Advertisement

Establishing Equalities of OLSEs and BLUEs Under Seemingly Unrelated Regression Models

Original Article
  • 3 Downloads

Abstract

Seemingly unrelated regression models (SURMs) are extensions of linear regression models which allow correlated errors between regression equations. The purpose of this article is to reconsider some fundamental problems on the performance and connection of ordinary least-squares estimators (OLSEs) and the best linear unbiased estimators (BLUEs) of parametric functions under an SURM. Motivated by a variety of known results and facts on the equivalence of OLSEs and BLUEs under general linear models, this article collects a list of necessary and sufficient conditions for OLSEs to be BLUEs under an SURM and presents a variety of statistical interpretations on the equivalence of OLSEs and BLUEs under the SURM.

Keywords

SURM OLSE BLUE Equivalence Statistical interpretation 

Mathematics Subject Classification

62F11 62H12 62J05 

Notes

Acknowledgements

The author wishes to thank two referees for their helpful comments and suggestions on this article.

References

  1. 1.
    Alalouf IS, Styan GPH (1979) Characterizations of estimability in the general linear model. Ann Stat 7:194–200MathSciNetCrossRefGoogle Scholar
  2. 2.
    Baksalary JK, Trenkler G (1989) The efficiency of OLS in a seemingly unrelated regressions model. Econ Theory 5:463–465CrossRefGoogle Scholar
  3. 3.
    Baltagi BH (1988) The efficiency of OLS in a seemingly unrelated regressions model. Econ Theory 4:536–537CrossRefGoogle Scholar
  4. 4.
    Bartels R, Fiebig DG (1991) A simple characterization of seemingly unrelated regressions models in which OLS is BLUE. Am Stat 45:137–140MathSciNetGoogle Scholar
  5. 5.
    Drygas H (1970) The coordinate-free approach to Gauss–Markov estimation. Springer, BerlinCrossRefGoogle Scholar
  6. 6.
    Gan S, Sun Y, Tian Y (2017) Equivalence of predictors under real and over-parameterized linear models. Commun Stat Theory Methods 46:5368–5383MathSciNetCrossRefGoogle Scholar
  7. 7.
    Graybill FA (1961) An introduction to linear statistical models, vol I. McGraw-Hill, New YorkzbMATHGoogle Scholar
  8. 8.
    Jiang B, Sun Y (2018) On the equality of estimators under a general partitioned linear model with parameter restrictions. Stat Pap 1:1–20.  https://doi.org/10.1007/s00362-016-0837-9 CrossRefGoogle Scholar
  9. 9.
    Jiang B, Tian Y (2017a) On additive decompositions of estimators under a multivariate general linear model and its two submodels. J Multivariate Anal 162:193–214MathSciNetCrossRefGoogle Scholar
  10. 10.
    Jiang B, Tian Y (2017b) Decomposition approaches of a constrained general linear model with fixed parameters. Electron J Linear Algebra 32:232–253MathSciNetCrossRefGoogle Scholar
  11. 11.
    Jiang B, Tian Y, Zhang X (2017) On decompositions of estimators under a general linear model with partial parameter restrictions. Open Math 15:1300–1322MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kurata H, Matsuura S (2016) Best equivariant estimator of regression coefficients in a seemingly unrelated regression model with known correlation matrix. Ann Inst Stat Math 68:1–19MathSciNetCrossRefGoogle Scholar
  13. 13.
    Lu C, Sun Y, Tian Y (2018) Two competing linear random-effects models and their connections. Stat Pap 59:1101–1115MathSciNetCrossRefGoogle Scholar
  14. 14.
    Penrose R (1955) A generalized inverse for matrices. Proc Camb Philos Soc 51:406–413CrossRefGoogle Scholar
  15. 15.
    Puntanen S, Styan GPH (1989) The equality of the ordinary least squares estimator and the best linear unbiased estimator, with comments by Kempthorne, O., Searle, S.R., and a reply by the authors. Am Stat 43:153–164Google Scholar
  16. 16.
    Rao CR (1973) Linear statistical inference and its applications, 2nd edn. Wiley, New YorkCrossRefGoogle Scholar
  17. 17.
    Searle SR (1971) Linear models. Wiley, New YorkzbMATHGoogle Scholar
  18. 18.
    Sun Y, Ke R, Tian Y (2013) Some overall properties of seemingly unrelated regression models. Adv Stat Anal 98:1–18zbMATHGoogle Scholar
  19. 19.
    Sun Y, Jiang B, Jiang H (2018) Computations of predictors/estimators under a linear random-effects model with parameter restrictions. Commun Stat Theory Methods.  https://doi.org/10.1080/03610926.2018.1476714 CrossRefGoogle Scholar
  20. 20.
    Tian Y (2010) Estimations of parametric functions under a system of linear regression equations with correlated errors. Acta Math Sin Engl Ser 26:1927–1942MathSciNetCrossRefGoogle Scholar
  21. 21.
    Tian Y, Zhang X (2016) On connections among OLSEs and BLUEs of whole and partial parameters under a general linear model. Stat Probab Lett 112:105–112MathSciNetCrossRefGoogle Scholar
  22. 22.
    Zellner A (1962) An efficiency method of estimating seemingly unrelated regression equations and tests for aggregation bias. J Am Stat Assoc 57:348–368CrossRefGoogle Scholar
  23. 23.
    Zellner A (1963) Estimators for seemingly unrelated regression equations: some exact finite sample results. J Am Stat Assoc 58:977–992MathSciNetCrossRefGoogle Scholar
  24. 24.
    Zellner A, Huang DS (1962) Further properties of efficient estimators for seemingly unrelated regression equations. Int Econ Rev 3:300–313 ZXCrossRefGoogle Scholar
  25. 25.
    Zhao L, Xu X (2017) Generalized canonical correlation variables improved estimation in high dimensional seemingly unrelated regression models. Stat Probab Lett 126:119–126MathSciNetCrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing 2018

Authors and Affiliations

  1. 1.Fuxin Higher Training CollegeFuxinChina

Personalised recommendations