Skip to main content
SpringerLink
Log in

Optimal estimator under risk matrix in a seemingly unrelated regression model and its generalized least squares expression

  • Regular Article
  • Published:
Statistical Papers Aims and scope Submit manuscript

Abstract

A set of multiple regression models whose error terms have possibly contemporaneous correlations is called a seemingly unrelated regression model. In this paper, a best equivariant estimator of the regression vector under risk matrix is established in a seemingly unrelated regression model. It should be noted that an estimator optimal with respect to risk matrix remains optimal under a broad range of quadratic loss functions. A generalized least squares expression of our estimator is also presented.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Alkhamisi MA (2010) Simulation study of new estimators combining the SUR ridge regression and the restricted least squares methodologies. Stat Pap 51(3):651–672

    Article  MathSciNet  Google Scholar 

  • Alkhamisi MA, Golam Kibria BM, Shukur Ghazi (2019) Estimation of SUR model with Var(\(p\)) disturbances. Commun Stat 5(4):432–453

    Google Scholar 

  • Ando T, Zellner A (2010) Hierarchical Bayesian analysis of the seemingly unrelated regression and simultaneous equation models using a combination of direct Monte Carlo and importance sampling techniques. Bayesian Anal 5(1):65–96

    Article  MathSciNet  Google Scholar 

  • Breusch TS, Pagan AR (1980) The Lagrange multiplier test and its application to model specification in econometrics. Rev Econ Stud 47(1):239–253

    Article  MathSciNet  Google Scholar 

  • Costa N, Lourenco J, Pereira ZL (2012) Responses modeling and optimization criteria impact on the optimization of multiple quality characteristics. Comput Ind Eng 62(4):927–935

    Article  Google Scholar 

  • Derringer G, Suich R (1980) Simultaneous optimization of several response variables. J Qual Technol 12(4):214–219

    Article  Google Scholar 

  • Gong L (2019) Establishing equalities of OLSEs and BLUEs under seemingly unrelated regression models. J Stat Theory Pract 13:5

    Article  MathSciNet  Google Scholar 

  • Gupta AK, Kabe DG (1998) A note on a result for two SUR models. Stat Pap 39(4):417–421

    Article  MathSciNet  Google Scholar 

  • Heijmans R, Neudecker H (1998) Estimation of the SURE model. Stat Pap 39(4):423–430

    Article  MathSciNet  Google Scholar 

  • Hejazi TH, Esfahani MS, Mahootchi M (2015) Optimization of degree of conformance in multiresponse-multistage systems with a simulation-based metaheuristic. Qual Reliabil Eng Int 31(4):645–658

    Article  Google Scholar 

  • Jiang H, Qian J, Sun Y (2020) Best linear unbiased predictors and estimators under a pair of constrained seemingly unrelated regression models. Stat Probab Lett 158:108669

    Article  MathSciNet  Google Scholar 

  • Kariya T (1981a) Bounds for the covariance matrices of Zellner fs estimator in the SUR model and the 2SAE in a heteroscedastic model. J Am Stat Assoc 76(376):975–979

    Article  MathSciNet  Google Scholar 

  • Kariya T (1981b) Tests for the independence between two seemingly unrelated regression equations. Ann Stat 9(2):381–390

    Article  MathSciNet  Google Scholar 

  • Kariya T (1985) A nonlinear version of the Gauss-Markov theorem. J Am Stat Assoc 80(390):476–477

    Article  MathSciNet  Google Scholar 

  • Kariya T, Kurata H (2004) Generalized least squares. Wiley, New York

    Book  Google Scholar 

  • Kariya T, Toyooka Y (1985) Nonlinear versions of the Gauss-Markov theorem and GLSE. In: Krishnaiah PR (ed) Multivariate analysis VI. North-Holland, Amsterdam, pp 345–354

    Google Scholar 

  • Kurata H (1999) On the efficiencies of several generalized least squares estimators in a seemingly unrelated regression model and a heteroscedastic model. J Multiva Anal 70(1):86–94

    Article  MathSciNet  Google Scholar 

  • Kurata H (2004) One-sided tests for independence of seemingly unrelated regression equations. J Multivar Anal 90(2):393–406

