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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.

##### MSC:
 62J05 Linear regression; mixed models 62H12 Estimation in multivariate analysis
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##### References:
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