Abstract Two-stage linear model is the combination of two periods of linear regression models, in which, the second one has more regressors than the first one. In this paper, we consider the equivalence between estimations of partial parametric functions under a two-stage linear model. In this case, estimations of the same partial unknown parameters in the contexts of the two-stage linear model may have different expressions and properties, and thus it would be of interest to consider the relationship between these inference results. Since estimations of parametric functions under two-stage linear model involve certain complicated algebraic operations of matrices and their generalized inverses, it is a tedious task to deal with these estimation expressions and their statistical properties. To overcome this difficulties, a powerful tool, the matrix rank methodology was utilized in the past decades to manipulate various complicated matrix expressions that involve generalized inverses of matrices. In this paper, we use generalized inverses of matrices and matrix rank methodology to establish a general estimation theory on connections between the best linear unbiased estimations of partial parametric functions under two-stage linear model. The theoretical and methodological innovation research work can serve as general reference on characterizing relations between different estimations under other parametric regression models and facilitate the use of these models in applied research.
Bo Jiang (Tue,) studied this question.