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doi: 10.3934/jimo.2022079
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## Statistical analysis of a linear regression model with restrictions and superfluous variables

 a. School of Economics and Management, Jiangxi University of Finance and Economics, Nanchang, Jiangxi, China b. College of Business and Economics, Shanghai Business School, Shanghai, China

*Corresponding author: Ruixia Yuan

Received  December 2021 Revised  April 2022 Early access May 2022

In statistical analysis of regression models, we frequently come across the cases in which two or more competing statistical models are assumed under the given data, while the corresponding statistical inference results are not necessarily the same. In this paper, we consider some comparison problems in the statistical inference of two given competing statistical models, where one is is a true and the other is assumed to be misspecified with inclusion of some superfluous variables. We shall derive the best linear unbiased estimators of unknown parametric vectors under the two competing models using some precise matrix analytic tools, discuss various statistical properties of the estimators, and analyze the connections among the estimators under the two models.

Citation: Wenming Li, Yongge Tian, Ruixia Yuan. Statistical analysis of a linear regression model with restrictions and superfluous variables. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022079
##### References:
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Styan, Around a formula for the rank of a matrix product with some statistical applications, In Lecture Notes in Pure and Appl. Math., Dekker, 139 (1993), 1–18. [7] P. Bhimasankaram and S. R. Jammalamadaka, Updates of statistics in a general linear model: A statistical interpretation and applications, Commun. Statist. Simul. Comput., 23 (1994), 789-801.  doi: 10.1080/03610919408813199. [8] K. L. Chu, J. Isotalo, S. Puntanen and G. P. H. Styan, On decomposing the Watson efficiency of ordinary least squares in a partitioned weakly singular linear model, Sankhyā, 66 (2004), 634-651. [9] K. L. Chu, J. Isotalo, S. Puntanen and G. P. H. Styan, Some further results concerning the decomposition of the Watson efficiency in partitioned linear models, Sankhyā, 67 (2005), 74-89. [10] H. Drygas, The Coordinate-free Approach to Gauss–Markov Estimation, Springer-Verlag, Berlin-New York 1970. [11] H. Drygas, A note on the inverse-partitioned-matrix method in linear regression analysis, Linear Algebra Appl., 67 (1985), 275-277.  doi: 10.1016/0024-3795(85)90201-0. [12] M. Dube, Mixed regression estimator under inclusion of some superfluous variables, Test, 8 (1999), 411-418.  doi: 10.1007/BF02595878. [13] M. Dube, V. K. Srivastava, H. Toutenburg and P. Wijekoon, Stein-rule estimator under inclusion of superfluous variables in linear regression models, Commun. Statist. Theory Meth., 20 (1991), 2009-2022.  doi: 10.1080/03610929108830618. [14] S. Gan, C. Lu and Y. Tian, Computation and comparison of estimators under different linear random-effects models, Commun. Statist. Simul. Comput., 49 (2020), 1210-1222.  doi: 10.1080/03610918.2018.1493507. [15] S. Gan, Y. Sun and Y. Tian, Equivalence of predictors under real and over-parameterized linear models, Commun. Statist. Theory Meth., 46 (2017), 5368-5383.  doi: 10.1080/03610926.2015.1100742. [16] J. Groß and S. 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show all references

