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Matrix differential calculus with applications in the multivariate linear model and its diagnostics

Author

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  • Liu, Shuangzhe
  • Leiva, Víctor
  • Zhuang, Dan
  • Ma, Tiefeng
  • Figueroa-Zúñiga, Jorge I.

Abstract

Matrix differential calculus is a powerful mathematical tool in multivariate analysis and related areas such as econometrics, environmetrics, geostatistics, predictive modeling, psychometrics, and statistics in general. One of the key contributions to its development was the introduction of the differential approach, which has led to a significant number of applications. In this paper, we present a study of this approach to matrix differential calculus with some of its key results along with illustrative examples. We also present new applications of this approach in the multivariate linear model: namely in efficiency comparisons, sensitivity analysis, and local influence diagnostics.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:jmvana:v:188:y:2022:i:c:s0047259x21001275
    DOI: 10.1016/j.jmva.2021.104849
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    References listed on IDEAS

    as
    1. Liu, Shuangzhe & Ma, Tiefeng & Polasek, Wolfgang, 2014. "Spatial system estimators for panel models: A sensitivity and simulation study," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 101(C), pages 78-102.
    2. Francisco J. A. Cysneiros & Víctor Leiva & Shuangzhe Liu & Carolina Marchant & Paulo Scalco, 2019. "A Cobb–Douglas type model with stochastic restrictions: formulation, local influence diagnostics and data analytics in economics," Quality & Quantity: International Journal of Methodology, Springer, vol. 53(4), pages 1693-1719, July.
    3. Yonghui Liu & Ruochen Sang & Shuangzhe Liu, 2017. "Diagnostic analysis for a vector autoregressive model under Student-super-′s t-distributions," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 71(2), pages 86-114, May.
    4. Brinkhuis, Jan, 2015. "On the use of coordinate-free matrix calculus," Journal of Multivariate Analysis, Elsevier, vol. 133(C), pages 377-381.
    5. S. Liu & T. Ma & A. SenGupta & K. Shimizu & M.-Z. Wang, 2017. "Influence Diagnostics in Possibly Asymmetric Circular-Linear Multivariate Regression Models," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 79(1), pages 76-93, May.
    6. H. Neudecker, 1967. "On matrix procedures for optimizing differentiable scalar functions of matrices," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 21(1), pages 101-107, March.
    7. 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.
    8. Heinz Neudecker & Shuangzhe Liu, 2001. "Some statistical properties of Hadamard products of random matrices," Statistical Papers, Springer, vol. 42(4), pages 475-487, October.
    9. Jan R. Magnus & Andrey L. Vasnev, 2007. "Local sensitivity and diagnostic tests," Econometrics Journal, Royal Economic Society, vol. 10(1), pages 166-192, March.
    10. Kollo, T. & Neudecker, H., 1993. "Asymptotics of Eigenvalues and Unit-Length Eigenvectors of Sample Variance and Correlation Matrices," Journal of Multivariate Analysis, Elsevier, vol. 47(2), pages 283-300, November.
    11. Heinz Neudecker & Shuangzhe Liu, 2001. "Statistical properties of the Hadamard product of random vectors," Statistical Papers, Springer, vol. 42(4), pages 529-533, October.
    12. Magnus, J.R. & Neudecker, H., 1980. "The elimination matrix : Some lemmas and applications," Other publications TiSEM 0e3315d3-846c-4bc5-928e-f, Tilburg University, School of Economics and Management.
    13. Turkington,Darrell A., 2013. "Generalized Vectorization, Cross-Products, and Matrix Calculus," Cambridge Books, Cambridge University Press, number 9781107032002, November.
    14. Magnus, Jan R., 1985. "On Differentiating Eigenvalues and Eigenvectors," Econometric Theory, Cambridge University Press, vol. 1(2), pages 179-191, August.
    15. Khatri, C. G. & Rao, C. Radhakrishna, 1981. "Some extensions of the Kantorovich inequality and statistical applications," Journal of Multivariate Analysis, Elsevier, vol. 11(4), pages 498-505, December.
    16. Carolina Marchant & Víctor Leiva & Francisco José A. Cysneiros & Juan F. Vivanco, 2016. "Diagnostics in multivariate generalized Birnbaum-Saunders regression models," Journal of Applied Statistics, Taylor & Francis Journals, vol. 43(15), pages 2829-2849, November.
    17. Caro-Lopera, Francisco J. & Leiva, Víctor & Balakrishnan, N., 2012. "Connection between the Hadamard and matrix products with an application to matrix-variate Birnbaum-Saunders distributions," Journal of Multivariate Analysis, Elsevier, vol. 104(1), pages 126-139, February.
    18. Yonghui Liu & Chaoxuan Mao & Víctor Leiva & Shuangzhe Liu & Waldemiro A. Silva Neto, 2022. "Asymmetric autoregressive models: statistical aspects and a financial application under COVID-19 pandemic," Journal of Applied Statistics, Taylor & Francis Journals, vol. 49(5), pages 1323-1347, April.
    19. Magnus, J.R. & Neudecker, H., 1985. "Matrix differential calculus with applications to simple, Hadamard, and Kronecker products," Other publications TiSEM 1b2f1740-bfd1-4ea5-986c-9, Tilburg University, School of Economics and Management.
    20. Fukang Zhu & Shuangzhe Liu & Lei Shi, 2016. "Local influence analysis for Poisson autoregression with an application to stock transaction data," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 70(1), pages 4-25, February.
    21. von Rosen, Dietrich, 1989. "Maximum likelihood estimators in multivariate linear normal models," Journal of Multivariate Analysis, Elsevier, vol. 31(2), pages 187-200, November.
    22. Víctor Leiva & Shuangzhe Liu & Lei Shi & Francisco José A. Cysneiros, 2016. "Diagnostics in elliptical regression models with stochastic restrictions applied to econometrics," Journal of Applied Statistics, Taylor & Francis Journals, vol. 43(4), pages 627-642, March.
    23. Shuangzhe Liu & Chris Heyde, 2008. "On estimation in conditional heteroskedastic time series models under non-normal distributions," Statistical Papers, Springer, vol. 49(3), pages 455-469, July.
    24. Neudecker, H., 1980. "A comment on "minimization of functions of a positive simidefinite matrix A subject to AX = 0"," Journal of Multivariate Analysis, Elsevier, vol. 10(1), pages 135-139, March.
    25. Liu, Shuangzhe & Neudecker, Heinz, 2009. "On pseudo maximum likelihood estimation for multivariate time series models with conditional heteroskedasticity," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(8), pages 2556-2565.
    26. Magnus, Jan R., 2010. "On the concept of matrix derivative," Journal of Multivariate Analysis, Elsevier, vol. 101(9), pages 2200-2206, October.
    27. repec:wop:ubisop:0081 is not listed on IDEAS
    28. H. Neudecker, 1981. "On the matrix formulation of Kaiser's varimax criterion," Psychometrika, Springer;The Psychometric Society, vol. 46(3), pages 343-345, September.
    29. Magnus, J.R. & Neudecker, H., 1979. "The commutation matrix : Some properties and applications," Other publications TiSEM d0b1e779-7795-4676-ac98-1, Tilburg University, School of Economics and Management.
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    2. Saulo, Helton & Vila, Roberto & Cordeiro, Shayane S. & Leiva, Víctor, 2023. "Bivariate symmetric Heckman models and their characterization," Journal of Multivariate Analysis, Elsevier, vol. 193(C).

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