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Three Scores and 15 Years (1948-2023) of Rao's Score Test: A Brief History

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  • Anil K. Bera
  • Yannis Bilias

Abstract

Rao (1948) introduced the score test statistic as an alternative to the likelihood ratio and Wald test statistics. In spite of the optimality properties of the score statistic shown in Rao and Poti (1946), the Rao score (RS) test remained unnoticed for almost 20 years. Today, the RS test is part of the ``Holy Trinity'' of hypothesis testing and has found its place in the Statistics and Econometrics textbooks and related software. Reviewing the history of the RS test we note that remarkable test statistics proposed in the literature earlier or around the time of Rao (1948) mostly from intuition, such as Pearson (1900) goodness-fit-test, Moran (1948) I test for spatial dependence and Durbin and Watson (1950) test for serial correlation, can be given RS test statistic interpretation. At the same time, recent developments in the robust hypothesis testing under certain forms of misspecification, make the RS test an active area of research in Statistics and Econometrics. From our brief account of the history the RS test we conclude that its impact in science goes far beyond its calendar starting point with promising future research activities for many years to come.

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  • Anil K. Bera & Yannis Bilias, 2024. "Three Scores and 15 Years (1948-2023) of Rao's Score Test: A Brief History," Papers 2406.19956, arXiv.org.
    • Handle: RePEc:arx:papers:2406.19956
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    References listed on IDEAS

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    1. Malabika Koley & Anil K. Bera, 2024. "To use, or not to use the spatial Durbin model? – that is the question," Spatial Economic Analysis, Taylor & Francis Journals, vol. 19(1), pages 30-56, January.
    2. Davidson, Russell & MacKinnon, James G, 1987. "Implicit Alternatives and the Local Power of Test Statistics," Econometrica, Econometric Society, vol. 55(6), pages 1305-1329, November.
    3. Bera, Anil K. & Montes-Rojas, Gabriel & Sosa-Escudero, Walter, 2010. "General Specification Testing With Locally Misspecified Models," Econometric Theory, Cambridge University Press, vol. 26(6), pages 1838-1845, December.
    4. Fang, Ying & Park, Sung Y. & Zhang, Jinfeng, 2014. "A simple spatial dependence test robust to local and distributional misspecifications," Economics Letters, Elsevier, vol. 124(2), pages 203-206.
    5. BERA, Anil K. & BYRON, Ray, 1983. "A note on the effects of linear approximation on hypothesis testing," LIDAM Reprints CORE 530, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    6. Berndt, Ernst R & Savin, N Eugene, 1977. "Conflict among Criteria for Testing Hypotheses in the Multivariate Linear Regression Model," Econometrica, Econometric Society, vol. 45(5), pages 1263-1277, July.
    7. repec:cup:cbooks:9780521252805 is not listed on IDEAS
    8. Andrea Bucci, 2024. "A sequential test procedure for the choice of the number of regimes in multivariate nonlinear models," Papers 2406.02152, arXiv.org.
    9. Bera, A.K. & Ullah, A., 1991. "RAO's Score Test in Econometrics," Other publications TiSEM 667d7827-c9d6-4f00-8183-6, Tilburg University, School of Economics and Management.
    10. Breusch, T S, 1978. "Testing for Autocorrelation in Dynamic Linear Models," Australian Economic Papers, Wiley Blackwell, vol. 17(31), pages 334-355, December.
    11. Breusch, T S, 1979. "Conflict among Criteria for Testing Hypotheses: Extensions and Comments," Econometrica, Econometric Society, vol. 47(1), pages 203-207, January.
    12. Saikkonen, Pentti, 1989. "Asymptotic relative efficiency of the classical test statistics under misspecification," Journal of Econometrics, Elsevier, vol. 42(3), pages 351-369, November.
    13. T. S. Breusch & A. R. Pagan, 1980. "The Lagrange Multiplier Test and its Applications to Model Specification in Econometrics," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 47(1), pages 239-253.
    14. Mukerjee, Rahul, 1990. "Comparison of tests in the multiparameter case I. Second-order power," Journal of Multivariate Analysis, Elsevier, vol. 33(1), pages 17-30, April.
    15. Yannis Bilias, 2000. "Sequential testing of duration data: the case of the Pennsylvania 'reemployment bonus' experiment," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 15(6), pages 575-594.
    16. Godfrey, Leslie G, 1978. "Testing for Higher Order Serial Correlation in Regression Equations When the Regressors Include Lagged Dependent Variables," Econometrica, Econometric Society, vol. 46(6), pages 1303-1310, November.
    17. Breusch, T S & Pagan, A R, 1979. "A Simple Test for Heteroscedasticity and Random Coefficient Variation," Econometrica, Econometric Society, vol. 47(5), pages 1287-1294, September.
    18. Anselin, Luc & Bera, Anil K. & Florax, Raymond & Yoon, Mann J., 1996. "Simple diagnostic tests for spatial dependence," Regional Science and Urban Economics, Elsevier, vol. 26(1), pages 77-104, February.
    19. Byron, Ray P & Bera, Anil K, 1983. "Least Squares Approximations to Unknown Regression Functions: A Comment," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 24(1), pages 255-260, February.
    20. Godfrey, Leslie G., 1978. "Testing for multiplicative heteroskedasticity," Journal of Econometrics, Elsevier, vol. 8(2), pages 227-236, October.
    21. Savin, N Eugene, 1976. "Conflict among Testing Procedures in a Linear Regression Model with Autoregressive Disturbances," Econometrica, Econometric Society, vol. 44(6), pages 1303-1315, November.
    22. Malabika Koley & Anil K. Bera, 2022. "Testing for spatial dependence in a spatial autoregressive (SAR) model in the presence of endogenous regressors," Journal of Spatial Econometrics, Springer, vol. 3(1), pages 1-46, December.
    23. Luc Anselin & Raymond J. G. M. Florax, 1995. "New Directions in Spatial Econometrics: Introduction," Advances in Spatial Science, in: Luc Anselin & Raymond J. G. M. Florax (ed.), New Directions in Spatial Econometrics, chapter 1, pages 3-18, Springer.
    24. Koenker, Roger, 1981. "A note on studentizing a test for heteroscedasticity," Journal of Econometrics, Elsevier, vol. 17(1), pages 107-112, September.
    25. Bera, Anil K. & Byron, Ray, 1983. "A note on the effects of linear approximation on hypothesis testing," Economics Letters, Elsevier, vol. 12(3-4), pages 251-254.
    26. Andre Lucas, 1998. "Inference on cointegrating ranks using lr and lm tests based on pseudo-likelihoods," Econometric Reviews, Taylor & Francis Journals, vol. 17(2), pages 185-214.
    27. Bera, Anil K. & Yoon, Mann J., 1993. "Specification Testing with Locally Misspecified Alternatives," Econometric Theory, Cambridge University Press, vol. 9(4), pages 649-658, August.
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