İki SUR Model Altında Ön Tahmin Edicilerin Kovaryans Matrisleri Üzerine Bazı Notlar

Abstract

Görünürde ilişkisiz regresyon (SUR) modelleri denklemler arasında hataların ilişkili olduğu çoklu regresyon denklemlerinin ele alındığı lineer regresyon modellerinin uzantılarıdır. Bu çalışmada, SUR modelleri altında ön tahmin problemi ele alınmıştır. İki SUR modeli altında tüm bilinmeyen vektörlerin en iyi lineer yansız ön tahmin edicilerinin (BLUP’ larının) istatistiksel özellikleri üzerine çeşitli sonuçlar verilmiştir. Özellikle, matrislerin bazı rank formülleri kullanılarak iki model altında BLUP’ların kovaryans matrisleri üzerine bazı sonuçlar elde edilmiştir.

Keywords

• BLUP,kovaryans matrisi,rank,SUR model

Some Notes on Covariance Matrices of Predictors under two SUR Models

Abstract

Seemingly unrelated regression (SUR) models are extensions of linear regression models by considering multiple regression equations with correlated errors among equations. In this study, prediction problem under SUR models are considered. Several results are given on statistical properties of the best linear unbiased predictors (BLUPs) of all unknown vectors under two SUR models. Especially, some results established on covariance matrices of BLUPs under two models by using some rank formulas of matrices.

Keywords

• BLUP,covariance matrix,rank,SUR model

References

1. Gong, L. (2019). Establishing equalities of OLSEs and BLUEs under seemingly unrelated regression Models. Journal of Statistical Theory and Practice, 13:5.
2. Hou, J., Zhao, Y. (2019). Some remarks on a pair of seemingly unrelated regression models. Open Math., 17, 979–989.
3. 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.
4. Sun, Y., Ke, R., Tian, Y. (2014). Some overall properties of seemingly unrelated regression models. Adv. Stat. Anal., 98 (2), 103–120.
5. Zellner, A. (1962). An efficient method of estimating seemingly unrelated regressions and tests for aggregation bias. J. Am. Stat. Assoc., 57, 348–368.
6. Zellner, A. (1963). Estimators for seemingly unrelated regression equations: some exact finite sample results. J. Am. Stat. Assoc., 58, 977–992.
7. Zellner, A., Huang, D. S. (1962). Further properties of efficient estimators for seemingly unrelated regression equations. Int. Econ. Rev., 3, 300–313.
8. Baksalary, J. K., Kala, R. (1979). On the prediction problem in the seemingly unrelated regression equations model. Statistics, 10, 203–208.

9. Baksalary, J. K., Trenkler, G. (1989). The efficiency of OLS in a seemingly unrelated regressions model. Econ. Theory, 5, 463–465.
10. Dwivedi, T. D., Srivastava, V. K. (1978). Optimality of least squares in the seemingly unrelated regression model. J. Econ., 7, 391–395.
11. Foschi, P., Kontoghiorghes, E. J. (2002). Seemingly unrelated regression model with unequal size observations: computational aspects. Comput. Stat. Data Anal., 41, 211–229.
12. Liu, A. Y. (2002). Efficient estimation of two seemingly unrelated regression equations. Journal of Multivariate Analysis, 82, 445–456.
13. Srivastava, V. K., Giles, D. E. A. (1987). Seemingly Unrelated Regression Equations Model. Marcel Dekker, New York.
14. Tian, Y. (2015). A new derivation of BLUPs under random-effects model. Metrika, 78, 905–918.
15. Alalouf, I. S., Styan, G. P. H. (1979). Characterizations of estimability in the general linear model. Ann. Stat., 7, 194–200.
16. Goldberger, A. S. (1962). Best linear unbiased prediction in the generalized linear regression models. J. Amer. Stat. Assoc., 57, 369–375.
17. Rao, C. R. (1973). Representations of best linear unbiased estimators in the Gauss–Markoff model with a singular dispersion matrix. J. Multivariate Anal., 3, 276–292.
18. Marsaglia, G., Styan, G. P. H. (1974). Equalities and inequalities for ranks of matrices. Linear Multilinear Algebra, 2, 269–292.
19. Tian, Y. (2017). Matrix rank and inertia formulas in the analysis of general linear models. Open Math., 15, 126–150.