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Estimation in singular linear models with stepwise inclusion of linear restrictions

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  • Ren, Xingwei

Abstract

In this paper, we consider the general linear model ℳ={y,Xβ,Σ}, without any rank assumptions to the model matrix X and covariance matrix Σ, and its two restricted models ℳr1={y,Xβ|A1β=r1,Σ} and ℳr12={y,Xβ|Aβ=r,Σ}, where r=(r1′,r2′)′ and A=(A1′,A2′)′. We give the necessary and sufficient conditions for the BLUEs to equal under ℳ and ℳr1, as well as under ℳr1 and ℳr12. We also derive that the BLUEs under ℳr1 are superior to the BLUEs under ℳ, and that the BLUEs under ℳr12 are superior to the BLUEs under ℳr1 in the sense of the covariance matrix.

Suggested Citation

  • Ren, Xingwei, 2016. "Estimation in singular linear models with stepwise inclusion of linear restrictions," Journal of Multivariate Analysis, Elsevier, vol. 148(C), pages 60-72.
    • Handle: RePEc:eee:jmvana:v:148:y:2016:i:c:p:60-72
    DOI: 10.1016/j.jmva.2016.02.018
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    References listed on IDEAS

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    1. repec:ebl:ecbull:v:3:y:2002:i:1:p:1-7 is not listed on IDEAS
    2. Ren, Xingwei, 2014. "On the equivalence of the BLUEs under a general linear model and its restricted and stochastically restricted models," Statistics & Probability Letters, Elsevier, vol. 90(C), pages 1-10.
    3. B. Ravikumar & Surajit Ray & N. Eugene Savin, 2000. "Robust Wald Tests in SUR Systems with Adding-up Restrictions," Econometrica, Econometric Society, vol. 68(3), pages 715-720, May.
    4. Dent, Warren T., 1980. "On restricted estimation in linear models," Journal of Econometrics, Elsevier, vol. 12(1), pages 49-58, January.
    5. Harry Haupt & Walter Oberhofer, 2002. "Fully restricted linear regression: A pedagogical note," Economics Bulletin, AccessEcon, vol. 3(1), pages 1-7.
    6. Yongge Tian, 2010. "On equalities of estimations of parametric functions under a general linear model and its restricted models," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 72(3), pages 313-330, November.
    7. Dong, Baomin & Guo, Wenxing & Tian, Yongge, 2014. "On relations between BLUEs under two transformed linear models," Journal of Multivariate Analysis, Elsevier, vol. 131(C), pages 279-292.
    8. Rao, C. Radhakrishna, 1973. "Representations of best linear unbiased estimators in the Gauss-Markoff model with a singular dispersion matrix," Journal of Multivariate Analysis, Elsevier, vol. 3(3), pages 276-292, September.
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