL.
    12. Christian M. Hafner & Oliver Linton & Haihan Tang, 2016. "Estimation of a Multiplicative Covariance Structure," CeMMAP working papers 23/16, Institute for Fiscal Studies.
    13. Linton, O. & Tang, H., 2020. "Estimation of the Kronecker Covariance Model by Quadratic Form," Cambridge Working Papers in Economics 2050, Faculty of Economics, University of Cambridge.
    14. Liu, Shuangzhe & Leiva, Víctor & Zhuang, Dan & Ma, Tiefeng & Figueroa-Zúñiga, Jorge I., 2022. "Matrix differential calculus with applications in the multivariate linear model and its diagnostics," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    15. Christian M. Hafner & Oliver Linton & Haihan Tang, 2016. "Estimation of a multiplicative covariance structure in the large dimensional case," CeMMAP working papers 52/16, Institute for Fiscal Studies.
    16. Albert Satorra, 1990. "Robustness issues in structural equation modeling: a review of recent developments," Quality & Quantity: International Journal of Methodology, Springer, vol. 24(4), pages 367-386, November.
    17. Hafner, C. M. & Linton, O., 2016. "Estimation of a Multiplicative Covariance Structure in the Large Dimensional Case," Cambridge Working Papers in Economics 1664, Faculty of Economics, University of Cambridge.
    18. Stavrev, Emil & Kambourov, Gueorgui, 1999. "Share Equations versus Double Logarithmic Functions in the Estimation of Income, Own- and Cross-Price Elasticities," Transition Economics Series 7, Institute for Advanced Studies.
    19. Abi Jaber, Eduardo & Bouchard, Bruno & Illand, Camille, 2019. "Stochastic invariance of closed sets with non-Lipschitz coefficients," Stochastic Processes and their Applications, Elsevier, vol. 129(5), pages 1726-1748.
    20. Filipiak, Katarzyna & Klein, Daniel & Mokrzycka, Monika, 2024. "Discrepancy between structured matrices in the power analysis of a separability test," Computational Statistics & Data Analysis, Elsevier, vol. 192(C).
    21. Katarzyna Filipiak & Daniel Klein & Anuradha Roy, 2015. "Score test for a separable covariance structure with the first component as compound symmetric correlation matrix," Working Papers 0148mss, College of Business, University of Texas at San Antonio.
    22. Stavrev, Emil & Kambourov, Gueorgui, 1999. "Estimation of Income, Own- and Cross-Price Elasticities. An Application for Bulgaria," Transition Economics Series 6, Institute for Advanced Studies.
    23. Maddalena Cavicchioli, 2016. "Weak VARMA representations of regime-switching state-space models," Statistical Papers, Springer, vol. 57(3), pages 705-720, September.

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