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Effect of adding regressors on the equality of the BLUEs under two linear models. (English) Zbl 1178.62068
Summary: We consider the estimation of regression coefficients in two partitioned linear models, shortly denoted as \({\mathcal M}_{12}=\{{\mathbf y},{\mathbf X}_1{\pmb\beta}_1+{\mathbf X}_2{\pmb\beta}_2,{\mathbf V}\}\), and \(\underline {{\mathcal M}}_{12}=\{{\mathbf y},{\mathbf X}_1{\pmb\beta}_1+{\mathbf X}_2{\pmb\beta}_2,\underline {{\mathbf V}}\}\), which differ only in their covariance matrices. We call \({\mathcal M}_{12}\) and \(\underline {{\mathcal M}}_{12}\) full models, and correspondingly, \({\mathcal M}_i=\{{\mathbf y},{\mathbf X}_i{\pmb\beta}_i,{\mathbf V}\}\) and \(\underline{\mathcal M}_i=\{{\mathbf y},{\mathbf X}_i{\pmb\beta}_i,{\mathbf V}\}\) small models. We give a necessary and sufficient condition for the equality between the best linear unbiased estimators (BLUEs) of \({\mathbf X}_1{\pmb\beta}_1\) under \({\mathcal M}_{12}\) and \(\underline{\mathcal M}_{12}\). In particular, we consider the equality of the BLUEs under the full models assuming that they are equal under the small models.

62J05 Linear regression; mixed models
62H12 Estimation in multivariate analysis
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