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On Some Characterizations of Probability Distributions with Applications in Econometrics: A Centennial Tribute to CR Rao

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  • B. L. S. Prakasa Rao

    (CR RAO Advanced Institute of Mathematics, Statistics and Computer Science)

  • T. Krishna Kumar

    (Business Analytics, Rockville Analytics)

Abstract

CR Rao, the living legend in statistics, is celebrating his birth centenary year in 2020–21. One of the major areas in which Rao made significant contributions to statistics is characterization problems. The main objective of this paper is to connect the work of Rao and his followers on characterization to the underlying fundamental econometric problem. In doing so, the authors hope to draw the attention of the econometricians to the characterization method as a tool to further advance the less attempted problem of econometric modeling with errors in variables. The authors also hope that this paper would draw the attention of statisticians to the more interesting characterization problems arising in econometrics, there by providing fields of application to a class of characterization problems that otherwise remain mostly theoretical. Ragnar Frisch, the founder of econometrics, conceived the econometric model as a system of linear structural equations. Noting how one can use the reduced form equations to determine the joint distribution of the endogenous variables, given the distribution of the exogenous variables, Frisch asked the inverse question: under what distributional assumptions on the exogenous variables, does there exist a system of linear structural equations? Keeping this in the background, CR Rao answered that question in 1943 and 1947 using the characterization method and further advanced the characterization method in statistics. Rao and his followers, however, did not connect their work with the underlying econometric problem. The authors extend the econometric problem posed by Frisch to a two endogenous and two exogenous variables case and establish the existence of two linear structural equations and their identification, as an example of the potential benefits from this association between econometrics and statistics.

Suggested Citation

  • B. L. S. Prakasa Rao & T. Krishna Kumar, 2021. "On Some Characterizations of Probability Distributions with Applications in Econometrics: A Centennial Tribute to CR Rao," Journal of Quantitative Economics, Springer;The Indian Econometric Society (TIES), vol. 19(2), pages 181-205, June.
  • Handle: RePEc:spr:jqecon:v:19:y:2021:i:2:d:10.1007_s40953-021-00234-2
    DOI: 10.1007/s40953-021-00234-2
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    References listed on IDEAS

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    1. Vinod, H. D., 1982. "Maximum entropy measurement error estimates of singular covariance matrices in undersized samples," Journal of Econometrics, Elsevier, vol. 20(2), pages 163-174, November.
    2. Olav Bjerkholt & Ariane Dupont, 2010. "Ragnar Frisch's Conception of Econometrics," History of Political Economy, Duke University Press, vol. 42(1), pages 21-73, Spring.
    3. Bjerkholt, Olav, 2005. "Frisch'S Econometric Laboratory And The Rise Of Trygve Haavelmo'S Probability Approach," Econometric Theory, Cambridge University Press, vol. 21(3), pages 491-533, June.
    4. Krishnaji, N, 1970. "Characterization of the Pareto Distribution Through a Model of Underreported Incomes," Econometrica, Econometric Society, vol. 38(2), pages 251-255, March.
    5. Khatri, C. G. & Rao, C. Radhakrishna, 1972. "Functional equations and characterization of probability laws through linear functions of random variables," Journal of Multivariate Analysis, Elsevier, vol. 2(2), pages 162-173, June.
    6. Lancaster, Tony, 2000. "The incidental parameter problem since 1948," Journal of Econometrics, Elsevier, vol. 95(2), pages 391-413, April.
    7. Vinod, Hrishikesh D., 2006. "Maximum entropy ensembles for time series inference in economics," Journal of Asian Economics, Elsevier, vol. 17(6), pages 955-978, December.
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    Cited by:

    1. B. L. S. Prakasa Rao & H. D. Vinod, 2022. "Introduction to the Special Issue in Honor of Professor C. R. Rao," Journal of Quantitative Economics, Springer;The Indian Econometric Society (TIES), vol. 20(1), pages 1-6, September.
    2. T. Krishna Kumar, 2023. "Professor C. R. Rao, the Founder President of the Indian Econometric Society is Awarded the International Prize in Statistics, 2023," Journal of Quantitative Economics, Springer;The Indian Econometric Society (TIES), vol. 21(3), pages 473-480, September.

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    More about this item

    Keywords

    Linear structural equations; Characterization; Stable law; Errors in variables models; Confluence analysis; Identification;
    All these keywords.

    JEL classification:

    • C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General
    • C18 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Methodolical Issues: General
    • C50 - Mathematical and Quantitative Methods - - Econometric Modeling - - - General

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