##### References:
 [1] I. S. Alalouf and G. P. H. Styan, Characterizations of estimability in the general linear model, Ann. Statist., 7 (1979), 194-200. [2] J. K. Baksalary, A study of the equivalence between a Gauss–Markoff model and its augmentation by nuisance parameters, Ser. Statist., 15 (1984), 3-35.  doi: 10.1080/02331888408801743. [3] J. K. Baksalary and R. Kala, Best linear unbiased estimation in the restricted general linear model, Ser. Statist., 10 (1979), 27-35.  doi: 10.1080/02331887908801464. [4] J. K. Baksalary and P. R. Pordzik, Inverse-partitioned-matrix method for the general Gauss–Markov model with linear restrictions, J. Statist. Plann. Inference, 23 (1989), 133-143.  doi: 10.1016/0378-3758(89)90084-0. [5] J. K. Baksalary and P. R. Pordzik, Implied linear restrictions in the general Gauss–Markov model, J. Statist. Plann. Inference, 30 (1992), 237-248.  doi: 10.1016/0378-3758(92)90084-6. [6] J. K. Baksalary and G. P. H. Styan, Around a formula for the rank of a matrix product with some statistical applications, In Lecture Notes in Pure and Appl. Math., Dekker, 139 (1993), 1–18. [7] P. Bhimasankaram and S. R. Jammalamadaka, Updates of statistics in a general linear model: A statistical interpretation and applications, Commun. Statist. Simul. Comput., 23 (1994), 789-801.  doi: 10.1080/03610919408813199. [8] K. L. Chu, J. Isotalo, S. Puntanen and G. P. H. Styan, On decomposing the Watson efficiency of ordinary least squares in a partitioned weakly singular linear model, Sankhyā, 66 (2004), 634-651. [9] K. L. Chu, J. Isotalo, S. Puntanen and G. P. H. Styan, Some further results concerning the decomposition of the Watson efficiency in partitioned linear models, Sankhyā, 67 (2005), 74-89. [10] H. Drygas, The Coordinate-free Approach to Gauss–Markov Estimation, Springer-Verlag, Berlin-New York 1970. [11] H. Drygas, A note on the inverse-partitioned-matrix method in linear regression analysis, Linear Algebra Appl., 67 (1985), 275-277.  doi: 10.1016/0024-3795(85)90201-0. [12] M. Dube, Mixed regression estimator under inclusion of some superfluous variables, Test, 8 (1999), 411-418.  doi: 10.1007/BF02595878. [13] M. Dube, V. K. Srivastava, H. Toutenburg and P. Wijekoon, Stein-rule estimator under inclusion of superfluous variables in linear regression models, Commun. Statist. Theory Meth., 20 (1991), 2009-2022.  doi: 10.1080/03610929108830618. [14] S. Gan, C. Lu and Y. Tian, Computation and comparison of estimators under different linear random-effects models, Commun. Statist. Simul. Comput., 49 (2020), 1210-1222.  doi: 10.1080/03610918.2018.1493507. [15] S. Gan, Y. Sun and Y. Tian, Equivalence of predictors under real and over-parameterized linear models, Commun. Statist. Theory Meth., 46 (2017), 5368-5383.  doi: 10.1080/03610926.2015.1100742. [16] J. Groß and S. Puntanen, Estimation under a general partitioned linear model, Linear Algebra Appl., 321 (2000), 131-144.  doi: 10.1016/S0024-3795(00)00028-8. [17] N. Güler and M. E. Büyükkaya, Notes on comparison of covariance matrices of BLUPs under linear random-efects model with its two subsample models, Iran. J. Sci. Tech. Trans. A, 43 (2019), 2993-3002.  doi: 10.1007/s40995-019-00785-3. [18] N. Güler and M. E. Büyükkaya, Rank and inertia formulas for covariance matrices of BLUPs in general linear mixed models, Commun. Statist. Theory Meth., 50 (2021), 4997-5012.  doi: 10.1080/03610926.2019.1599950. [19] C. R. Hallum, T. O. Lewis and T. L. Boullion, Estimation in the restricted general linear model with a positive semidefinite covariance matrix, Commun. Statist., 1 (1973), 157-166.  doi: 10.1080/03610927308827014. [20] S. J. Haslett and S. Puntanen, Effect of adding regressors on the equality of the BLUEs under two linear models, J. Statist. Plann. Inference, 140 (2010), 104-110.  