    Article  MathSciNet  Google Scholar 

  • Kurata H, Kariya T (1996) Least upper bound for the covariance matrix of a generalized least squares estimator in regression with applications to a seemingly unrelated regression model and a heteroscedastic model. Ann Stat 24(4):1547–1559

    Article  MathSciNet  Google Scholar 

  • 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(4):705–723

    Article  MathSciNet  Google Scholar 

  • Liu A (2002) Efficient estimation of two seemingly unrelated regression equations. J Multivar Anal 82(2):445–456

    Article  MathSciNet  Google Scholar 

  • Ma T, Ye R (2010) Efficient improved estimation of the parameters in two seemingly unrelated regression models. J Stat Plan Inference 140(9):2749–2754

    Article  MathSciNet  Google Scholar 

  • Matsuura S, Kurata H (2020) Covariance matrix estimation in a seemingly unrelated regression model under Stein’s loss. Stat Methods Appl 29(1):79–99

    Article  MathSciNet  Google Scholar 

  • Ng VM (2002) Robust Bayesian inference for seemingly unrelated regressions with elliptical errors. J Multivar Anal 83(2):409–414

    Article  MathSciNet  Google Scholar 

  • Percy DF (1992) Predictions for seemingly unrelated regressions. J R Stat Soc B 54:243–252

    Google Scholar 

  • Riggs KR, Young PD, Young DM (2016) Confidence ellipsoids for the primary regression coefficients in two seemingly unrelated regression models. Stat Methodol 32:1–13

    Article  MathSciNet  Google Scholar 

  • Shah HK, Montgomery DC, Carlyle WM (2004) Response surface modeling and optimization in multiresponse experiments using seemingly unrelated regressions. Qual Eng 16(3):387–397

    Article  Google Scholar 

  • Srivastava VK, Giles DEA (1987) Seemingly unrelated regression equations models. Dekker, New York

    MATH  Google Scholar 

  • Sun Y, Ke R, Tian Y (2014) Some overall properties of seemingly unrelated regression models. AStA Adv Stat Anal 98(2):103–120

    Article  MathSciNet  Google Scholar 

  • Zellner A (1962) An efficient method of estimating seemingly unrelated regressions and tests for aggregation bias. J Am Stat Assoc 57(298):348–368

    Article  MathSciNet  Google Scholar 

  • Zellner A (1963) Estimators for seemingly unrelated regression equations: some exact finite sample results. J Am Stat Assoc 58(304):977–992

    Article  Google Scholar 

  • Zellner A, Ando T (2010) A direct Monte Carlo approach for Bayesian analysis of the seemingly unrelated regression model. J Economet 159(1):33–45

    Article  MathSciNet  Google Scholar 

  • Zhao L, Wang X (2019) Seemingly unrelated extreme learning machine. Neurocomputing 355:134–142

    Article  Google Scholar 

  • Zhao L, Xu X (2017) Generalized canonical correlation variables improved estimation in high dimensional seemingly unrelated regression models. Stat Probab Lett 126:119–126

    Article  MathSciNet  Google Scholar 

  • Zhao L, Yan L, Xu X (2018) High-correlated residuals improved estimation in the high-dimensional SUR model. Commun Stat 47(6):1583–1605

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank anonymous reviewers for many constructive and useful comments. Matsuura’s portion of this work is supported by JSPS KAKENHI Grant Number 20K11713. Kurata’s portion of this work is supported by JSPS KAKENHI Grant Number 19K11853.

Author information

Authors and Affiliations

  1. Faculty of Science and Technology, Keio University, Tokyo, Japan

    Shun Matsuura

  2. Graduate School of Arts and Sciences, The University of Tokyo, Tokyo, Japan

    Hiroshi Kurata

Authors
  1. Shun Matsuura
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hiroshi Kurata
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shun Matsuura.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Matsuura, S., Kurata, H. Optimal estimator under risk matrix in a seemingly unrelated regression model and its generalized least squares expression. Stat Papers 63, 123–141 (2022). https://doi.org/10.1007/s00362-021-01232-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00362-021-01232-5

Keywords

Mathematics Subject Classification

Search

Navigation