IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2406.19956.html
   My bibliography  Save this paper

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

Author

Listed:
  • 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.

Suggested Citation

  • 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, revised Oct 2024.
    • Handle: RePEc:arx:papers:2406.19956
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2406.19956
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    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. White,Halbert, 1996. "Estimation, Inference and Specification Analysis," Cambridge Books, Cambridge University Press, number 9780521574464, January.
    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Taşpınar, Süleyman & Doğan, Osman & Bera, Anil K., 2017. "GMM gradient tests for spatial dynamic panel data models," Regional Science and Urban Economics, Elsevier, vol. 65(C), pages 65-88.
    2. Dastoor, Naorayex K., 1997. "Testing for conditional heteroskedasticity with misspecified alternative hypotheses," Journal of Econometrics, Elsevier, vol. 82(1), pages 63-80.
    3. Taspinar, Suleyman & Dogan, Osman & Bera, Anil K., 2017. "GMM Gradient Tests for Spatial Dynamic Panel Data Models," MPRA Paper 82830, University Library of Munich, Germany.
    4. MacKinnon, James G, 1992. "Model Specification Tests and Artificial Regressions," Journal of Economic Literature, American Economic Association, vol. 30(1), pages 102-146, March.
    5. Doğan, Osman & Taşpınar, Süleyman & Bera, Anil K., 2021. "A Bayesian robust chi-squared test for testing simple hypotheses," Journal of Econometrics, Elsevier, vol. 222(2), pages 933-958.
    6. Russell Davidson & Victoria Zinde‐Walsh, 2017. "Advances in specification testing," Canadian Journal of Economics/Revue canadienne d'économique, John Wiley & Sons, vol. 50(5), pages 1595-1631, December.
    7. Federico Zincenko & Walter Sosa-Escudero & Gabriel Montes-Rojas, 2014. "Robust tests for time-invariant individual heterogeneity versus dynamic state dependence," Empirical Economics, Springer, vol. 47(4), pages 1365-1387, December.
    8. Bera, Anil K. & Doğan, Osman & Taşpınar, Süleyman, 2018. "Simple tests for endogeneity of spatial weights matrices," Regional Science and Urban Economics, Elsevier, vol. 69(C), pages 130-142.
    9. Brown, Kenneth & Cribari-Neto, Francisco, 1992. "On Hypothesis Testing: A Selective Look at the Lagrange Multiplier, Likelihood Ratio and Wald Tests," Brazilian Review of Econometrics, Sociedade Brasileira de Econometria - SBE, vol. 12(2), November.
    10. Dogan, Osman & Taspinar, Suleyman & Bera, Anil K., 2017. "Simple Tests for Social Interaction Models with Network Structures," MPRA Paper 82828, University Library of Munich, Germany.
    11. Bera, Anil K. & Montes-Rojas, Gabriel & Sosa-Escudero, Walter, 2009. "Testing under local misspecification and artificial regressions," Economics Letters, Elsevier, vol. 104(2), pages 66-68, August.
    12. Bera Anil K. & Bilias Yannis & Yoon Mann J. & Taşpınar Süleyman & Doğan Osman, 2020. "Adjustments of Rao’s Score Test for Distributional and Local Parametric Misspecifications," Journal of Econometric Methods, De Gruyter, vol. 9(1), pages 1-29, January.
    13. L. G. Godfrey & M. R. Veal, 2000. "Alternative approaches to testing by variable addition," Econometric Reviews, Taylor & Francis Journals, vol. 19(2), pages 241-261.
    14. 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.
    15. Anil K. Bera & Osman Doğan & Süleyman Taşpınar & Monalisa Sen, 2020. "Specification tests for spatial panel data models," Journal of Spatial Econometrics, Springer, vol. 1(1), pages 1-39, December.
    16. Godfrey, Leslie G & Orme, Chris D, 1996. "On the Behavior of Conditional Moment Tests in the Presence of Unconsidered Local Alternatives," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 37(2), pages 263-281, May.
    17. 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.
    18. Adrian C. Darnell, 1994. "A Dictionary Of Econometrics," Books, Edward Elgar Publishing, number 118.
    19. Montes-Rojas, Gabriel & Sosa-Escudero, Walter, 2011. "Robust tests for heteroskedasticity in the one-way error components model," Journal of Econometrics, Elsevier, vol. 160(2), pages 300-310, February.
    20. Bera, Anil K. & Sosa-Escudero, Walter & Yoon, Mann, 2001. "Tests for the error component model in the presence of local misspecification," Journal of Econometrics, Elsevier, vol. 101(1), pages 1-23, March.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:2406.19956. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.