doi: 10.1016/j.jspi.2009.06.010. [21] J. S. Hodges, Richly Parameterized Linear Models: Additive, Time Series, and Spatial Models Using Random Effects, CRC Press, Boca Raton, FL, 2014. [22] J. Isotalo, S. Puntanen and G. P. H. Styan, Effect of adding regressors on the equality of the OLSE and BLUE, Internat. J. Statist. Sci., 6 (2007), 193-201. [23] S. R. Jammalamadaka and D. Sengupta, Changes in the general linear model: a unified approach, Linear Algebra Appl., 289 (1999), 225-242.  doi: 10.1016/S0024-3795(97)10047-7. [24] S. R. Jammalamadaka and D. Sengupta, Inclusion and exclusion of data or parameters in the general linear model, Stat. Probab. Lett., 77 (2007), 1235-1247.  doi: 10.1016/j.spl.2007.03.008. [25] B. Jiang and Y. Sun, On the equality of estimators under a general partitioned linear model with parameter restrictions, Stat. Papers, 60 (2019), 273-292.  doi: 10.1007/s00362-016-0837-9. [26] B. Jiang and Y. Tian, Decomposition approaches of a constrained general linear model with fixed parameters, Electron. J. Linear Algebra, 32 (2017), 232-253.  doi: 10.13001/1081-3810.3428. [27] B. Jiang and Y. Tian, On additive decompositions of estimators under a multivariate general linear model and its two submodels, J. Multivariate Anal., 162 (2017), 193-214.  doi: 10.1016/j.jmva.2017.09.007. [28] B. Jiang and Y. Tian, On equivalence of predictors/estimators under a multivariate general linear model with augmentation, J. Korean Stat. Soc., 46 (2017), 551-561.  doi: 10.1016/j.jkss.2017.04.001. [29] B. Jiang, Y. Tian and X. Zhang, On decompositions of estimators under a general linear model with partial parameter restrictions, Open Math., 15 (2017), 1300-1322.  doi: 10.1515/math-2017-0109. [30] H. Jiang, J. Qian and Y. Sun, Best linear unbiased predictors and estimators under a pair of constrained seemingly unrelated regression models, Stat. Probab. Lett., 158 (2020), 108669.  doi: 10.1016/j.spl.2019.108669. [31] S. J. Jun and J. Pinkse, Adding regressors to obtain efficiency, Econometric Theory, 25 (2009), 298-301.  doi: 10.1017/S0266466608090567. [32] K. Kadiyala, Mixed regression estimator under misspecification, Economics Lett., 21 (1986), 27-30.  doi: 10.1016/0165-1765(86)90115-1. [33] H. Kurata, A generalization of Rao's covariance structure with applications to several linear models, J. Multivariate Anal., 67 (1998), 297-305.  doi: 10.1006/jmva.1998.1771. [34] C. Lu, S. Gan and Y. Tian, Some remarks on general linear model with new regressors, Stat. Prob. Lett., 97 (2015), 16-24.  doi: 10.1016/j.spl.2014.10.015. [35] C. Lu, Y. Sun and Y. Tian, A comparison between two competing fixed parameter constrained general linear models with new regressors, Statistics, 52 (2018), 769-781.  doi: 10.1080/02331888.2018.1469021. [36] C. Lu, Y. Sun and Y. Tian, Two competing linear random-effects models and their connections, Stat. Papers, 59 (2018), 1101-1115.  doi: 10.1007/s00362-016-0806-3. [37] J. R. Magnus and J. Durbin, Estimation of regression coefficients of interest when other regression coefficients are of no interest, Econometrica, 67 (1999), 639-643.  doi: 10.1111/1468-0262.00040. [38] A. Markiewicz and S. Puntanen, Admissibility and linear sufficiency in linear model with nuisance parameters, Stat. Papers, 50 (2009), 847-854.  doi: 10.1007/s00362-009-0256-2. [39] A. Markiewicz and S. Puntanen, All about the $\perp$ with its applications in the linear statistical models, Open Math., 13 (2015), 33-50.  doi: 10.1515/math-2015-0005. [40] G. Marsaglia and G. P. H. Styan, Equalities and inequalities for ranks of matrices, Linear Multilinear Algebra, 2 (1974/75), 269-292.  doi: 10.1080/03081087408817070. [41] G. Marsaglia and G. P. H. Styan, Rank conditions for generalized inverses of partitioned matrices, Sankhyā Ser. A, 36 (1974), 437–442. [42] T. 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